Interactions between a Particle Covered by an Ion-Penetrable

charge model where the effect of the sizes of all the charged species is neglected. ... case where a particle is covered by an ion-penetrable charged ...
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Langmuir 2002, 18, 2789-2794

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Interactions between a Particle Covered by an Ion-Penetrable Charged Membrane and a Charged Surface: A Modified Gouy-Chapman Theory Yung-Chih Kuo Department of Chemical Engineering, National Chung Cheng University, Chia-Yi, Taiwan 62102, Republic of China

Meng-Ying Hsieh and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Received August 14, 2001. In Final Form: November 25, 2001 The interaction between a colloidal particle and a surface plays an important role in various interfacial phenomena in both industrial and biological processes. Previous efforts are mainly based upon a pointcharge model where the effect of the sizes of all the charged species is neglected. Here, we consider the case where a particle is covered by an ion-penetrable charged membrane, which mimics a biological entity, by taking this effect into account. We show that the sizes of the charged species are in the estimation of the electrical interaction force between particle and surface; the general trend of the result obtained is consistent with the experimental observations reported in the literature. The effects of the following key factors on the energy barrier for the particle-surface interaction are examined: the surface potential, the fixed charge density and the thickness of membrane layer, and the sizes of charged species.

1. Introduction Resolution of the interactions between a colloidal particle and a surface in an electrolyte solution plays an important role in the understanding of various interfacial phenomena in both industrial and biological processes. For instance, both the adsorption of microbes on the microcarriers in a bioreactor and the deposition of platelets onto an injured vessel wall of an organism involve particle-surface interaction. Most of the previous efforts considered two types of contribution to such interaction, namely, the repulsive electrical double-layer force and the attractive London-van der Waals force, which led to the landmark Derjaguin-Landau-Verwey-Overbeek (DLVO) theory.1,2 In this conventional theory, a primitive point-charge model (PCM), governed by the PoissonBoltzmann equation, describes the electrostatic term.3,4 Historically, the first successful attempt on the way toward a quantitative analysis of the charge density around a surface and the electrical interaction is the Gouy-Chapman theory (GCT) proposed nearly 90 years ago for a uniform dielectric continuum medium.5,6 Grahame7 extended the original version of GCT to the cases for 1:2 and 2:1 electrolytes. Since the hard-core exclusion effect is completely ignored in the classic GCT, it was predicted through modern statistical mechanics that the infinite attraction of an ionic species by the image charges * To whom correspondence should be addressed. Fax: 886-223623040. E-mail: [email protected]. (1) Derjaguin, B. V.; Landau, L. Acta Physiochim. (USSR) 1941, 14, 633. (2) Verwey, E. J.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (3) Hsu, J. P.; Kuo, Y. C. J. Chem. Soc., Faraday Trans. 1993, 89, 1229. (4) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1995, 171, 254. (5) Gouy, G. J. Phys. Radium 1910, 9, 457. (6) Chapman, D. L. Philos. Mag. 1913, 25, 475. (7) Grahame, D. C. J. Chem. Phys. 1953, 21, 1054.

on the polarized surface gives rise to the collapse of the electrolytic system.8 Preventing the calamity resulting from the structural contradiction, a distance of closest approach and a repulsion core of the ions were drawn by Stern9 and Grahame,10-12 respectively. This improvement is equivalent to the concept of the parallel plate capacitor.13 Moreover, the calamity resulting from the structural contradiction occurs because it was predicted through modern statistical mechanics that the infinite attraction of an ionic species by the image charges on the polarized surface gives rise to the collapse of the electrolytic system.13 Following Stern, by a simple change of variables in the Gouy-Chapman distribution functions, Henderson and Blum14 studied the system comprising spherical ions near a uniformly charged hard wall. They treated the strong electrolyte as hard spheres with equal diameter and symmetric charge while the ions interact with the wall. However, the hard-core interactions among the ions in the liquid solution are neglected. They referred to this approximation as a modified Gouy-Chapman theory (MGCT), which is a revised Stern model for electrolyte ions near an interface. A short time later, Torrie and Valleau15 demonstrated the feasibility of an MGCT by the Monte Carlo simulation in the examination of a doublelayer structure. To estimate the ionic distribution in an electrical diffuse layer, an MGCT with unequal ionic radii was investigated by Valleau and Torrie.16 They predicted that a concentration-dependent potential of zero charge (8) Lieb, E. H.; Lebowitz, J. L. Lectures in Theoretical Physics; University of Colorado Press: Boulder, 1972; Vol. 14. (9) Stern, O. Z. Elektrochem. 1924, 30, 508. (10) Grahame, D. C. Chem. Rev. 1947, 41, 441. (11) Grahame, D. C. J. Am. Chem. Soc. 1954, 76, 4819. (12) Grahame, D. C. J. Am. Chem. Soc. 1957, 79, 2093. (13) Bockris, J. O’M.; Reddy, A. K. Modern Electrochemistry; Plenum Press: New York, 1970. (14) Henderson, D.; Blum, L. J. Phys. Chem. 1978, 69, 5441. (15) Torrie, G. M.; Valleau, J. P. Chem. Phys. Lett. 1979, 65, 343. (16) Valleau, J. P.; Torrie, G. M. J. Chem. Phys. 1982, 76, 4623.

10.1021/la011293s CCC: $22.00 © 2002 American Chemical Society Published on Web 03/09/2002

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and a double layer with zero charge on the surface occur due to asymmetry of the closest approach to a charged surface. The modification to the cases for unequal ionic radii and 1:1, 2:2, 1:2, and 2:1 binary electrolytes in the liquid phase near a surface was presented by Bhuiyan et al.17 They found that even in the absence of specific adsorption, a nonzero potential at zero charge on the surface can occur. Recently, we proposed an MGCT for the inclusion of an ion-penetrable charged membrane over a rigid uncharged core to reflect more realistic behaviors of microorganisms in the physiological environment.18 It was proven that a reverse in electrical potential may occur near a membrane core if cations are smaller than both anions and fixed groups in a negatively charged membrane. We further examined the influences of ionic diameters on the double-layer properties of an ionpenetrable charged membrane.19 It was concluded that the potential drop across a membrane increases with the size of counterions but decreases with that of co-ions. For the effects of ionic sizes on the net penetration charge into a membrane, the reverse is true. Kuo and Hsu20 applied an MGCT to derive analytical expressions for the electrostatic interaction force and energy between two identical ion-penetrable charged membranes under the condition of the linearized Poisson-Boltzmann equation. For membranes with constant fixed charge density, the GCT underestimates the interaction force and energy. The reverse is true for membranes with constant membrane potential. In this study, the electrical interaction force and the potential-energy barrier between a particle covered by an ion-penetrable charged membrane and a planar charged surface are investigated by taking the various radii of charged species into account. In particular, the effects of the surface potential of a surface, the thickness of the membrane, the density of fixed charge, and the sizes of charged species are examined. The results obtained here are closely relevant to the qualitative assessment for adsorption processes such as deposition of proteins or nucleic acids on a collector surface, hemodialysis, and anchorage-dependent cell culture. 2. Theory Figure 1 shows a schematic representation of the system under consideration. Both the particle covered by an ionpenetrable membrane and the charged surface are immersed in an a:b electrolyte solution. Let σf, σan, and σca be the effective diameters of fixed groups, anions, and cations, respectively. The scaled symbols X ) κr, Xca ) κσca/2, Xan ) κσan/2, Xi ) κσf/2, Xo ) D - Xi, Xan′ ) D + L - Xan, and Xca′ ) D + L - Xca are introduced, κ and r being the reciprocal Debye length and the distance, respectively. D and L denote the scaled membrane thickness and the scaled distance between the membrane and the charged surface, 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 seven regions: (I) X < Xca, which comprises the charge-free region (-∞ < X < 0) and the inner uncharged membrane (0 < X < Xca), X being the scaled distance; (II) Xca < X < Xan, which (17) Bhuiyan, L. B.; Blum, B.; Henderson, D. J. Chem. Phys. 1983, 78, 442. (18) Hsu, J. P.; Kuo, Y. C. J. Chem. Phys. 1999, 111, 4807. (19) Kuo, Y. C.; Hsu, J. P. J. Phys. Chem. B 1999, 103, 9743. (20) Kuo, Y. C.; Hsu, J. P. Langmuir 2000, 16, 6233.

Kuo et al.

Figure 1. Schematic representation of the system under consideration. Xca and Xan are the effective most interior locations of positive and negative charges, respectively; Xi and Xo are the locations of inner and outer planes of fixed charge; Xan′ and Xca′ are the location of closest approach between anions and the charged surface and that between cations and the charged surface. Region I denotes the charge-free region in the membrane layer and the membrane core; II is the region close to the membrane core where only cations are present; III is the region close to the membrane core where both cations and anions are present; IV is the region in the membrane where all charged species are present. Region V represents the region comprising the outer uncharged membrane and the diffuse double layer; VI is the region close to the charged surface where only cations are present; VII is the charge-free region close to the charged surface.

contains cations only; (III) Xan < X < Xi, which contains both cations and anions; (IV) Xi < X < Xo, which contains all charged species; (V) Xo < X < Xan′, which includes the outer uncharged membrane (Xo < X < D) and the liquid phase containing cations and anions (D < X < Xan′); (VI) Xan′ < X < Xca′, which denotes the region containing cation only, close to the charged surface; (VII) Xca′ < X < D + L, which represents the charge-free region, close to the charged surface. Here, Xi and Xo are the location of the inner plane of fixed charge and that of the outer plane of fixed charge, respectively. Note that the fixed charges are present in the region Xi e X e Xo. We assume that the spatial variation in the scaled electrical potential, φ, can be described by the PoissonBoltzmann equation,

d2φ [-u exp(-aφ) + v exp(bφ) - wN] ) a+b dX2

(1)

where φ ) eφ/kBT, κ2 ) e2a(a + b)n0a/0rkBT, and the scaled concentration of fixed charge N ) ZN0NA/an0a. In these expressions, φ is the electrical potential, e represents 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, and kB and T denote the Boltzmann constant and the absolute temperature, respectively. For convenience, a region vector (w,v,u) is defined: (w,v,u) ) (0,0,0), (0,0,1), (0,1,1), (1,1,1), (0,1,1), (0,0,1), and (0,0,0) represent regions I through VII, respectively. The boundary conditions associated with eq 1 are assumed as

dφ f 0 as X f -∞ r,c dX dφ (dX )

r,m

X)X+ ca

dφ (dX )

) r,m

X)Xca

(1a)

and φ(X+ ca) ) φ(Xca)

X ) Xca (1b)

Particle Interacting with a Charged Surface

dφ (dX )

r,m

dφ (dX )

) r,m

X)X+ an

X)Xan

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and φ(X+ an) ) φ(Xan)

wg )

X ) Xan (1c) dφ dX

( )

r,m

X)X+ i

dφ dX

( )

) r,m

and

X)Xi

φ(X+ i )

)

φ(Xi )

X ) Xi (1d) dφ (dX )

r,m

X)X+ o

dφ (dX )

) r,m

and φ(X+ o ) ) φ(Xo )

X)Xo

X ) Xo (1e) dφ (dX )

r,dl

+ X)Xan′

dφ (dX )

) r,dl

X)Xan′

+ and φ(Xan′ ) ) φ(Xan′ )

X ) Xan′ (1f) dφ dX

( )

r,dl

+ X)Xca′

dφ dX

( )

) r,dl

X)Xca′

X)D+L

(1h)

In these expressions, φ0 is the scaled electrical potential on a charged surface, r,c, r,m, and r,dl denote the relative permittivities of hard core, membrane, and liquid solution, respectively, with r,c < r,m < r,dl, in general. Equations 1a and 1h imply that the system under consideration is at electroneutrality. We consider the general case in which the thickness of the membrane layer and that of the double layer can be comparable. This implies that the potential at the membrane core may not reach the Donnan potential.21 Since the fixed charge density is constant, neither the potential at the membrane core nor the potential at the membrane-liquid interface is constant, that is, both of them vary with L. The electrical interaction force between a particle and a planar charged surface, F, can be calculated by22,23

1 1 a + b dφ 2 F ) (ebφ - 1) + (e-aφ - 1) + CRT b a 2 dX

( ) i

∫φ0 N dφ

(2)

where R is the ideal gas constant and C ) an0a/NA is the charge concentration at the bulk liquid phase. The scaled electrical energy, wel, can be evaluated by22,23

F(λ)

∫X∞ ∫l∞ kBT dλ dl

wel ) 2πRc3

(3)

c

where Xc ) r/Rc, Rc being the particle radius, and λ and l are dummy variables. The scaled van der Waals potential, wVDW, can be evaluated by24

wVDW )

[

)]

-A132 1 + 2H H + ln 6kBT 2H(1 + H) 1+H

(

(4)

where H ) (X - D)/[2(κRc + D)]. The scaled gravitational potential, wg, can be expressed as25 (21) Hsu, (22) Hsu, (23) Hsu, (24) Hsu, (25) Kuo,

(5)

where ∆F denotes the mass-density difference between a particle and the surrounding liquid, g is the gravity, and h represents the closest distance between the membrane layer and the charged surface. To measure the adhesion of colloidal particles in practice, the planar charged surface is usually placed under the bottom of a suspension.26,27 In this case, the gravitational force contributes attractive energy. The scaled total potential energy, wTOT, is the sum of the scaled electrical potential, the scaled van der Waals potential, and the scaled gravitational potential, that is,

wTOT ) wel + wVDW + wg

(6)

The scaled potential-energy barrier between a particle and a charged surface, ∆wTOT, is defined as

+ and φ(Xca′ ) ) φ(Xca′ )

X ) Xca′ (1g) φ ) φ0

4 πR 3∆Fgh 3kBT c

J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1994, 166, 208. J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1996, 183, 194. J. P.; Kuo, Y. C. Langmuir 1997, 13, 4372. J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1996, 183, 184. Y. C.; Hsu, J. P.; Chen, D. F. Langmuir 2000, 17, 3466.

∆wTOT ) wTOT,max - wTOT,min

(7)

where wTOT,max and wTOT,min are the scaled total potential energy at the primary maximum and that at the secondary minimum, respectively. 3. Results and Discussion The behaviors of the electrostatic interaction force and the potential-energy barrier between a particle and a surface based on the present MGCT model are examined through numerical simulation. For the following illustrations, the temperature and the Hamaker constant are fixed, and therefore the van der Waals potential is a function of the separation distance between particle and charged surface only. Since the gravitational potential is also only a function of the separation distance as shown in eq 5, the effects of parameters on the system under consideration are mainly electrical. The variations in the scaled electrostatic interaction force between the particle and the surface as a function of the scaled separation distance between them under various conditions are presented in Figures 2-4. For comparison, the results based on the corresponding PCM are also shown in these figures. The total amount of fixed charge is constant in Figures 2-4. This implies that N0(D - 2Xi) is fixed in the present model and N0D is fixed in the corresponding PCM. Figure 2 shows that the electrostatic force between a particle and a charged surface decreases monotonically with the separation distance between them. Note that N0(D - 2Xi) ) 5 × 10-4 C/m3 in the present model and N0D ) 5 × 10-4 C/m3 in the corresponding PCM. As can be seen in Figure 2, the classic PCM underestimates the electrostatic force. This is because if the size of fixed charge is considered, its concentration in region IV is higher than that of the corresponding PCM, and the electrical field generated by the membrane is stronger. In the present model, if D + L < 1.5, the decrease of the absolute value of the electrostatic potential in the membrane phase compensates the increase of the absolute value of the electrostatic potential in the small region of the double layer. The classic PCM only predicts the increase of the absolute value of the electrostatic potential in both membrane phase and double-layer region as D +L < 1.5. Therefore, if D + L < 1.5, the deviation of PCM is small as justified in Figure (26) Weiss, L.; Harlos, J. P. J. Theor. Biol. 1972, 37, 169. (27) Ruckenstein, E.; Marmur, A.; Rakower, S. R. Thromb. Haemostasis 1976, 36, 334.

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Figure 2. Variation in the scaled electrostatic interaction force between a particle and a charged surface as a function of the scaled separation distance between them for the case φ0 ) -1.6. Solid curve, present model, N0 ) 5 × 10-3 C/m3, Xca ) 0.05, Xan ) 0.35, Xi ) 0.45; dashed curve, classic PCM, N0 ) 5 × 10-4 C/m3. Key: a ) 1, b ) 1, I ) 1 × 10-3 M, D ) 1, r ) 78, and T ) 298.15 K.

Figure 3. Variation in the scaled electrostatic interaction force between a particle and a charged surface as a function of the scaled separation distance between them for the case φ0 ) -0.26. Solid curve, present model, N0 ) 2 × 10-5 C/m3, Xca ) 0.05, Xan ) 0.10, Xi ) 0.20; dashed curve, classic PCM, N0 ) 1.2 × 10-5 C/m3. Key: same as Figure 2.

2. Intuitively, the effect of the sizes of the charged species becomes less significant as the scale of the system is larger. This is justified in Figure 2, where the difference between the present model and the corresponding PCM is small if D + L > 4, that is, the effect of sizes of charged species is negligible if the separation distance is large. The absolute value of φ0 in Figure 3 is higher than that of Figure 2, and the qualitative behavior of Figure 3 is similar to that of Figure 2. For the case of Figure 3, N0(D - 2Xi) ) 1.2 × 10-5 C/m3 for the present model and N0D ) 1.2 × 10-5 C/m3 for the corresponding PCM. This figure suggests that the deviation of PCM decreases with the separation distance. A comparison between Figures 2 and 3 suggests that the electrostatic interaction force of the latter is much smaller than that of the former. This is because both φ0 and N0 of Figure 3 are much smaller than those of Figure 2. Although a smaller N0 allows more coions to penetrate into the membrane, the membrane of

Kuo et al.

Figure 4. Variation in the scaled electrostatic interaction force between a particle and a charged surface as a function of the scaled separation distance between them for the case φ0 ) -1.6. Solid curve, present model, N0 ) 1.6 × 10-3 C/m3, Xca ) 0.05, Xan ) 0.35, Xi ) 0.45; dashed curve, classic PCM, N0 ) 1.6 × 10-4 C/m3. Key: same as Figure 2.

the present MGCT can offer less space for co-ions than that of the corresponding PCM. Also, the N0 of the present model is larger than that of the corresponding PCM (5:3). Therefore, the total amount of co-ions of the former, which contributes to the repulsive force, is less than that of the latter. Therefore, for a fixed separation distance, the classic PCM will overestimate the electrostatic force, as shown in Figure 3. This is opposed to the result presented in Figure 2. The question arises of whether the classic PCM overestimates or underestimates the electrostatic interaction force as the magnitude of the electrostatic interaction force is between that of Figure 2 and that of Figure 3. The value of N0 in Figure 4 is smaller than that of Figure 2. In Figure 4, N0(D - 2Xi) ) 1.6 × 10-4 C/m3 for the present model and N0D ) 1.6 × 10-4 C/m3 for the corresponding PCM. This figure shows that the curve predicted by the present model and that by the corresponding PCM intersect each other at a critical separation distance. If the separation distance is smaller than this critical value, the PCM overestimates the electrostatic interaction force, and the reverse is true if the separation distance is greater than the critical value. The critical separation distance shifts to a larger value as the parameter for the electrostatic interaction, such as φ0 or N0, reduces. Figure 5 shows the variation in the scaled energy barrier between a particle and a surface as a function of the scaled electrical potential on the surface. Since a higher surface potential leads to a stronger electrostatic interaction, the higher the absolute value of the scaled electrical potential on the surface, the higher the scaled potential barrier between a particle and a surface, as illustrated in Figure 5. As can be seen in this figure, the thicker the membrane, the lower the energy barrier. This figure suggests that the scaled energy barrier and the scaled surface potential be nonlinearly correlated. In Figures 6 and 7, since the fixed charge appears merely in the region IV, the N0(D - 2Xi) value is constant to maintain the condition for a fixed total amount of the fixed charge. The variation in the scaled energy barrier between a particle and a surface as a function of the scaled membrane thickness is illustrated in Figure 6. Note that since the fixed charge appears only in region IV, the condition of constant total amount of the fixed charge requires that N0(D - 2Xi) is fixed. Also, the thicker the membrane, the lower the fixed charge density in region IV. As mentioned above, the lower the fixed charge density,

Particle Interacting with a Charged Surface

Figure 5. Variation in the scaled energy barrier between a particle and a charged surface as a function of the scaled electrical potential on a charged surface for the case N0 ) 2 × 10-5 C/m3, Xca ) 0.05, Xan ) 0.10, and Xi ) 0.20. Curve 1, D ) 1.0; curve 2, D ) 1.3. Key: a ) 1, b ) 1, I ) 1 × 10-3 M, r ) 78, T ) 298.15 K, A132 ) 1.95 × 10-21 J, Rc ) 1 × 10-6 m, and ∆F ) 70 kg/m3.

Figure 6. Variation in the scaled energy barrier between a particle and a charged surface as a function of the scaled membrane thickness for the case (D - 2Xi)N0 ) 1.2 × 10-5 C/m3, Xca ) 0.05, Xan ) 0.10, and Xi ) 0.20. Curve 1, φ0 ) -0.40; curve 2, φ0 ) -0.26. Key: same as Figure 5.

the weaker the electrostatic repulsion. Therefore, the thicker the membrane, the smaller the scaled energy barrier, as can be seen in Figure 6. This figure also reveals that if D < 1.2, the scaled energy barrier decreases dramatically with D. This is because the influence of N0 on the energy barrier becomes significant if N0 > 1.5 × 10-5 C/m3. If N0 < 1.5 × 10-5 C/m3, or D > 1.2, the energy barrier becomes relatively insensitive to the variation in membrane thickness. An approximate inverse relationship between the scaled energy barrier and the scaled membrane thickness can be drawn. As pointed out in Figure 6, the higher the electrical potentials on the charged surface, the higher the energy barrier. The variation in the scaled energy barrier between a particle and a charged surface as a function of the fixed charge density is plotted in Figure 7. Since a higher N0 can result in a stronger electrostatic interaction, the higher the N0, the higher the energy barrier between a particle and a charged surface, as can be seen in Figure 7. The relationship between N0 and ∆wTOT is almost linear under the conditions examined. The higher the electrical potentials on the charged surface, the higher the energy barrier as also presented in Figure 7.

Langmuir, Vol. 18, No. 7, 2002 2793

Figure 7. Variation in the scaled energy barrier between a particle and a charged surface as a function of the fixed charge density for the case (D - 2Xi)N0 ) 1.2 × 10-5 C/m3, φ0 ) -0.26, Xca ) 0.05, Xan ) 0.10, and Xi ) 0.20. Curve 1, φ0 ) -0.40; curve 2, φ0 ) -0.26. Key: same as Figure 5.

Figure 8. Variation in the scaled energy barrier between a particle and a charged surface as a function of the scaled average radius of cations for the case φ0 ) -0.26, D ) 1, N0 ) 2 × 10-5 C/m3, and Xi ) 0.45. Curve 1, Xan ) 0.45; curve 2, Xan ) 0.40. Key: same as Figure 5.

The variation in the scaled energy barrier between a particle and a charged surface as a function of the scaled effective radius of cations is given in Figure 8. As revealed in this figure, the larger the cations, the higher the energy barrier between a particle and a charged surface. This is because the greater the size of the cations, the greater the region I and the smaller the region II. Since high negative electrical potential occurs in region I, the smaller the cations, the more effective the compensation of this interior membrane potential. Also if Xca is greater than 0.23, the larger the anion, the higher the energy barrier. However, if Xca is smaller than 0.23, the reverse is true. Under the conditions of Figure 8, the energy barrier is nonlinearly dependent on the effective radius of cations. Colic et al.28 and Chapel29 presented the influence of monovalent ion size on repulsive force between silica surfaces. They found that as Cl- is the common anion, the strength of repulsive force follows the sequence from weakest to strongest with Li+ < Na+ < K+ < Cs+. They emphasize that the repulsive force is related to the size of bare counterions. The same order about the relation of monovalent cation size was also observed for the repulsion force between alumina (28) Colic, M.; Fishe, M. L.; Franks, G. V. Langmuir 1998, 14, 6107. (29) Chapel, J. P. Langmuir 1994, 10, 4237.

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Kuo et al.

Figure 9. Variation in the scaled energy barrier between a particle and a charged surface as a function of the scaled average radius of anions for the case φ0 ) -0.26, D ) 1, N0 ) 2 × 10-5 C/m3, and Xca ) 0.05. Curve 1, Xi ) 0.28; curve 2, Xi ) 0.20. Key: same as Figure 5.

Figure 10. Variation in the scaled energy barrier between a particle and a charged surface as a function of the scaled average radius of fixed groups for the case φ0 ) -0.26, D ) 1, N0 ) 2 × 10-5 C/m3, and Xan ) 0.10. Curve 1, Xca ) 0.08; curve 2, Xca ) 0.05. Key: same as Figure 5.

surfaces.30 This order is consistent with the result of Figure 8, but the corresponding PCM predicts no difference in the repulsion force among various sizes of cations with the same valence. The variation in the scaled energy barrier between a particle and a charged surface as a function of the scaled effective radius of anions is given in Figure 9. As revealed in this figure, the larger the anions, the higher the energy barrier between a particle and a charged surface. This is because the greater the size of the anions, the greater the region II and the smaller the region III. The alteration of the space in the membrane gives rise to the decrease of the electrostatic potential and the increase of the electrostatic interaction energy. As indicated in Figure 9, the larger the fixed groups, the higher the energy barrier. Under the conditions of Figure 9, the energy barrier is roughly linearly dependent on the effective radius of anions. Shahgaldian and Coleman31 found that as Na+ is the common cation, the collapse pressure of a p-dodecanoylcalix[4]arene monolayer follows the sequence from lowest to highest with BF4- < I- < CH3CO2-. Since the repulsive force leads to the collapse pressure, the order of anion size may correspond to the phenomena. This order is consistent with the result of Figure 9. However, the corresponding PCM predicts no difference in the electrical interaction among various sizes of anions with the same valence. Figure 10 shows the variation in the scaled energy barrier between a particle and a charged surface as a function of the scaled effective radius of fixed groups. As can be seen in this figure, the larger the effective radius of fixed groups, the higher the energy barrier. Since the larger the fixed groups, the smaller the region IV, the fixed charge is more concentrated in this region as Xi increases. A stronger electrical field arising from the fixed charge in the membrane causes a higher energy barrier. The larger the cations, the higher the energy barrier as revealed in Figure 10. According to Figure 10, the energy barrier is nonlinearly dependent on Xi. This is mainly due to the fact that the fixed charge density in region IV increases nonlinearly with the increase in Xi.

Although both the van der Waals potential and the gravitational potential contribute to the attractive energy between the particle and the charged surface, they have different distance dependence as expressed in eqs 4 and 5. For the cases examined, if the separation distance is 7 times the double-layer thickness, the gravitational potential is about 10% of the van der Waals potential, and if the separation distance is 50 times the double-layer thickness, the van der Waals potential becomes about 9% of the gravitational potential. This implies that the gravitational force dominates the attraction of colloidal particles toward a surface if the separation distance is larger than 50 times the double-layer thickness. The electrical repulsive force is nearly negligible if the separation distance is greater than about 10 times the doublelayer thickness and becomes comparable to the van der Waals attractive force if the separation distance is smaller than about 5 times the double-layer thickness.

(30) Colic, M.; Franks, G. V.; Fisher, M. L.; Lange, F. F. Langmuir 1997, 13, 3129. (31) Shahgaldian, P.; Coleman, A. W. Langmuir 2001, 17, 6851.

4. Conclusion In summary, the interaction between a colloidal particle covered by an ion-penetrable membrane layer and a rigid, negatively charged surface is studied taking the effect of the sizes of all the charged species into account. The results of numerical simulation reveal that the sizes of the charged species can play a significant role. Depending upon the separation distance and the level of the interaction, the deviation of the classic point-charge model from the present model in the electrical interaction force can be either positive or negative. The energy barrier for the particle-surface interaction is high under the following conditions: (i) The absolute electrical potential on the surface is high. (ii) The fixed charge density in the membrane layer is high. (iii) The membrane is thin. (iv) The radius of cations is large. (v) The radius of anions is large. (vi) The radius of fixed groups in the membrane layer is large. The general trend is consistent with the experimental observations reported in the literature, which cannot be predicted by the corresponding point-charge model. Acknowledgment. This work is supported by the National Science Council of the Republic of China. LA011293S