Langmuir 2000, 16, 6233-6239
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Electrical Interactions between Two Ion-Penetrable Charged Membranes: Effect of Sizes of Charged Species Yung-Chih Kuo Department of Chemical Engineering, Eastern College of Technology and Commerce, 110 Tung-Fung Rd, Hu-Nei, Kaohsiung, Taiwan 829, ROC
Jyh-Ping Hsu*,† Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, ROC Received September 15, 1999. In Final Form: April 3, 2000 A diffuse double-layer model for evaluation of the electrical interactions between two particles each covered by an ion-penetrable charged membrane in an electrolyte solution is proposed by taking the sizes of charged species into account. We show that the ionic sizes have a significant effect on the electrical interaction force between two particles. For constant fixed charge density, the corresponding point charge model will underestimate the interaction force, and the reverse is true for constant membrane potential. The deviation due to the assumption of point charges is more serious for constant fixed charge density than that for constant membrane potential. It is found that the smaller the counterions, the greater the interaction force, and the reverse is true for co-ions.
Introduction The electrical contribution plays a significant role in understanding the structure of a suspension and the thermodynamics of an ionic system.1,2 Since the pairwise interactions are dominant in a dilute suspension, it is crucial to quantify the double-layer interaction between two particles. The classic Derjaguin-Landau-VerweyOverbeek theory3,4 for the pairwise interaction between two identically charged particles in an electrolyte solution, a central canon in colloid science, provided a solid foundation for the electrostatic interaction. Although more sophisticated approaches such as molecular simulations5 and those that consider particle-particle and particleion interactions, for example, Percus-Yevick,6 hipernetted chain,7 mean spherical approximation,8 and optimized cluster expansions,9 have been proposed, the electrical double layer is often described by a mean field level of the relatively simple Poisson-Boltzmann equation because it often yields sufficiently accurate results.10,11 Recently, a great deal of theoretical interest has been attracted to a system comprising fixed charge materials immersed in an electrolyte solution.11-15 Materials of this * To whom all correspondence should be addressed. † Fax: 886-2-23623040. E-mail:
[email protected]. (1) van Aken, G. A.; Lekerkerker, H. N. W.; Overbeek, J. Th. G.; De Bruyn, P. L. J. Phys. Chem. 1990, 94, 8468. (2) Kuo, Y. C.; Hsu, J. P. Chem. Phys. 1998, 236, 1. (3) Derjaguin, B. V.; Landau, L. Acta Physiochim. (USSR) 1941, 14, 633. (4) Verwey, E. J.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (5) Rivera, S. R.; Sorensen, T. S. Mol. Simul. 1994, 13, 115. (6) Henderson, D.; Abraham, F. F.; Baker, J. A. Mol. Phys. 1976, 31, 1291. (7) Rasaiah, J. C.; Friedman, H. L. J. Chem. Phys. 1968, 48, 2742. (8) Lebowitz, J. L.; Percus, J. K. Phys. Rev. 1966, 144, 251. (9) Stell, G.; Lebowitz, J. L. J. Chem. Phys. 1968, 49, 3706. (10) Friedrichs, M.; Zhou, R.; Edinger, S. R.; Friesner, R. A. J. Phys. Chem. 1999, 103, 3, 3057. (11) Kuo, Y. C.; Hsu, J. P. J. Phys. Chem B. 1999, 103, 9743. (12) Ohshima, H.; Kondo, T. J. Theor. Biol. 1987, 128, 187. (13) Ohshima, H.; Kondo, T. Biophys. Chem. 1988, 32, 161. (14) Kuo, Y. C.; Hsu, J. P. J. Chem. Phys. 1995, 102, 1806.
type play a significant role in a variety of fields in practice, including chemical engineering, material processing, environmental engineering, and biotechnology. Typical example includes polyelectrolyte gels, ion-exchange resins, synthetic membranes, and biological cells, to name a few. For charged membranes in an electrolyte solution, three ionic species are encountered: the fixed charges contained in membrane matrix and the mobile counterions and coions in the liquid phase. In such a system, the membrane phase is usually considered to be microscopically homogeneous for a sufficiently long time scale due to the random thermal motion of its component parts.16,17 For the majority of the previous studies,12-17 however, the ionic species were regarded as point charges. The effect of various zones of closest approach between the charged species and the membrane hard core on double-layer properties was investigated recently.11,18 It was found that the spatial variation in electrical potential was dramatically influenced by the sizes of charged species.18 The ionic sizes have a significant effect on the apparent electrical field due to the presence of the membrane and on the net penetration charges into the membrane even if the potential is low.11 In this study the electrostatic interaction force and interaction energy between two identical charged membranes in an electrolyte solution are derived by taking the finite sizes of charged species into account. Within the framework of the linearized Poisson-Boltzmann equation, an analytical expression for electrical potential is obtained. Two idealized models for membrane, namely, constant membrane potential and constant fixed charge density, are considered to justify the method proposed. (15) Hsu, J. P.; Kuo, Y. C. J. Chem. Phys. 1995, 103, 465. (16) Manning, G. S. J. Chem. Phys. 1967, 47, 3377. (17) Hsu, J. P.; Kuo, Y. C. J. Chem. Soc., Faraday Trans. 1995, 91, 1223. (18) Hsu, J. P.; Kuo, Y. C. J. Chem. Phys. 1999, 111, 4807.
10.1021/la991223+ CCC: $19.00 © 2000 American Chemical Society Published on Web 06/29/2000
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free region and region II, [Xc, Xan], is equal to the effective scaled radius of anion. The interval including the chargefree region, regions II and III, [Xc, Xi], is equal to the effective scaled radius of fixed group. The interval for the outer uncharged membrane, [Xo, d + Xc], is also equal to the effective scaled radius of fixed group. On the basis of the above description, 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 being respectively the reciprocal Debye length and the distance. On the basis of the Gauss theory of electric field, 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, XL, and d are, respectively, the scaled location of the uncharged core-membrane interface, 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 membranes, and the scaled thickness of membrane phase. The scaled size of the fixed group is Xi - Xc. 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 comprising the outer uncharged membrane and the diffuse double layer.
Analysis A schematic representation of the system under consideration is shown in Figure 1. We consider two identical planar particles, and each comprises a rigid, uncharged core and an ion-penetrable membrane layer of scaled thickness d in an a:b electrolyte solution. The symmetric nature of the system suggests that only the interval [-∞, XL] needs to be considered, XL being the location of the midplane between two particles. The membrane layer contains uniformly distributed, negatively fixed charge. Let σf, σan, and σca be the effective diameters of fixed group, anion, and cation, respectively. Without loss of generality, we assume that σf > σan > σca; the present analysis, however, does not limit to this condition. 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. With reference to Figure 1, the system is divided into five regions. Region I represents the domain X < Xca, which comprises the charge-free region (-∞ < X < Xc) and the inner uncharged membrane (Xc < X < Xca), X being the scaled distance. Region II is represented by Xca < X < Xan, which contains cations only. Region III is described by Xan < X < Xi, which contains both cations and anions. Region IV is Xi < X < Xo, which contains all charged species, and region V is Xo < X, which includes the outer uncharged membrane (Xo < X < Xc + d) and the liquid phase (XL > X > Xc + d). Here, Xi and Xo denote respectively the locations of the inner and the outer planes of the fixed charge and Xc, Xca, and Xan represent respectively the location of the uncharged core-membrane interface and the most interior locations of positive and negative charges. Xi - Xc represents the effective scaled radius of fixed group. It is worth noting that the five regions exist naturally, provided that the sizes of charged spices are finite. In the arrangement of charged species, the closest approach between each species and the membrane-core interface is assumed to equal to its effective radius. The interval for the charge-free region, [Xc, Xca], is equal to the effective scaled radius of caion. The interval including the charge-
d2φ [-u exp(-aφ) + v exp(bφ) - wN] ) a+b dX2
(1)
where φ ) eφ/kBT, κ2 ) e2a(a + b)n0a/0rkBT, and N ) ZN0NA/an0a. In these expressions, φ, N0, Z, NA, e, 0, r, n0a, kB, and T denote respectively the electrical potential, the density and the valence of fixed groups, the Avogadro number, the elementary charge, the permittivity of a vacuum, the relative permittivity of the system, the number concentration of cations in the bulk liquid phase, the Boltzmann constant, and the absolute temperature. For convenience, a vector for the presence/absence of (1) fixed charge, (2) anion, and (3) 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-V. Here, since only the averaged effect resulted from the plane perpendicular to the r-direction is considered, the fixed charge can be assumed to distribute smoothly from the viewpoint of r even in a small region. The limitation of the validation of eq 1 is that the variation of the electrical field over the plane perpendicular to the r-direction is sufficiently small. If the amount of fixed charge in a membrane is small, a considerable amount of counterions and co-ions may penetrate into the membrane. In the case of low membrane potential, the effect of the finite sizes of the charged species is, therefore, significant and eq 1 can be approximated by
2 d2φ ) [(ua + vb)φ + v - u - wN] 2 ak dX
(2)
where k ) 2(1 + b/a). The boundary conditions associated with this equation are assumed as
dφ f 0 as X f -∞ dX dφ (dX )
) X)Xca+
dφ (dX )
)
X)Xan+
dφ (dX ) dφ (dX )
dφ (dX ) dφ (dX )
dφ (dX ) dφ )( ) dX
X)Xca-
X)Xan-
) X)Xi+
X)Xo+
X)Xi-
X)Xo-
(2a)
φ(X+ ca) ) φ(Xca) at X ) Xca
(2b) φ(X+ an) ) φ(Xan) at X ) Xan
(2c) φ(X+ i ) ) φ(Xi ) at X ) Xi (2d) φ(X+ o ) ) φ(Xo ) at X ) Xo
dφ f 0 as X f XL dX
(2e) (2f)
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Equations 2a,f imply that the system under consideration is at electroneutrality. Solving eq 2 associated with eqs 2a-f yields
φ ) φc region I
(x )
φ ) C1 sinh
2 X + C2 cosh k
(3)
(x )
2 1 X + region II k a (4)
φ ) C3 sinh(X) + C4 cosh(X) region III φ ) C5 sinh(X) + C6 cosh(X) +
ak [C cosh(XL) + C4 sinh(XL)] 2C9 3
(7)
(8)
∫0
FR ) -
(aena - benb) dφ
(9)
where φL is the midplane potential between two particles, and 0
(9a)
0
(9b)
na ) na (1 - aφ) nb ) nb (1 + bφ)
In these expressions na and nb represent respectively the number concentrations of cation and anion. Note that in the case of low membrane potential the potential on the midplane between two particles is also low. Substituting eqs 9a,b into eq 9, we obtain, after integrating,
FR )
a2k 0 n k Tφ 2 4 a B L
(10)
where φL is the scaled electrical potential on the midplane between two particles. The electrical interaction energy, VR, can be evaluated by
VR )
∫(X∞ -X )FR dXL
2 κ
L
c
(11)
Let us consider two idealized cases, which are used most frequently in the relevant studies, namely, constant potential and constant charge density. In the former the fixed charge density is regulated by the separation distance between two particles. From eq 7, φL can be expressed as, after rearrangement,
φL ) (C3 + C4C10/C9) sinh(XL) + (C4 + C3C10/C9) cosh(XL) (12) where C10 is defined in the Appendix.
(13)
where
sinh(Xo) - sinh(Xi) + cosh(Xo) - cosh(Xi) sinh(Xo) - sinh(Xi) - cosh(Xo) + cosh(Xi) E ) (C4 + C3)D + C4 - C3
(6)
where C9 is defined in the Appendix. The electrical interaction force between two particles FR can be evaluated by φL
∑
D)
where C1-C8 are defined in the Appendix and φc is the scaled electrical potential in the uncharged core of a particle, or the scaled membrane potential. It can be shown that N0 and φc satisfy the implicit relation (Appendix)
N)
a2kn0akBT ∞ VR ) E2 (-D)j-1 exp[-2j(XL - Xc)] 4κ j)1
(5)
2N region IV ak
φ ) C7 exp(-X) + C8 exp(X) region V
Substituting eq 12 into eq 10 yields an explicit expression for the electrical interaction force. Substituting this expression into eq 11 leads to, after integration (Appendix),
(13a) (13b)
For the case of constant fixed charge density the membrane potential is regulated by the separation distance between two particles. Solving eq 8 for φc yields
φc )
1 F1 cosh(XL) + F2 sinh(XL) + a G1 cosh(XL) + G2 sinh(XL)
(14)
where F1, F2, G1, and G2 are defined in Appendix. Equation 7 leads to
φL ) [-F1 + G1(φc - 1/a)] sinh(XL) + [-F2 + G2(φc - 1/a)] cosh(XL) (15) Substituting eqs 14 and 15 into eq 10 yields an explicit expression for the electrical interaction force. Substituting this expression into eq 11 leads to, after integration (Appendix), ∞ a2kna0kBT H2 (-K)j-1 exp[-2j(XL - Xc)] VR ) 4κ j)1
∑
(16)
H ) (F1 - F2) - (F1 + F2)K K ) (G1 - G2)/(G1 + G2) Results and Discussion A typical illustration for the thickness of each region proposed in the present model is human erythrocyte whose peripheral zone contains a lipid layer with averaged thickness of 15 nm surrounded by a saline solution. Here, the fixed charge is assumed to be mainly arises from the dissociated of the carboxyl groups in the surface layer. The radii of Na+, Cl-, and the carboxyl groups are about 0.098, 0.181, and 0.35 nm, respectively, and the radii of hydrated Na+, Cl-, and the carboxyl groups are approximately 0.3, 0.4, and 0.5 nm, respectively. Therefore the thickness of regions I-V is 0.3, 0.1, 0.1, 14, and 0.5 nm, respectively. Although the thickness of regions I and II for mobile ions with hydration shells around them is thicker than that for nonhydrated ions, the analysis procedure for both is the same. Since the present model is based on the averaged electrical field resulted from the plane perpendicular to the r-direction, the changes of the hydration structure in different regions does not influence the model presented if the electrolyte concentration is fixed. The ionic sizes of charged species were not considered in the analysis of Ohshima and Kondo.12,15 Available results which take the effect of ionic sizes into account are not for membrane systems.6-10 The present analysis is the first attempt to investigate the interaction between two membranes by taking the sizes of charged species into account. It can be shown that the present model
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Figure 2. Variation of the scaled interaction force between two particles as a function of the scaled half separation distance between them for the case d ) 1. Key: solid curves, present model; dash curves, classic point-charge model. Curve 1: the case of constant membrane potential with φc ) -0.5. Curve 2: the case of constant fixed charge density with φc ) -0.5 at (XL - Xc) f ∞ and a fixed N evaluated by eq. A5. a ) 2, b ) 1, Xca - Xc ) 0.05, Xan - Xc ) 0.1, and Xi - Xc ) 0.2.
Kuo and Hsu
Figure 4. Variation of the scaled interaction force between two particles as a function of the scaled half separation distance between them for the case d ) 0.5. Key: same as Figure 2.
Figure 5. Variation of the scaled interaction energy between two particles as a function of the scaled half separation distance between them for the case d ) 0.5. Key: same as Figure 2. Figure 3. Variation of the scaled interaction energy between two particles as a function of the scaled half separation distance between them for the case d ) 1. Key: same as Figure 2.
reduces to the classic point-charge model (PCM) for an ion-penetrable charged membrane if Xca - Xc, Xan - Xc, and Xi - Xc all vanish. That is, if the sizes of all charged species are infinitely small, the corresponding PCM can be recovered as a special case of the present model. Also, as the thickness of membrane layer approaches zero, the present model reduces to the case of two rigid surfaces. The effect of ionic sizes on the electrical interaction force and the interaction energy between two particles are examined through numerical simulation. Figures 2 and 3 show respectively the variations of the scaled interaction force and the scaled interaction energy as a function of the scaled half separation distance between two particles. As can be seen from these figures, both the scaled interaction force and interaction energy decrease with the increase in XL - Xc. The interaction force and the interaction energy for the case of constant potential are smaller than those for the case of constant charge density. In the former, the classic PCM overestimates both the interaction force and the interaction energy, and the reverse is true for the latter. This implies that the difference between the two idealized models predicted by the classic PCM is smaller than that predicted by the present models. Figures 2 and 3 also reveal that the deviation due to the assumption of point charges is more
serious for the case of constant charge density than that for the case of constant potential. The variations of the scaled interaction force and the scaled interaction energy between two particles as a function of the scaled half separation distance between them for a membrane thickness different than that used in Figures 2 and 3 are illustrated in Figures 4 and 5, respectively. A comparison between Figures 2 and 3 and Figures 4 and 5 shows that the thinner the membrane the larger the interaction force and interaction energy. This is because, for a thinner membrane, a greater fixed charge density is necessary to maintain a specific constant membrane potential. Since the fixed charge density contained in a thinner membrane is greater than that in a thicker membrane for the case of constant fixed charge density, the former yields a larger interaction force and interaction energy. Figures 6 and 7 present respectively the variations of the scaled interaction force and the scaled interaction energy between two particles as a function of the scaled effective radius of cations for the case (Xan - Xc) ) 0.1. The leftmost points represent the results for the case where counterions are treated as point charges, and the rightmost points are the results for the case where counterions and co-ions have the same size with (Xca - Xc) ) (Xan - Xc) ) 0.1; that is, region II vanishes. As can be seen from Figures 6 and 7 both the interaction force and the interaction energy decrease with the increase in Xca - Xc. That is, the larger the size of cation, the smaller the interaction force
Ion-Penetrable Charged Membranes
Figure 6. Variation of the scaled interaction force between two particles as a function of the scaled effective radius of cations for the case Xan - Xc ) 0.1. Key: - -, d ) 1.1; s, d ) 1; - - -, d ) 0.5. Curve 1: constant membrane potential with φc ) -0.5. Curve 2: constant fixed charge density with φc ) -0.5 at (XL - Xc) f ∞ and a fixed N evaluated by eq A5. a ) 2, b ) 1, Xi - Xc) 0.2, and XL - Xc ) 1.5.
Langmuir, Vol. 16, No. 15, 2000 6237
Figure 8. Variation of the scaled interaction force between two particles as a function of the scaled effective radius of anions for the case Xca - Xc ) 0.05. Key: same as Figure 6.
Figure 9. Variation of the scaled interaction energy between two particles as a function of the scaled effective radius of anions for the case Xca - Xc ) 0.05. Key: same as Figure 6. Figure 7. Variation of the scaled interaction energy between two particles as a function of the scaled effective radius of cations for the case Xan - Xc ) 0.1. Key: same as Figure 6.
and the lower the interaction energy between two particles. Since the electrical potential decreases with the position variable in region II, the smaller this region (the larger the size of cation), the smaller the repulsive force between two particles. Figures 6 and 7 also suggest that the thinner the membrane the larger the interaction force and the interaction energy between two particles. The effect of cation size on the variation of the interaction force and interaction energy between two particles is greater for the case of constant fixed charge density than that for the case of constant membrane potential. Figures 8 and 9 illustrate respectively the variations of the scaled interaction force and the scaled interaction energy between two particles as a function of the scaled effective radius of anions for the case (Xca - Xc) ) 0.05. The leftmost points represent the results for the case where counterions and co-ions have the same size with (Xca - Xc) ) (Xan - Xc) ) 0.05; that is, region II vanishes. The rightmost points are the results for the case where coions and fixed groups have the same size with (Xan - Xc) ) (Xi - Xc) ) 0.2; that is, region III vanishes. Figures 8 and 9 reveal that both the interaction force and interaction energy increase with Xan - Xc. That is, the larger the size of anion the greater the interaction force and the higher the interaction energy between two particles. Since the electrical potential decreases (increases) with the position
variable in region II (III), the greater (smaller) the regions II (III), (the larger the size of anion), the larger the repulsive force between two particles. A comparison between Figures 6 and 7 with Figures 8 and 9 suggests that the effect of ionic sizes on both the interaction force and the interaction energy is more significant for the case of constant charge density than that for the case of constant potential. The same conclusions as those drawn for the case the size of cations are varying, Figures 2 through 5, can be drawn from Figures 6 through 9. Figures 8 and 9 also suggest that the thicker the membrane the smaller the interaction force and the interaction energy between two particles. The effect of anion size on the variations of the interaction force and the interaction energy is more significant for the case of constant fixed charge density than that for the case of constant membrane potential. In a study of the influence of ionic size on short-range repulsive forces between silica surfaces, Colic et al.19 found that among the lyotropic series of monovalent electrolytes, LiCl, NaCl, KCl, and CsCl, lithium produces short-range repulsive interparticle forces with the longest extent at low ionic strength. This is consistent with the present theoretical result in that the smaller the counterions the stronger the electrostatic repulsive force. It was also observed that both the viscosity and the rate of sedimentation of a slurry correlate with ionic size.19 Apparently, these observations cannot be explained by a PCM. (19) Colic, M.; Fisher, M. L.; Franks, G. V. Langmuir 1998, 14, 6107.
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The present model can be used to predict various fundamental properties of a colloidal dispersion such as the rate of coagulation, the critical coagulation concentration, and the stability ratio. For example, if particles remain at constant charge density, the rate of coagulation predicted by the present model can be much slower than that predicted by the corresponding PCM if the membrane layer is sufficiently thin.
A1 ) (2/k)1/2 cosh(Xan) cosh[(2/k)1/2Xan] - sinh (Xan) sinh[(2/k)1/2Xan] (A1k) A2 ) (2/k)1/2 cosh(Xan) sinh[(2/k)1/2Xan] - sinh (Xan) cosh[(2/k)1/2Xan] (A1m) B1 ) cosh(Xan) sinh[(2/k)1/2Xan] - (2/k)1/2 sinh (Xan) cosh[(2/k)1/2Xan] (A1n)
Conclusions
B2 ) cosh(Xan) cosh[(2/k)1/2 Xan] - (2/k)1/2 sinh
The classic point-charge model (PCM) for an ionpenetrable charged membrane in an electrolyte solution is modified to take the effect of finite sizes of charged species on double-layer interactions into account. We show that the difference in the interaction force between constant potential model and constant charge density model based on the present analysis is larger than that based upon the corresponding PCM. The difference in the interaction force based on the present analysis from that based on the corresponding PCM is smaller for constant potential model than for constant charge density model. The interaction force for the case of constant charge density is more sensitive to ionic sizes than that for the case of constant potential. The larger the counterions the smaller the interaction force between two particles, and the reverse is true for co-ions.
(Xan) sinh[(2/k)1/2Xan] (A1o) F1, F2, G1, and G2 used in the text are defined by
F1 ) {(2N/k)[sinh(Xo) - sinh(Xi)] + sinh(Xan)}/a (A2a) F2 ) -{(2N/k)[cosh(Xo) - cosh(Xi)] + cosh(Xan)}/a (A2b) G1 ) A2 cosh[(2/k)1/2Xca] - A1 sinh[(2/k)1/2Xca]
(A2c)
G2 ) B2 cosh[(2/k)1/2Xca] - B1 sinh[(2/k)1/2Xca]
(A2d)
C7 and C8 can also be expressed as Acknowledgment. This work is supported by the National Science Council of the Republic of China.
C7 ) [C5 + C6 tanh(Xo)]/[1 - tanh(Xo) coth(XL)] C8 ) [C6 + C5 tanh(Xo)]/[1 - coth(Xo) tanh(XL)]
Appendix
(A3a) (A3b)
The values of C1-C10 in the text are defined by
(
C1 ) - φc -
(
C2 ) φc -
) (x ) ) (x ) 1 sinh a
1 cosh a
2 X k ca
2 X k ca
(A1a)
Substituting eqs A3a and A3b into eq 7 and invoking the latter part of eq 2e yields
N)
(A1b)
C3 ) -[sinh(Xan)]/a + A1C1 + A2C2
(A1c)
C4 ) [cosh(Xan)]/a + B1C1 + B2C2
(A1d)
C 5 ) C3 +
2N sinh(Xi) ak
(A1e)
C6 ) C 4 -
2N cosh(Xi) ak
(A1f)
(ak/2)(βC3 + RC4) R cosh(Xi) - β sinh(Xi) - Rβ
(A4)
R ) cosh(Xo) tanh(XL) - sinh(Xo)
(A4a)
β ) cosh(Xo) - sinh(Xo) coth(XL)
(A4b)
where
C7 ) C3 - (2N/ak)[sinh(Xo) - sinh(Xi)]
(A1g)
C8 ) C4 - (2N/ak)[cosh(Xo) - cosh(Xi)]
(A1h)
C9 ) [sinh(Xo) - sinh(Xi)] cosh(XL) - [cosh(Xo) cosh(Xi)] sinh(XL) (A1i) C10 ) [cosh(Xo) - cosh(Xi)] cosh(XL) - [sinh(Xo) sinh(Xi)] sinh(XL) (A1j)
Equation A4, which is equivalent to eq 8, is an alternative expression for the scaled fixed charge density. Note that eqs 8 and A4 become
N)
(ak/2)(C3 + C4) as [sinh(Xo) - sinh(Xi) - cosh(Xo) + cosh(Xi)] XL f ∞ (A5)
This expression is the N - φc relation for an isolated membrane.
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Expanding C10/C9 in terms of exp(-2jXL), j ) 0, 1, ..., yields
C10 C9
φc )
∞
) -1 + 2
(-1)j+1Dj exp[-(2j + 1)XL] ∑ j)1
From eq 14, φc can be expanded as follows:
1 a
+ (G1 + G2)-1{(F1 + F2) +
Substituting eq A6 into eq 12 and collecting terms of the same order of exp(-XL), we obtain
H
(-D)j exp[-(2j + 1)XL] ∑ j)0
(-K)j exp[-(2j + 2)XL]} ∑ j)0
(A8)
Substituting eq A8 into eq 15 and collecting terms of the same order of exp(-XL), we obtain ∞
∞
φL ) E
∞
(A6)
φL ) H
(A7)
Substituting eqs A7 and 10 into eq 11, eq 13 is recovered, after integration.
(-K)j exp[-(2j + 1)XL] ∑ j)0
(A9)
Substituting eqs A9 and 10 into eq 11, eq 16 can be recovered. LA991223+
