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Langmuir 2006, 22, 3174-3179
Steric and Bridging Interactions between Two Plates Induced by Grafted Polyelectrolytes Haohao Huang and Eli Ruckenstein* Department of Chemical and Biological Engineering, State UniVersity of New York at Buffalo, Buffalo, New York 14260 ReceiVed October 17, 2005. In Final Form: January 3, 2006 If colloidal particles are grafted with a polymer, then the grafted chains can provide steric repulsion between them. If some of the grafted polymer chains are also adsorbed to a second particle, then a bridging force is generated as well. For uncharged plates and polymer, the following contributions to the free energy of the system have been taken into account in the calculation of the interaction force: (i) the Flory-Huggins expression for the mixing free energy of the grafted chains with the liquid; (ii) the entropy loss due to the connectivity of the polymeric segments; (iii) the van der Waals interactions between the segments and the plates; and (iv) the free energy of adsorption of the polymer segments of the grafted chains on the other plate. For charged plates, the electrostatic free energy as well as the free energy of the electrolyte are included in the total free energy of the system. By minimizing the free energy with respect to the segment concentration and, when it is the case, with respect to the electrical potential, equations for the segment number density distribution and for the electrical potential are obtained, on the basis of which the interactions between two plates grafted with polymer chains that can be also adsorbed on the other plate were calculated. The interaction thus obtained includes steric and bridging forces.
1. Introduction When two plates with polymer chains grafted to their surface approach each other, a steric repulsion is generated between the two brushes as soon as the tips of the grafted chains begin to contact each other.1,2 The repulsive steric interactions can increase the stability of a colloidal system. When the distance between the plates becomes less than twice the thickness of the polymer brush, the polymer chains on the two plates begin to interdigitate. The increase of the segment concentration in the interdigitation domain leads to repulsion. When opposite polymer brushes contact each other, the brushes become compressed in addition to their interdigitation. Therefore, the steric repulsion includes entropic and elastic contributions. However, if some of the grafted chains are also adsorbed on the second plate, then the polymers draw the plates toward each other, resulting in an attractive bridging force.3,4 The bridging force may induce the coagulation of the colloidal particles. The bridging was extensively examined experimentally,5-7 theoretically,8-10 and by Monte Carlo simulations.11-14 Of course, the steric and bridging forces are simultaneously present between the plates, and their effects are not expected to be additive. Many theories were developed to describe the distribution of segments between two polymer brushes. On the basis of a lattice * To whom correspondence should be addressed. E-mail: feaeliru@ acsu.buffalo.edu. Tel: (716) 645-2911 ext. 2214. Fax: (716) 645-3822. (1) de Gennes, P. G. AdV. Colloid Interface Sci. 1987, 27, 189. (2) Roan, J. R; Kawakatsu, T. J. Chem. Phys. 2002, 116, 7283. (3) Iler, R. K. J. Colloid Interface Sci. 1971, 37, 364. (4) Dickinson, E.; Eriksson, L. AdV. Colloid Interface Sci. 1991, 34, 1. (5) Luckham, P. F.; Klein, J. J. Chem. Soc., Faraday Trans. 1984, 80, 865. (6) Claesson, P. M.; Ninham, B. W. Langmuir 1992, 8, 1506. (7) Lowack, K.; Helm, C. A. Macromolecules 1998, 31, 823. (8) Varoqui, R. J. Phys. II 1993, 3, 1097. (9) Podgornik, R. J. Phys. Chem. 1992, 96, 884. (10) Borukhov, I.; Andelman, D.; Orland, H. J. Phys. Chem. B 1999, 103, 5042. (11) Akesson, T.; Woodward, C.; Jonsson, B. J. Chem. Phys. 1989, 91, 2461. (12) Jimenez, J.; de Joannis, J.; Bitsanis, I.; Rajagopalan, R. Macromolecules 2000, 33, 7157. (13) Misra, S.; Mattice, W. L. Macromolecules 1994, 27, 2058. (14) van Giessen, S.; Szleifer, I. J. Chem. Phys. 1995, 102, 9069.
model, Scheutjens et al. developed a discrete self-consistent theory to describe the steric repulsion.15-18 A similar model involving interdigitation was employed by Li and Ruckenstein.19,20 The assumption of free jointed chains used by Scheutjens et al. was replaced by the latter authors with the assumption of a correlation between neighboring bonds. Another kind of self-consistent theory used a continuum diffusionrepresentationtodescribethedistributionofsegments.21-23 The segments were assumed to be subjected to an external potential of a mean field. An analytical approximation of the latter self-consistent theory was suggested by Milner et al. (the MWC model)24 on the basis of the observation that at high stretching the partition function of the brush is dominated by the classical path as the most probable distribution. Under this assumption, it was found that the self-consistent field is parabolic and leads to a parabolic distribution of the monomer density. Similar theories for polyelectrolyte brushes25-27 also adopted the parabolic distribution approximation. The polyelectrolyte molecules affect in a more complex manner than the neutral ones the steric and bridging interactions between colloidal particles because they affect the electrical potential of the electrical double layer,28,29 which in turn affects the conformation of the chains and the overall electrostatic interactions. (15) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (16) Leermakers, F. A. M.; Sheutjens, J. M. H. M. J. Chem. Phys. 1988, 89, 3264. (17) Leermakers, F. A. M.; Sheutjens, J. M. H. M. J. Chem. Phys. 1988, 89, 6912. (18) Wijmans, C. M.; Leermakers, F. A. M.; Fleer, G. J. Langmuir 1994, 10, 4514. (19) Li, B. Q.; Ruckenstein, E. J. Chem. Phys. 1997, 106, 280. (20) Ruckenstein, E.; Li, B. Q. J. Chem. Phys. 1997, 107, 932. (21) Edwards, S. F. Proc. Phys. Soc. 1965, 85, 613. (22) Borukhov, I.; Andelman, D.; Orland, H. J. Phys. Chem. B 1999, 103, 5042. (23) Huang, H. H.; Ruckenstein, E. AdV. Colloid Interface Sci. 2004, 112, 37. (24) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (25) Miklavic, S. J.; Marcelja, S. J. Phys. Chem. 1988, 92, 6718. (26) Misra, S.; Varanasi, S.; Varanasi, P. P. Macromolecules 1989, 22, 4173. (27) Zhulina, E. B.; Borisov, O. V. J. Chem. Phys. 1997, 107, 5952.
10.1021/la0527947 CCC: $33.50 © 2006 American Chemical Society Published on Web 03/01/2006
Steric and Bridging Interactions between Two Plates
Langmuir, Vol. 22, No. 7, 2006 3175
A statistical model by Manciu and Ruckenstein,30,31 which assumed that the chain distributions are generated by all possible random walks and employed an average free energy for all of the chain distributions that end at a certain distance x from the interface, showed that by accounting for the most likely chain distribution and discarding all other chain distributions one arrives at the same scaling law as does the MWC model. However, when all possible distributions (with their appropriate Boltzmann weights) have been taken into account, a departure from the MWC model results, most notably, at high grafting density, when the brush becomes more step-like than parabolic-like.30 With the assumption of a step function for the distribution of the segment density and using the ground-state dominance approximation, Alexander32 and de Gennes1 suggested a scaling analysis to calculate the interaction between two brushes. An equation is derived in the present article for the free energy of the system by using various contributions to the free energy. For uncharged plates and polymer, the following contributions were considered: (i) the Flory-Huggins expression for the mixing free energy of the grafted chains with the liquid; (ii) the entropy loss due to the connectivity of the segments of the chains; (iii) the van der Waals interaction between segments and plates; and (iv) the free energy of adsorption of the segments of the grafted polymers on the other plate. For charged surfaces and polymers, the electrostatic free energy and the free energy of the electrolyte were also added to obtain the total free energy of the system. The minimum free energy with respect to the segment density and, when it was the case, with respect to the electrical potential provides two differential equations for the segment density and electrical potential. The minimum of the free energy with respect to the segment density of the polymer near the surface provided the concentration of the segments that are subject to adsorption. Because both the steric and bridging interactions occur simultaneously, these two interactions are coupled and should not be calculated separately. Some polymer chains are grafted on only one plate and contribute to the steric repulsion; the chains that are also adsorbed on the interacting plate contribute to bridging. The goal of the present article is to calculate the interaction between two plates grafted with polymer chains, the segments of which can be also adsorbed on the interacting second plate.
βFp )
[
2
∫0D - 2l13τV2n2 + 6l13wV3n3 - Un + l6 (dφ dx )
[
AHl′3 1 1 + U) 3kT x3 (D - x)3
]
(1)
The polymer contribution contains the Flory-Huggins mixing free energy of the segments with the solvent molecules, the van der Waals interaction -UkT between segments and the two plates, and the entropy loss caused by the connectivity of the segments 8,23
(2)
(3)
where AH is the Hamaker constant and l′ is the diameter of a segment that is assumed to be spherical. Close to the surface, there is a Born repulsion that we consider that generates an infinite repulsion over a distance of 3.5 Å. The double-layer contribution Fd contains the entropic Fent of the ions and the electrostatic contributions Fele. The entropic contribution of the electrolyte to the free energy of the system is due to the change in the entropy of ions between the plates compared to that in the reservoir.
βFent )
[(
xi
∫0 ∑ ci ln x i D
)
1+ cw ln
i,b
∫0 ∑ i D
(
∑i xi
∑i
1-
ci
ci ln
ci,b
]
dx ≈
xi,b
)
- ci + ci,b dx (4)
where ci is the concentration of ion i, cw is the concentration of water, xi is the mole fraction of ion i, and the subscript b indicates the concentration in the bulk. The electrostatic free-energy contribution is the sum of the electrostatic energy of the system and the chemical contribution of the surface. For constant surface potential ψ0, the chemical contribution is given by
Fchem ) -2σψ0
Two plates grafted with polymer chains on their surface are immersed in an electrolyte solution. The two plates are considered to be located at x ) 0 and D, respectively, and the system is considered to be in contact with a large reservoir containing an electrolyte solution free of polymer. Compared to the concentration of the ions of the electrolyte, the concentration of the counterions dissociated from the grafted polyelectrolyte is neglected. The free energy F of the system contains a polymer (Fp), a double layer (Fd), and an adsorption (Fa) contribution.
(28) Ohshima, H. Colloid Polym. Sci. 1999, 277, 535. (29) Hoagland, D. A. Macromolecules 1990, 23, 2781. (30) Manciu, M.; Ruckenstein, E. Langmuir 2004, 20, 8155. (31) Manciu, M.; Ruckenstein, E. Langmuir 2004, 20, 6490. (32) Alexander, S. J. Phys. 1977, 38, 983.
dx
In eq 2, n and V are the local number density and the volume of a polymer segment, τ is the dimensionless excluded volume parameter, w is the second virial coefficient, φ ) xn, l is the Kuhn length of a segment, β ) 1/kT, k is the Boltzmann constant, and T is the absolute temperature. For the van der Waals interactions between plates and segments in the solution, one can write the expression
2. Theoretical Framework
F ) Fp + F d + F a
]
2
(5)
where σ is the surface charge density. Therefore, the electrostatic free-energy contribution can be expressed as
1 Fele ) 0 2
∫0D(dψ dx )
1 ) 0 2
∫0D(dψ dx )
2
dx - 2σψ0
2
dx + 20ψ0
dψ | dx x)0
(6a)
Using Green’s theorem, eq 6a can be rewritten as
1 Fele ) 0 2 ) -0
( ) ∫[
∫0D dψ dx D
0
ψ
2
dx - 0
[
2
∫0D ψ ddxψ2 + (dψ dx )
( )]
d2ψ 1 dψ 2 + dx dx2 2 dx
]
2
dx (6b)
where is the dielectric constant, 0 is the vacuum permittivity, and ψ is the electrical potential. The optimization of the total free energy of the system with respect to the electrical potential provides the following relation
3176 Langmuir, Vol. 22, No. 7, 2006
Huang and Ruckenstein
between the electrical potential and the charge density (Poisson equation)
F d2ψ )2 dx 0
(7)
where the charge density in the solution F includes the charges from the electrolyte ions and the polyelectrolyte +
-
F ) e(c - c + Rφ ) 2
dx ∫0D[eψ(c+ - c- + Rφ2) - 21(dψ dx ) ] 2
(9)
The adsorption contribution to the free energy is a result of the replacement of solvent molecules on the surface by adsorbed polymer segments. The adsorption contribution is given by
2Γ∆Gads Fads ) N0
(s0 - s)Kadsns 1 + Kadsns
(11)
where s0 is the total number of adsorption sites on the surface per unit area, s is the grafting density of the polymer chains. ns is the segment concentration near the plates and Kads is the adsorption constant (RT ln Kads ) -∆Gads, R being the gas constant and T the temperature in K taken 300 K). All of the segments between the two plates belong to the grafted polymer chains, and the plates are in contact with an electrolyte solution free of polymer. Consequently, the total number of segments in the solution between the two plates is constant. If the grafting density of the polymer chains is denoted by s, then the total number of segments in the solution is 2sN, where N is the number of segments per chain. This constraint must be taken into account in the minimization of the free energy with respect to the segment density. Because the number of segments in solution is much larger than the number of segments adsorbed, the number of the latter segments can be neglected. Hence, one can write
2sN ≈
∫0D φ2 dx
(12)
The minimization of the free energy with respect to φ under the (33) Farrokhpay, S.; Morris, G. E.; Fornasiero, D.; Self, P. J. Colloid Interface Sci. 2004, 274, 33.
(13)
where λ is a Lagrange multiplier, which is provided by eq 12. The minimization of the free energy with respect to ψ leads to the equation
1 d2ψ ) [2ce sinh(βeψ) - Reφ2] dx2 0
(14)
Equation 13 provides the segment density of the polymer chains subjected to the external electrical potential, ψ, and the van der Waals interaction with the plates, -UkT. Equation 14 represents a modified Poisson-Boltzmann equation in which the first term on the right-hand side accounts for the charges of the small ions of the salt and the second term accounts for the charges of the polyelectrolyte chains. When the polymer is not charged, eqs 13 and 14 can be solved independently. Because of symmetry, there are two boundary conditions at the middle distance between them
(10)
where Γ is the number of segments adsorbed per unit area, ∆Gads is the molar adsorption free energy of the polymer, and N0 is Avogadro’s number. Typical values for ∆Gads are -13 kJ/mol for poly(acrylic acid) on alumina and -10 to -16 kJ/mol for polyacrylamide on talc.33 Here the adsorption free energy of a polymer molecule is assumed to be the same as the adsorption free energy of a segment. The number of segments adsorbed per unit area can be calculated using the Langmuir adsorption isotherm
Γ)
l2 d2φ 1 ) -τl3φ3 + wl6φ5 - Uφ + Rβeψφ + λφ 2 6 dx 2
(8)
c+ and c- are the concentrations of the cations and anions, respectively, and R is the charge of each segment. Substituting eq 7 into eq 6b, one obtains
Fele )
constraint (eq 12) and considering that V ) l3 leads to
dψ (x ) D/2) ) 0 dx
(15)
dφ (x ) D/2) ) 0 dx
(16)
At the surface of the charged plates, the boundary condition is
ψ ) ψ0
(17)
for a constant potential of the plates. The segment concentrations at z ) 0 and D, which are the same, are selected by trial and error such as to minimize the free energy. The Lagrange multiplier is determined also by trial and error using eq 12. The profiles of the segment concentration and the electrical potential for the minimum free energy allow one to calculate the interaction force f between the two plates, which is given by the expression
f)-
δF δD
( { [ ]}
)
1 1 4 6 2 ) kT l3τφD/2 - wl6φD/2 + kTUD/2φD/2 + 2 6 eψD/2 ∂(φ2D/(1 + Kadsφ2D)) 2c cosh + 2s0KadsRT ln Kads kT ∂D (18) 3. Calculation of the Steric Interaction If the grafted polymer is not adsorbed on the second plate, then the grafted polymer provides only steric interaction between the plates. This repulsive steric interaction increases the stability of the colloidal system. To calculate the steric interaction between two plates, the adsorption constant Kads in eq 11 should be taken to be zero in this case. Figure 1 presents the steric interaction between two plates for various graft densities. As the grafting density increases, the segment density at the middle between plates increases. Therefore, the steric interaction between the two brushes increases. As shown in Figure 2, the steric repulsion increases as the polymer chain becomes longer. As the length of the polymer
Steric and Bridging Interactions between Two Plates
Figure 1. Steric interaction between two plates for various grafting densities. τ ) 0.3, w ) 1, N ) 200, l ) 5 Å, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, and s0 ) 0.8 nm-2. s/s0 is equal to the following values: (1) 0.01; (2) 0.02; (3) 0.03; (4) 0.04; and (5) 0.05.
Figure 2. Steric interaction between two plates for various chain lengths. τ ) 0.3, w ) 1, l ) 5 Å, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, and s0 ) 0.8 nm-2. The ratio of the number of grafting sites to the number of adsorption sites is 0.02. N is equal to the following values: (1) 150; (2) 200; (3) 300; and (4) 500.
Langmuir, Vol. 22, No. 7, 2006 3177
Figure 4. Steric interactions between two cross cylinders for τ ) 0.3, w ) 1, l ) 7.77 Å, N ) 430, s0 ) 1.66 nm-2, R ) 0, ψ0 ) 0, and s/s0 ) 0.00779. (1) de Gennes’s model; (2) the present model; (3) Ruckenstein and Li’s model. The experimental data (o) are from ref 34.
concentration of the segments against the distance to the surface of the plate can be well approximated by a step function. As the distance between the two plates increases, the concentration of the polymer segments decreases. Our model confirms one of the basic assumption of the Alexander-de Gennes theory, namely, that the segment distribution can be approximated by a step function. For comparison purposes, the expressions for the interaction between two uncharged plates grafted with uncharged chains suggested by de Gennes1 and Ruckenstein and Li20 are given below and compared with results of the present calculations. Two effects, the osmotic and the elastic contributions, were taken into account in the de Gennes analysis for the interaction between two parallel plates1
f=
{( ) ( ) }
kT 2L0 d3 D
9/4
-
D 2L0
3/4
(19)
where d is the average separation between two graft sites and L0 is the brush thickness. Using a lattice model, Ruckenstein and Li derived a more rigorous expression for the steric interaction between two parallel plates.20 The interaction force p between two crossed cylinders of radius R, for which experimental data are available,34 can be obtained from the interaction free energy F between two parallel plates using the Derjaguin approximation
p ) 2πF R
(20)
A comparison between the three expressions for the steric interaction p is made in Figure 4, which shows that for polystyrene the present model is in better agreement with the more rigorous approach of Ruckenstein and Li than the equation of de Gennes. Figure 3. Distribution of polymer segments between two plates at various distances. τ ) 0.2, w ) 1, l ) 5 Å, ψ0 ) 0.005 V, R ) 2e, N ) 200, s0 ) 4 nm-2, and c ) 0.01 M. The ratio of the number of grafting sites to the number of adsorption sites is 0.1. Only half the distance is plotted. (1) 200; (2) 300; (3) 400; and (4) 600 Å.
increases, the number of segments between the two plates increases, and this results in an increase in the free energy and in the repulsive force. Figure 3 presents the distribution of polymer segments between the two plates for various distances. The distribution of the
4. Steric Bridging Interactions Induced by a Polyelectrolyte If the grafted polymer molecules are also adsorbed on the second plate, then the force between the two plates includes both bridging and steric interactions. In this case, the adsorption contributions to the total free energy must be included among the free-energy contributions. When the polymer is charged, eqs (34) Taunton, H. J.; Toprakcioglu, C.; Fetters, L. J.; Klein, J. Macromolecules 1990, 23, 571.
3178 Langmuir, Vol. 22, No. 7, 2006
Figure 5. Interaction between two plates at various grafting densities. τ ) 0.3, w ) 1, N ) 200, l ) 5 Å, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, s0 ) 0.8 nm-2, and ∆Gads ) -13.25 kJ/mol. s/s0 is equal to the following values: (1) 0.01; (2) 0.02; (3) 0.03; (4) 0.04; and (5) 0.05.
Figure 6. Distribution of polymer segments between two plates for various grafting densities. τ ) 0.2, w ) 1, l ) 5 Å, ψ0 ) 0.005 V, R ) 2e, c ) 0.01 M, N ) 200, and ∆Gads ) -13.25 kJ/mol. s/s0 is equal to the following values: (1) 0.05; (2) 0.1; (3) 0.2; and (4) 0.3 and s0 ) 4 nm-2.
13 and 14 are not independent and have to be solved simultaneously. The segment concentration ns at the surface was selected by minimizing the free energy of the system with respect to ns. 4.1. Effect of the Grafting Density of the Polymer. The grafting density of the polymer affects the total number of segments and the adsorption of the segments on the plate surface. Figure 5 shows that as the grafting density of the segments increases the force becomes more repulsive. At relatively small values of the grafting density, the force decreases with increasing distance and passes through a minimum, after which it increases. Figure 5 shows that there are conditions under which the overall force becomes negative, and hence the bridging becomes dominant. Figure 6 shows that the shape of the segment concentration distribution can be described by a step function and that the uniform concentration becomes larger for higher grafting densities. Consequently, higher bridging is generated, and the attractive bridging force increases. However, the higher segment concentration also increases the steric interaction. Because the latter dominates, the overall interaction that includes both the bridging and the steric forces becomes more repulsive with increasing grafting density, at least for the parameters employed in the present calculations. 4.2. Effect of the Stiffness of the Chains. The stiffness of the chains, represented by the term l2/6 (dφ/dx)2 in eq 2, together
Huang and Ruckenstein
Figure 7. Interaction between two plates for various segment lengths l. τ ) 0.3, w ) 1, N ) 200, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, s0 ) 0.8 nm-2, and ∆Gads ) -13.25 kJ/mol. The ratio of the number of grafting sites to the number of adsorption sites is 0.02. l (nm) is equal to the following values: (1) 0.3; (2) 0.4; (3) 0.5; and (4) 0.8.
Figure 8. Interaction between two plates for various chain lengths. τ ) 0.3, w ) 1, l ) 5 Å, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, s0 ) 0.8 nm-2, and ∆Gads ) -13.25 kJ/mol. The ratio of the number of grafting sites to the number of adsorption sites is 0.02. N is equal to the following values: (1) 150; (2) 200; (3) 300; (4) 500; and (5) 800.
with the electrical and van der Waals interactions determines the conformation of the polyelectrolyte chains and hence their charge and segment concentration distributions in solution. A larger value of l means a stiffer polymer chain. The chains become more extended with increasing chain stiffness, and the repulsive force between plates increases with increasing stiffness (Figure 7). For stiffer chains, the contribution of bridging is smaller than for flexible chains because the concentration of segments near the plate surface decreases with increasing stiffness. 4.3. Effect of the Length of the Polymer. As the length N of the chain increases, the repulsive force becomes stronger (Figure 8). Obviously, the steric repulsion is expected to increase with increasing N. The adsorption of the polymer segments is expected to increase as well. However, for the conditions employed, the steric repulsion increases more than the bridging attraction. 4.4. Segment Charge through Dissociation Equilibrium. If the polyelectrolyte chains generate their charge through dissociation equilibrium, then
SM T M- + S+
(21)
where SM is an ionizable group of the polyelectrolyte chain, Mis the anion of the electrolyte and chain, and S+ is a dissociated site.
Steric and Bridging Interactions between Two Plates
Langmuir, Vol. 22, No. 7, 2006 3179
The fraction θ of the dissociated ionizable groups is given by
Kd ≡
θ[M-] 1-θ
(22)
where Kd is the dissociation equilibrium constant and [M-] is the anion concentration.
[M-] ) c exp
(eψ kT )
(23)
where c is the electrolyte concentration in the reservoir.
R ) ξeθ )
ξeKd Kd + c exp(eψ/kT)
(24)
where ξ is the number of ionizable groups per segment. Figure 9 presents the effect of the equilibrium constant Kd on the force. As Kd increases, the force becomes more repulsive. This happens because as the dissociation constant of the polyelectrolyte ionizable groups increases the segments of the polyelectrolyte become more positively charged. Therefore, the segment adsorption on the surfaces that are also positively charged becomes smaller. As a result, the bridging contribution becomes smaller, and the interaction becomes more repulsive with increasing dissociation constant.
5. Conclusions The interaction between two plates grafted with charged polymer chains was calculated by taking into account both the bridging and steric interactions. The free energy of the system was calculated by accounting for a number of contributions: (i) the Flory-Huggins free energy for the grafted chains and liquid; (ii) the van der Waals interactions between the segments and plates; (iii) the connectivity free energy of the segments; and (iv)
Figure 9. Interaction between two plates grafted with a polyelectrolyte for various dissociation equilibrium constants. τ ) 0.3, w ) 1, l ) 5 Å, ψ0 ) 0.01 V, c ) 0.01 M, s0 ) 0.8 nm-2, ∆Gads ) -13.25 kJ/mol, and ξ ) 1. The ratio of the number of grafting sites to the number of adsorption sites is 0.02. Kd is equal to the following values: (1) 1; (2) 10; (3) 100; and (4) 1000 mol/m3.
the adsorption free energy. For charged polymers and surfaces, an entropic contribution due to the electrolyte ions and an electrostatic contribution were included as well. The overall interaction becomes more repulsive with increasing grafting density of the polymer on the surface and with increasing length of the polymer chains. When the plates and the segments have the same sign of charge, the interaction is more repulsive for higher charge densities of the segments because both the steric interaction and the double-layer force are stronger and the bridging weaker. The interaction between two plates is more repulsive for flexible charged chains than for stiff charged chains. There are conditions under which the bridging attraction dominates the steric repulsion. This occurs at sufficiently low grafting densities as well as sufficiently low segment charges (when the charges of the segments and the plate surfaces have the same sign). LA0527947
