Surface Interactions in the Presence of Polyelectrolytes. A Simple

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Articles Surface Interactions in the Presence of Polyelectrolytes. A Simple Theory J. Ennis,*,†,‡ L. Sjo¨stro¨m,§ T. Åkesson,§ and Bo Jo¨nsson§ Research School of Chemistry, Australian National University, Canberra, ACT 0200, Australia, and Department of Theoretical Chemistry, Lund University, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden Received October 15, 1999. In Final Form: May 24, 2000 The interaction between two charged walls neutralized by grafted polyelectrolytes and mobile counterions is studied using a simplified model system, in which the charged monomers of the polyelectrolyte are replaced by “grafted” ions that interact with the grafting wall via a one-dimensional potential. Using a mean-field approximation, the model is solved numerically, and some analytic results are obtained for large separations. The behavior of the mean-field solution is checked against Monte Carlo simulations of grafted polyelectrolyte chains and is found to agree qualitatively. In salt-free systems there is generally a long-range repulsion due to free counterions, and then a short-range bridging attraction if sufficient polyelectrolyte is present. The addition of salt screens the long-range repulsion and usually lowers the pressure. A bridging attraction can occur at short-range and is significant mainly when the effective surface charge of the wall plus grafted polyelectrolyte is small.

1. Introduction The theory of charged colloids has a long history dating back to the early works by Gouy1 and Chapman.2 The basis for our understanding of such systems is still the so-called DLVO theory as laid out independently by Derjaguin and Landau3 and Verwey and Overbeek.4,5 This theory has met with considerable success and is capable of describing the stability of charged aggregates in solution under a variety of conditions. The DLVO theory has two main ingredients: a repulsive double-layer repulsion and an attractive van der Waals force, where the former is derived from the Poisson-Boltzmann (PB) equation. The theory is most often qualitatively correct and sometimes also quantitatively applicable. The PB equation is based on a mean-field approximation and as such it is likely to become less reliable in “strongly coupled” systems, i.e., when the electrostatic interactions increase. The most common situation is in systems with divalent ions, where the DLVO theory can become qualitatively incorrect.6 With this exception one can confidently use the theory in most aqueous solutions containing different charged aggregates and small counterions and salt. A completely new picture appears when a flexible * Australian National University. † To whom correspondence should be addressed. ‡ Current address: CSIRO Petroleum, P.O. Box 3000, Glen Waverley VIC 3150, Australia. § Lund University. (1) Gouy, G. J. Phys. 1910, 9, 457. (2) Chapman, D. L. London Edinburgh Philos. Mag. J. Sci. 1913, 25, 475. (3) Derjaguin, B. V.; Landau, L. Acta Phys. Chim. URSS 1941, 14, 633. (4) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier Publishing Company Inc.: Amsterdam, 1948. (5) Overbeek, J. G. J. Chem. Phys. 1987, 87, 4406. (6) Guldbrand, L.; Jo¨nsson, B.; Wennerstro¨m, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221.

polyelectrolyte is added to a charged colloidal dispersion. Then the simple picture with a double-layer repulsion is no longer valid, and we can expect that the addition of polyelectrolyte might lead to both attractive and repulsive forces in a quite complicated fashion. Notwithstanding these uncertainties, polyelectrolytes have become widely used in many technical formulations ranging from hair shampoo to pulp productionsthe list of usage could be made very long.7-9 Polyelectrolytes are also abundant in biological systems as, for example, in the compaction of DNA by spermine and spermidine,10,11 and they may interfere in the blood coagulation process.12 It is clear that their role is to modify the electrostatic interactions between other charged aggregates, but it is far from clear how this is done. Thus, there is a need for a simple and coherent theory of colloid stability in the presence of flexible polyelectrolytes. The PB equation is easy to solve numerically for charged aggregates immersed in a salt solution. There also exist a few special cases where it can be solved analytically,1,2,13 which is extremely valuable from the point of qualitative discussions. The more complex situation with polyelectrolytes present has been addressed in a few investigations, with the electrostatic contributions mostly analyzed in a mean-field way. The chain conformations can be treated via a step-weighted lattice model,14,15 as solutions of a partial differential “diffusion” equation with an external (7) Jones, G. D. Polyelectrolytes; Technomic: Westerport, CT, 1976. (8) Eriksson, B.; Ha¨rdin, A. M. In Flocculation in Biotechnology and Separation Systems; Elsevier Science Publishers: Amsterdam, 1987. (9) Sasaki, K. J.; Burnett, S. L.; Christian, S. D.; Tucker, E. E.; Scamehorn, J. F. Langmuir 1989, 5, 363. (10) Gosule, L. C.; Schellman, J. A. Biochemistry 1976, 259, 333. (11) Wilson, R. W.; Bloomfield, V. A. Biophys. Chem. 1979, 18, 2192. (12) Nevo, A.; deVries, A.; Katchalsky, A. Biochim. Biophys. Acta 1955, 17, 536. (13) Engstro¨m, S.; Wennerstro¨m, H. J. Phys. Chem. 1978, 82, 2711. (14) Van der Schee, H.; Lyklema, J. J. Phys. Chem. 1984, 88, 66616667.

10.1021/la9913654 CCC: $19.00 © 2000 American Chemical Society Published on Web 08/09/2000

Interactions in the Presence of Polyelectrolytes

potential,16-18 or by allowing for Boltzmann weighting of the continuous monomer-monomer and monomer-wall bonding potentials.19 Specialized theories have also been developed for the case of polyelectrolytes grafted on neutral surfaces or surfaces with charge of the same sign, where the polyelectrolyte layer is typically very extended20-23 (although there is a substantial literature on such “brushes”, we shall not refer to it in detail, since our model is for the contrary case in which the polyelectrolyte has charge of opposite sign to the surface, and the layer is thin). In most cases the numerical effort needed is substantial and analytical expressions are difficult to obtain. We know of only one case where asymptotic formulae have been derived and then for the case of a continuous chain.24 Monte Carlo (MC) simulations have also been performed for a variety of colloid-polyelectrolyte systems,19,25,26 and in general the MC results seem to support the findings from the mean-field treatment, which is a comforting and valuable circumstance on which we will build in this study. Our strategy is to create a model system that approximates the experimental situation of polyelectrolytes on surfaces of opposite charge and that is easy to solve numerically. The simplified model system will also be able to provide analytical asymptotic results. It should be emphasized that we are not looking for a quantitative theory, but a qualitative one containing the essential physical mechanisms. Thus we are not adding any “new” physics to the models used by other workers. However it should be noted that the some theoretical models use a continuous distribution of charge along the polyelectrolyte chain,18,27 whereas we employ a discrete distribution of charge, which is closer to reality and easier to simulate. The difference between these approaches becomes important at small separations where chain bridging occurs for the chain with discrete charges. It should be appreciated that most real systems containing polyelectrolytes are poorly characterized in several ways. With this in mind we will limit the discussion to the interaction of two infinite parallel surfaces carrying a uniform surface charge density. The surfaces can be neutralized by small ions but they can also be neutralized by flexible polyelectrolytes, and for simplicity we will assume that these are grafted to the surface at one end. Experimental studies of the adsorption of cationic polyelectrolytes on negatively charged mica surfaces indicate that the adsorption is to a large degree irreversible, with the thickness of the adsorbed layer depending mainly on the conditions under which adsorption takes place.28-30 The use of end-grafted polyelectrolytes in our (15) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman and Hall: New York, 1994. (16) Odijk, T. Macromolecules 1980, 13, 1542-1546. (17) Muthutkumar, M. J. Chem. Phys. 1987, 86, 7230-7235. (18) Borhukhov, I.; Andelman, D.; Orland, H. Europhys. Lett. 1995, 32, 499-504. (19) Åkesson, T.; Woodward, C.; Jo¨nsson, B. J. Chem. Phys. 1989, 91, 2461-2469. (20) Miklavic, S. J.; Marcˇelja, S. J. Phys. Chem. 1988, 92, 67186722. (21) Pincus, P. Macromolecules 1991, 24, 2912-2919. (22) Ross, R. S.; Pincus, P. Macromolecules 1992, 25, 2177-2183. (23) Zhulina, E. B.; Birshtein, T. M.; Borisov, O. V. Macromolecules 1995, 28, 1491-1499. (24) Podgornik, R. J. Phys. Chem. 1992, 96, 884-896. (25) Dahlgren, M. A. G.; Waltermo, Å.; Blomberg, E.; Claesson, P. M.; Sjo¨stro¨m, L.; et al. J. Phys. Chem. 1993, 97, 11769-11775. (26) Ennis, J.; Sjo¨stro¨m, L.; Åkesson, T.; Jo¨nsson, B. J. Phys. Chem. B 1998, 102, 2149-2164. (27) Podgornik, R. J. Phys. Chem. 1991, 95, 5249-5255. (28) Dahlgren, M. A. G.; Hollenberg, H. C. M.; Claesson, P. M. Langmuir 1995, 11, 4480-4485.

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model is convenient because it fixes the amount at the surface, and the focus of this study is on forces rather than on adsorption. It is also possible to do simulations on nongrafted chains for which the total amount of polyelectrolyte between the surfaces does not change as the surface-to-surface separation is varied. This approximation is sometimes called “restricted equilibrium”,31,32 and the simulation results are very close to the grafted case. The theoretical calculations can be carried out for the true equilibrium as well and yield very similar results, but simulations become quite time-consuming owing to the difficulties in controlling the chemical potential of the polyelectrolyte. The aim of this work is to create a simple theory for the interactions in a mixture containing charged aggregates and oppositely charged polyelectrolyte chains. Thus, we will focus on the osmotic pressure in such a system and pay rather little attention to the distribution of the different charged species. The ideal experiment to compare our results with will be some sort of surface force experiment using either the surface force apparatus33 or the atomic force microscope34 or an osmotic stress experiment.35 Phase diagrams of surfactant-polyelectrolyte mixtures provide another source for experimental comparison,36 as do studies on thin-liquid films.37 The outline of the paper is as follows. In section 2 we introduce the two models for the polyelectrolyte system that are used in the analytical work and in the simulations. In section 3 we discuss the mean-field approximation to the simple model and summarize the analytic results that can be obtained in various regimes of salt concentration and separation. The details of the Monte Carlo simulations on the more complex model are given in section 4. Then in section 5 we examine the behavior of the pressures obtained from the mean-field solution in a variety of cases, comparing the qualitative features with MC simulations at key points, and our main findings are summarized in section 6. 2. Models In a real system a charged aggregate can take on a variety of geometrical shapes, but we will restrict ourselves to a system consisting of two parallel infinite and uniformly charged walls. This is a convenient system for a theoretical analysis, but it appears in many experimental situations as well, e.g., a lamellar liquid crystalline system or charged membranes. The interaction of two planar surfaces can sometimes be directly transfered to the interaction of two curved surfaces if the curvature is much smaller than the interparticle separation (the Derjaguin approximation38). Figure 1a gives a schematic picture of the system. In most experiments the degree of polymerization for the polyelectrolyte chain can amount to thousands of monomer (29) Hoogeveen, N. G.; Cohen Stuart, M. A.; Fleer, G. J. J. Colloid Interface Sci. 1996, 182, 133-145. (30) Hoogeveen, N. G.; Cohen Stuart, M. A.; Fleer, G. J. J. Colloid Interface Sci. 1996, 182, 146-157. (31) Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1985, 18, 1882-1900. (32) Borukhov, I.; Andelman, D.; Orland, H. Macromolecules 1998, 31, 1665-1671. (33) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (34) Biggs, S.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1994, 90, 3415-3421. (35) Parsegian, V. A.; Rand, R. P.; Fuller, N. L.; Rau, D. C. Methods Enzymol. 1986, 127, 400. (36) Bagger-Jo¨rgensen, H.; Olsson, U.; Iliopoulos, I. Langmuir 1995, 11, 1934. (37) Bergeron, V.; Langevin, D.; Asnacios, A. Langmuir 1996, 12, 1550-1556. (38) Derjaguin, B. Kolloid-Z 1934, 69, 155.

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between two monovalent charges, separated by a distance rij, is given by

e2 u(rij) ) ( rij u(rij) ) ∞

rij > dhc rij E dhc

(1)

where e is the elementary charge, dhc is the diameter of the species, and is the dielectric permittivity, which in SI units equals 4π0r, where 0 is the permittivity of free space and r is the relative permittivity of the solvent. The plus sign is for like charges and the minus sign for unlike charges. All charged species interact with the hard charged walls, which are assumed to carry a uniform surface charge density, σ. The relevant equation for this interaction has been described in detail elsewhere.39 In simulation work, a polyelectrolyte is often represented by hard charged spheres connected by harmonic springs. The harmonic bonds correspond to an approximately Gaussian distribution of bond lengths and can be viewed as an averaging over the conformational freedom of the covalent structure that implicitly joins the charged sites. In this way we avoid the complications of the detailed atomic interaction within the chain and between different chains while still retaining the main features of a connected structure. Thus, the model consists of freely jointed chains where each monomeric unit carries a single charge. The force constant K of the harmonic potentials is related to the charge density of a chain; e.g., a weak force constant corresponds to a large separation between charges and so a weakly charged chain. The total interaction between two neighboring monomers then reads

u(ri,i+1) )

Figure 1. Schematic picture of a planar electric double layer. The polyelectrolytes are end-grafted to the walls, and the small co- and counterions are confined between the walls at z ) 0 and z ) h, but in equilibrium with a bulk solution of fixed salt concentration. (a, top) Model system used in the MC simulations and (b, bottom) simplified model system used in the mean-field calculations.

units or more. However, the observed surface forces change little as the degree of polymerization is varied.25 This is fortunate, since it is beyond our capability to treat very long chains in a simulation, and we have to restrict the chain length to a few tens of monomeric units. In simulations it is found that for chains adsorbed onto oppositely charged walls the interaction is only weakly dependent on the chain length, once the number of monomers is larger than 5-10. This is true for the cases when the surface charge is either undercompensated or only slightly overcompensated by the polyelectrolyte and does not hold for the “brush” regime of polyelectrolytes grafted onto neutral surfaces. The theoretical reason for this behavior will be explained below. The solvent is described by its dielectric permittivity, and the only way it enters the calculation is via a scaling of the electrostatic interactions. This approximation ignores short-range effects due to the solvent structure (although some of these phenomena are effectively incorporated in the monomer size). The pair interaction

e2 K + r2 ri,i+1 2 i,i+1

ri,i+1 > dhc

(2)

This model has been studied in the past using computer simulations, and it has also been solved in the framework of the PB approximation.19 Despite the seemingly drastic simplifications introduced in this model, it is still very difficult to solve numerically even within the mean-field approximation. One clue about how to simplify it further can be found in the fact that a dimer gives qualitatively the same result as a system containing much longer oligomers. This is so because the important bridging component in the pressure usually only involves a few monomers. Thus, the simplification we would like to advocate in this presentation is one where the polyelectrolyte chains pictured in Figure 1a are replaced by single monomers grafted to the charged walls, as shown in Figure 1b. We shall refer to this latter case as the simple model. This system can be solved quite easily within the PB approach, and analytical asymptotic results are also readily obtainable. At first sight this simplified model would appear to be simplistic, in that it omits the connectivity of the polyelectrolyte chain, replacing it with an effective interaction between the charged monomer and the grafting surface. However, there are a couple of observations that can be used to motivate this approach. On the experimental side it is found that the long-range interactions between surfaces neutralized by adsorbed polyelectrolytes can be fitted well to the theory for the interaction of two electrical (39) Jo¨nsson, B.; Wennerstro¨m, H.; Halle, B. J. Phys. Chem. 1980, 84, 2179.


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double layers.40 Thus the zeroth-order theory is that the polyelectrolyte simply changes the effective surface charge. However, it is also observed that at short range there is an attraction that can be significantly in excess of that predicted for van der Waals interactions, which is independent of chain length, and which can be shown to be consistent with a short-range bridging mechanism.25 As noted above, simulations demonstrate that only a few monomers are involved in this bridging. Our first-order simplified theory effectively allows for bridges with a single monomer (connected to the grafting surface with a harmonic spring potential). In this paper we show that this simple and tractable model is able to reproduce all the qualitative features of simulations that employ longer chains. 3. A Mean-Field Approximation Contact Theorem. Analytical and numerical solutions for the simple model (see Figure 1b) can be derived upon making a mean-field approximation. We first discuss this approximation in the context of the exact expression for the osmotic pressure and then derive the corresponding equations. For the interaction of two planar surfaces in the presence of small ions and polyelectrolytes, one can determine the osmotic pressure from a straightforward derivative of the free energy or the configurational integral with respect to the separation h. The resulting exact expression, usually referred to as a contact theorem, is

() h

∑i ni 2

Posm ) kBT

∑i

) kBT

+ Pcorr + Pcoll + Pbridge

ni(0 or h) -

2πσ2

+ Pgraft

(3)

(4)

where ni is the number density of species i, σ is the surface charge density, kB is Boltzmann’s constant, and T is the absolute temperature. The two expressions in eqs 3 and 4 are obtained by taking the derivative with respect to the midplane and one of the charged surfaces, respectively. One can of course take the derivative with respect to any plane parallel to the surfaces, but these two are particularly useful. Note that the partitioning into collision and correlation (electrostatic) terms is somewhat arbitrary. For the present model with hard core ions, it is unambiguous. The collision term Pcoll is repulsive and comes from hard core collisions at the midplane, while the Pcorr refers to electrostatic interactions across the midplane and is usually attractive. The collision term becomes important at high salt concentrations, while in the absence of divalent counterions the correlation term usually is negligible in comparison to the ideal term. Pbridge refers here to the attractive contribution due to monomermonomer bonds that span the midplane, and Pgraft to the attractive contribution from bonds grafting the polyelectrolyte to the wall. In the mean-field approximation the correlation and collision terms are identically zero so eq 3 becomes

Posm ) Pideal + Pbridge

(5)

where Pideal ) kBT∑ini(h/2). The net pressure is evaluated under the assumption that the salt is in equilibrium with the bulk; i.e., the chemical potential is the same. The polyelectrolytes, on the other hand, are grafted to the surfaces and are not in (40) Biggs, S.; Proud, A. D. Langmuir 1997, 13, 7202-7210.

equilibrium with the bulk. In the absence of extra salt, Posm is equal to the net osmotic pressure, while in the more common situation with added salt one has osm Posm - Posm net ) P bulk

(6)

It can be deduced from the above equations that in the absence of any polyelectrolyte the mean-field approximation leads to a repulsive pressure. When polyelectrolyte is added, the presence of a bridging term in eq 5 changes the result and the pressure can become attractive in the mean-field description. Thus, one sees already at this stage that the simple DLVO picture of a repulsive double-layer pressure immediately breaks down and one can expect a complex force behavior where the addition of salt, the amount of polyelectrolyte, and the surface charge density will affect the net osmotic pressure. It is perhaps worthwhile to clarify what we mean by “bridging” in the above description. The term Pbridge in eq 3, which appears when components of the pressure are identified at the midplane, arises from monomermonomer bonds crossing the midplane. It is not necessary that the polyelectrolyte is in contact with both surfaces, although this conception of bridging frequently occurs in the literature. It is also not the same as the “patch charge” mechanism that has been proposed, which arises from the heterogeneity of charge in the lateral direction due to adsorbed polyelectrolyte.41 Nor does it rely on nonelectrostatic interactions or effects of solvent quality to produce additional monomer-monomer or monomer-surface attractions. The mechanism of the bridging in this model, which depends only on electrostatic interactions and monomermonomer bonding interactions, can be qualitatively described as follows. At large separations it is energetically favorable for a single polyelectrolyte to be located near one surface, to which it has opposite charge, even though it loses entropy owing to the restriction on its configurational freedom. Here it is unfavorable for the chain to stretch across the midplane to the electrostatic potential well near the other surface, since this would require numerous monomers to be located in the region of high electrostatic potential. When the separation between the surfaces is reduced to a few monomer-monomer spacings, then the energetic cost of crossing the midplane is reduced, since only a few monomers are involved, and there is a gain in entropy as the chain now has greater configurational freedom. The chains that do cross the midplane contribute an attractive component to the net osmotic pressure. Mean-Field Equations. In deriving the equations for the mean-field approximation to the simple model, we shall assume a 1:1 electrolyte, since the mean-field approximation is known to be reasonable for small monovalent ions in water at room temperature. The generalization to any symmetric electrolyte is straightforward. Furthermore, we shall assume that the grafting ions, which take the place of the monomers of the polyelectrolyte, are monovalent cations. Let the grafted ions have local number densities nl(z) (for the ions grafted to the left-hand wall) and nr(z) (for ions grafted to the right-hand wall), where the z axis is in the direction across the slit, as in Figure 1. Let the number densities of the mobile ions be denoted by n+(z) for the cations and n-(z) for the anions. Denote by h the separation of the two walls, and denote by γ the proportion of the surface charge that is neutralized by the grafting ions. In the absence of salt, (41) Gregory, J. J. Colloid Interface Sci. 1976, 55, 35.

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we will assume 0 e γ e 1, whereas in the presence of salt we have γ g 0. However, our model is not suitable for polyelectrolyte brushes, which adopt more extended conformations. The simple model assumes a flat conformation appropriate for polyelectrolytes on surfaces of opposite charge in which the length and connectivity of the chain have little effect on the forces, so the region of validity is limited to γ not much greater than 1. The charge density on each wall will be denoted by σ. The grafting potential shall be assumed to be of the form kBT(x/d)2, where x is the distance from the grafting surface and d is the characteristic width of the grafting potential. This implies a relation between d and the force constant (eq 2)

2 d) K

1/2

()

(7)

It is relatively easy to extend the theory to cover other forms for the grafting potential, and these yield qualitatively similar results as long as the potential increases at least as fast as some positive power of x (the distance from the grafting surface) for large x. The electrostatic potential ψ(z) satisfies Poisson’s equation

4π ∇2ψ ) - F(z)

(8)

where F(z) is the charge density at z. Since the ions of species i are at equilibrium, the total average force on an ion must vanish, so for the grafted ions at the left wall (z ) 0)

z2 )0 d

()

ˆ (z) + kBT∇ kBT∇ ln nl(z) + e∇ψ

where ψ ˆ is an exact potential of mean force. The meanfield approximation consists of assuming that ψ ˆ ) ψ, i.e., the ions interact with each other only through the average electrostatic potential ψ. The validity of this approximation has been examined for interacting double layers42 and also for a system with grafted ions,26 and its limitations are well-understood; it neglects a variety of correlation effects owing to the presence of other ions,42 which give an additional attractive contribution to the osmotic pressure at short-range. Integrating the force balance equation (assuming ψ ) ψ), we obtain

nl(z) ) n∞l exp(-eψ/(kBT) - (z/d)2) where n∞l is a constant that depends on the surface density of the grafted ions at the left wall. Proceeding in a similar way for the other ions, and taking account of the symmetry, eq 8 becomes

4πe ∞ d2ψ (n+ exp(-eψ/(kBT)) - n∞- exp(eψ/(kBT)) + )2 dz n∞g exp(-eψ/(kBT))(exp(-(z/d)2) + exp(-((h z)/d)2))) (9) where e is the elementary charge and n∞+ and n∞- are the concentrations of the respective species at the point where ψ is taken to be zero. The walls are located at z ) 0 and z ) h. It is clear from this equation that at the mean-field (42) Carnie, S. L.; Torrie, G. M. Adv. Chem. Phys. 1984, LVI, 141.

level our simplified model is equivalent to an ansatz for the functional form of the monomer density profile. When there is salt present, it is natural to take the zero of potential to be out in the bulk solution, so that n∞+ ) n∞) n, where n is the salt concentration in bulk. When only counterions are present, then n∞- ) 0, and the zero of potential may be taken for convenience at the location of the potential minimum. The number density n∞g is a parameter that is determined in the course of the solution.For the grafted ions, we have the constraint

∫0hen∞g exp(-eψ/(kBT) - (z/d)2) dz ) -γσ

(10)

For the boundary conditions on ψ, we shall assume that the dielectric constant of the wall is negligible, so that

|

|

dψ 4πσ dψ ))dz z)0 dz z)h

(11)

The net osmotic pressure between the two walls, P, can be split into physically relevant components as discussed in the previous subsection. In the mean-field approximation, if this is done at the midplane, the result is eq 5. Pbridge is the average of the total grafting force per unit area on grafted ions whose grafting “bonds” cross the midplane, and this is always attractive. In the salt-free case, it is convenient to define a dimensionless surface charge by

4πσed g0 s)kBT while in the case of added salt it is more natural to define it by

4πσe S)κkBT where

κ2 )

8πne2 kBT

(12)

(13)

We have obtained some analytical results from the mean-field approximation for large separations, and these are discussed in the following subsections. However in order to obtain solutions for arbitrary separations and surface charge densities, we solve eq 9 numerically, using the Bulirsch-Stoer method43 and a shooting technique (with Newton-Raphson iteration) to satisfy the two boundary conditions (eq 11) and the charge balance constraint in eq 10, with the latter being evaluated at the same time as taking a Bulirsch-Stoer integration step. Large Separations: Counterions Only. In an earlier paper26 asymptotic results were derived for an asymmetric system (in which only one wall has grafted ions) by the method of matched asymptotic expansions. Fortunately, these results can be applied to the current symmetric system (with two walls bearing grafted ions) by noting that in the symmetric system with counterions only, the derivative of the potential at the midplane is zero by symmetry. At large separations, when virtually none of the grafting ions cross the midplane, the potential distribution in either half will be almost the same as in the asymmetric system when the wall without grafted ions is neutral (the corrections due to grafting ions crossing (43) Press, W. P.; Flannery, B. P.; Teukolsky, S. A.; Vettering, W. T. Numerical Recipes, The Art of Scientific Computing; Cambridge University Press: Cambridge, 1986.


Langmuir, Vol. 16, No. 18, 2000 7121

the midplane are exponentially small in h and can be neglected by comparison to the inverse powers of h). Thus for h . d we have where C satisfies

(kBT)2 d d P) C1-A +O 2 2 h h 2πe h

(

2

(( ) ))

(14)

change sign as S increases. Various analytical and numerical results were given in the earlier paper on the asymmetric system,26 and these can be applied here without change, since the effective surface charge is defined as a property of a single wall in contact with bulk electrolyte. At low surface charge densities, we have 2

2C

1/2

S* ) S(1 - γe(κd) /4) + O(S2)

h 1 tan C1/2 ) sn 2 d

(

)

where sn ) s(1 - γ) and A is a constant independent of h that depends on s and γ. Note that hsn/d is independent of d, so C depends on γ, σ, and h but not on d. Thus the leading term at large separations satisfies the usual meanfield equations for counterions only, with the charge on each wall reduced by a factor (1 - γ). When γ ) 1 there are no counterions present in the system, only grafted ions, and once the separation h is much greater than d, the range of the grafting potential, then the net pressure is exponentially small. If sn > 0, then at large separations

(

C ) π2 1 -

8d 48d2 + + O(h-3) snh s2 h2 n

)

(15)

Thus C goes to a numerical constant as h f ∞, and so asymptotically the pressure between the walls is always repulsive at large enough separations, and decays as h-2 with a coefficient independent of the surface charge densities and the properties of the grafted ions (unless γ ) 1, as discussed above). The first correction that depends on the details of the surfaces is at O(h-3). Expressions for the coefficient A in eq 14 can be obtained again from the results for the same coefficient in the asymmetric system,26 which we shall denote by Aasym. The relationship is simply that A ) 2Aasym, so it is enough to summarize the results obtained in the earlier paper. For the specific grafting potential used here, A is always positive and increases monotonically as the scaled surface charge density s increases. Thus the presence of the grafted layer always makes the net pressure less repulsive. As the scaled surface charge density s becomes larger, A approaches a finite limit, which is O(1) unless γ is very close to 1. If for example, one increases d at fixed h and σ, then s increases and so A tends to a limit, but the attractive correction to the pressure in eq 14 continues to increase owing to the factor of d. Large Separations: salt. When salt is included, the form of the pressure between the two walls at large separations h is given by

P ∼ 4kBTn(S*)2 exp(-κh)

(16)

where S* is the scaled effective surface charge of one wall. The effective surface charge for γ ) 0 (i.e., no grafted ions) is given at the mean-field level by

8 S* ) ((1 + S2/4)1/2 - 1) S

(17)

where S is the scaled surface charge density defined in eq 12. For small S we have S* ) S - S3/16 + O(S5), while for S f ∞, S* increases monotonically and we have S* f 4. The effective surface charge for γ > 0 (i.e., when part of the surface charge is neutralized by grafted polyelectrolyte), can be monotonically decreasing with increasing S for large enough γ, and in some circumstances it can

(18)

Now when γ ) exp(-(κd)2/4), then the leading term in eq 18 vanishes. This implies that by changing γ or by changing κd it is possible to achieve a situation in which the effective surface charge is close to zero, so that the leading repulsive term in the pressure (see eq 16) vanishes, and the interaction is dominated by shorter range terms such as bridging. This kind of phenomena is observed experimentally by changing the concentration of polyelectrolyte in the solution from which the layer is initially adsorbed40,44 or by allowing more time for adsorption to occur.45 As the adsorbed amount increases, the long-range double-layer repulsion decreases in strength until at a certain point, usually referred to as the charge neutralization point (cnp), the force is dominated by the van der Waals attraction and a short-range bridging attraction. As the adsorbed amount increases further, the doublelayer repulsion reappears and the surfaces are said to be overcompensated by the polyelectrolyte. It is clear from the above analysis in eq 18 that the charge neutralization point (i.e., the vanishing of the long-range double-layer interaction) in our model does not necessarily correspond to γ ) 1, although in low-salt conditions (when κd is small) this is very nearly true. This can be quantified further by an analytic result given in our earlier paper.26 If κd , 1, then the method of matched asymptotic expansions gives

S* )

8 ((1 + S2n/4)1/2 - 1) Sn Sγ(1 + S2n/2)/2 + O((κd)3) (19) (κd)2 (1 + S2n/4 + (1 + S2n/4)1/2)

where Sn ) S(1 - γ). The first term in eq 19 corresponds to the usual mean-field result (eq 17) with the surface charge density reduced by a factor of (1 - γ). This holds at very low salt, when d, the width of the grafting potential, is small compared to the Debye screening length. The first correction due to the finite width of the grafting potential, the (κd)2 term in eq 19, reduces the effective surface charge. If γ is close to 1 and so Sn is small, then this second term can dominate. 4. Monte Carlo Simulations The Monte Carlo simulations were performed in the canonical ensemble with the traditional Metropolis algorithm.46 Owing to the symmetry of the model system (Figure 1a), periodic boundary conditions were used in the lateral directions, i.e., parallel to the surfaces. Each charge interacts with all other charges within the simulation box, including the uniformly smeared out surface charge, according to the minimum image convention. Charged species outside the box are taken into account as an external potential, which is determined numerically from the average distribution inside the box in a standard manner.39 (44) Dahlgren, M. A. G.; Claesson, P. M.; Audebert, R. J. Colloid Interface Sci. 1994, 166, 343-349. (45) Claesson, P. M.; Ninham, B. W. Langmuir 1992, 8, 1406-1412. (46) Metropolis, N. A.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A.; Teller, E. J. Chem. Phys. 1953, 21, 1087-1097.

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The length of the polyelectrolytes has been chosen to be 20, but as mentioned before, the same results would be achieved with an even shorter oligoelectrolyte. The standard Metropolis algorithm with single-particle moves is usually very inefficient for chain simulations, and it is in general advisable to use another approach, e.g., a pivot algorithm.47,48 We have, however, found that the singlemove algorithm is competitive up to about 20 monomers, and above that the pivot algorithm should be the preferred choice. All charged species are treated as monovalent hard core ions with a common diameter of 4.25 Å or as monovalent point charges. The hard wall constraints are applied in relation to the center of each particle, which means that the position of the centers is restricted to 0 < z < h. The MC simulations were carried out with at least 120 charged particles some of which were connected so as create polyelectrolytes. More than 25 000 configurations per particle were used to equilibrate the system, and the production runs were approximately four times larger. In the presence of salt, the equilibrium with the bulk is determined by the chemical potential, which is obtained by a modified Widom technique49,50 both in the double layer and in the bulk. Since the net pressure is obtained as the difference between the pressures in the double layer and in the bulk, which are roughly of the same magnitude, a high precision is needed for these numbers in order to achieve accurate results. The chemical potential was determined by inserting five “ghost” particles every 100th configuration.

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Figure 2. Osmotic pressure (in mol L-1) as a function of separation and varying fraction of polyelectrolyte (γ is given in the figure) calculated by the mean-field model. R is the gas constant, and T is the absolute temperature. The width of the grafting potential, d, is chosen to be 10 Å. The area per surface charge is 50 Å2. No added salt. The circles represent pressures obtained from simulations with γ ) 0.5 and d ) 5.92 Å. The number of monomers per chain is 20.

5. Results and Discussions The following discussion will focus on the contribution to the stability of different aggregates and phases due to the presence of polyelectrolyte and added salt. The contribution from dispersion forces and the use of the Derjaguin approximation would also be needed to be considered for a more complete discussion of the stability of colloidal particles. We will use the (net) osmotic pressure in the lamellar system (see Figure 1) for this purpose. Most of the results comes from the mean-field treatment of the simplified model system in Figure 1b. We will, however, on crucial points compare with Monte Carlo simulations on the model in Figure 1a to ensure that the mean-field treatment actually provides a qualitatively correct picture of the underlying physics. In both the meanfield calculations and the simulations we use T ) 298.16 K and a relative dielectric constant of 78.47. 5.1. Salt-Free Systems. Effect of γ. Figure 2 shows force curves for a double layer with added polyelectrolyte, but without any salt. That is, only counterions are present, and a fraction of them γ is coming from the polyelectrolytes and the remaining fraction 1 - γ comes from the small monovalent counterions. The surface charge density is one charge per 50 Å2. We have included simulation results (represented by circles) for a system with 20 monomers per chain and γ ) 0.5. In a qualitative respect the simple mean-field model seems to reproduce the simulations quite well. The main difference is in the bridging regime, since bridging is facilitated if several monomers are involved. The width of the grafting potential used in the mean-field model, d, can be thought of as an effective parameter that mimics the behavior of the grafted chains in the MC simulations. As such, the value of d can be adjusted to fit (47) Lal, M. Mol. Phys. 1969, 17, 57. (48) Madras, N.; Sokal, A. D. J. Stat. Phys. 1988, 50, 109-186. (49) Widom, B. J. Chem. Phys. 1963, 39, 2808. (50) Svensson, B. R.; Woodward, C. E. Mol. Phys. 1988, 64, 247.

Figure 3. Osmotic pressure as a function of separation and varying surface charge density for d ) 10 Å. The amount of polyelectrolyte is γ ) 0.99. The area per surface charge (in Å2) is given in the figures. The inset shows an expanded view of the pressure at larger separations. No added salt.

the force curves obtained in a particular simulation. In the results shown in Figure 2, the value of d ) 10 Å has been chosen so that the mean-field pressures at large separations are roughly the same as those from a simulation with chains consisting of 20 monomers and a d value of 5.92 Å. In general we note that when γ f 0 the ordinary doublelayer repulsion is retrieved and that all force curves predict a repulsive pressure at large separation. This is to be expected from the mean-field solution and is also found in MC simulations. At a separation of the order of the monomer bond length, there appears an attractive pressure component, the magnitude of which increases with γ. This force is due to bonds crossing the midplane, and close to γ ) 1 it dominates over the repulsive contribution, as shown in Figure 2. Note that for γ equal to 1, as discussed above, the pressure is zero outside the range of the bridging attraction, which is here about 35 Å. Effect of Surface Charge Density. Figure 3 shows the effect on the pressure curves of changing the surface charge density at γ ) 0.99. Even at the lowest charge density


Figure 4. Osmotic pressure as a function of separation and varying chain flexibility. The parameter d indicates the flexibilityslarger d means a more flexible chain. The graphs are obtained for d ) 6, 10, and 14 Å. The area per surface charge is 50 Å2, and γ, the fraction of polyelectrolyte, is 0.99. No added salt.

shown there is some bridging, and the depth of the attractive minimum is increasing more than linearly with the surface charge density. Outside the bridging regime the leading term in the pressure is due to the free counterions, as shown in eq 14. Increasing the surface charge density at fixed γ means that the number of free counterions increases, corresponding to an increase in sn ) s(1 - γ). Examining eq 15, it can be seen that this leads to an increase in the total pressure, and this is borne out in the inset to Figure 3. At small separations, the pressure is dominated by the ideal pressure contribution, which depends on the total number of grafted and free ions, and thus in this regime the pressure also becomes more repulsive as the surface charge density increases. In experimental force measurements at low salt concentrations (which are very similar to the situation discussed here of counterions only), evidence of such a bridging attraction has been seen in the vicinity of the charge neutralization point for polyelectrolytes adsorbed on mica25,44,45,51,52 (the lattice charge density on bare mica is around 1 charge per 50 Å2). Effect of Chain Flexibility. A general result, for a system with γ close to unity, is that the range of the attraction increases as the polyelectrolyte becomes more flexible, i.e., when the distance between the charged monomers increases. Figure 4 demonstrates this result for γ ) 0.99 and a surface charge density of 50 Å2 per charge, and in addition it shows that while the range of the attraction increases it also becomes weaker. The findings are in agreement with simulation results.19 There is also experimental evidence to support this result. Comparing a polyelectrolyte with 10% of the monomers charged (which corresponds in our model to a large d value) to a similar polyelectrolyte with 100% of the monomers charged, it has been found that bridging is weaker in the 10% charged polyelectrolyte, but of longer range.53 5.2. In the Presence of Salt. Salt has in many cases a strong influence on the stability of colloidal dispersions. Electrostatic screening reduces the magnitude of pressure components of electrostatic origin, but salt also modifies the pressure in a less direct way. For instance, the (51) Marra, J.; Hair, M. L. J. Phys. Chem. 1988, 92, 6044-6051. (52) Bremmell, K. E.; Jameson, G. J.; Biggs, S. Colloids Surf., A 1998, 139, 199-211. (53) Kjellin, U. R. M.; Claesson, P. M.; Audebert, R. J. Colloid Interface Sci. 1997, 190, 476-484.

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Figure 5. Osmotic pressure as a function of separation and varying amount of polyelectrolyte (γ is given in the figure). The area per surface charge is 50 Å2, and d ) 10 Å. The salt concentration in the bulk is 100 mM.

conformational response of the polyelectrolytes and changes of the osmotic equilibrium of mobile species must be considered in order to achieve a complete description of the salt dependence. The behavior of the osmotic pressure can be quite complex owing to competing effects, but fortunately the behavior of its components is often more easily interpreted. Effect of Adding Salt. Addition of a small amount of salt causes only minor changes in the osmotic pressure, which, except for γ close to 1, becomes less repulsive (or more attractive). The most important effect is that the repulsion is of much shorter range when salt is added, and at large separations the h-2 dependence of the pressure turns into an exponential decay with the decay length, κ-1, as shown in eq 16. Equation 19 indicates that at low salt concentrations the long-range behavior is essentially a normal double-layer repulsion with the surface charge density multiplied by a factor of (1 - γ). Further increasing the electrolyte concentration in the bulk results in a richer behavior of the osmotic pressure as seen in Figure 5, which gives results for 100 mM salt. In most cases the pressure is further reduced, but one may easily find systems with γ around unity, which become more repulsive. The osmotic pressure outside the bridging regime is very small for γ ≈ 0.9 with a maximum pressure close to 1 mM at 40 Å. This agrees with the asymptotic behavior discussed above in relation to eqs 16, 18, and 19; i.e., for κd ∼ 1 there is a value of γ less than unity for which the effective surface charge S* changes sign, and so the long-range repulsion between the surface disappears. These asymptotic expressions for the pressure depend solely on the ideal contribution (there is no bridging at large separations) and are derived in the overlap approximation, in which the electrostatic potential distribution is assumed to be equal to the sum of the potential distributions due to two isolated walls. The effective surface charge S* is computed from the decay of the potential distribution away from an isolated wall with grafted ions in contact with a bulk salt reservoir, and the changes in S* with salt concentration can be explained in terms of free energy changes for the free and grafted ions. In Figure 5 we have also included one case with γ > 1, which corresponds to a situation where the surface charges are overcompensated by grafted ions. Electroneutrality is maintained by the presence of coions (a species with charge of the same sign as the surface charge). The magnitudes of both the harmonic pressure and the ideal

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Figure 6. Osmotic pressure as a function of the amount of polyelectrolyte at various salt concentrations. The area per surface charge is 50 Å2, and the separation is 40 Å. The meanfield results (the lines) use d ) 10 Å, while the simulation results (the symbols) are for chains with 20 connected monomers and d ) 5.92 Å. The inset shows an enlargement of the region around γ ) 0.7-1.0. Dot-dash and circles: 0 mM salt. Long dash and squares: 50 mM salt. Dashed line and triangles: 150 mM salt.

monomer pressure increase, in fact to a larger extent than expected, but the pressures balance each other such that the change in the total contribution from grafted ions is negligible. The large repulsion thus relates to the coions, the concentration of which is high in the midregion. Another way of illustrating how salt affects the osmotic pressure is presented in Figure 6, where the change in pressure with γ at fixed separation is shown for various bulk electrolyte concentrations. As seen in the figure, the mean-field results are qualitatively supported by simulations. At the chosen separation and with a small amount of salt, the osmotic pressure as a function of γ has a very sharp minimum close to γ ) 1. Decreasing γ from this value results in a rapidly increasing pressure due to a higher concentration of free counterions at the midplane, while the total pressure from the polyions is only affected to a small extent. The strength per bond crossing the midplane is unaffected by γ, but the probability of a chain bridging is reduced since the monomer density becomes more condensed at the walls for a smaller γ. On the other hand, the addition of salt, at a given γ, facilitates bridging. When the surface charges are overcompensated by grafted ions, i.e., γ > 1, the system responds with large repulsive forces (as seen also in Figure 5). Adding more salt to the system results in a less distinct minimum at a slightly higher pressure and a lower value of γ. The latter observation agrees with the predictions of eqs 18 and 19, since the minimum pressure occurs where S* ) 0, and the critical value of γ is predicted to fall as the salt concentration (and so κ) increases. Both the simulations and the mean-field approximation show that adding salt initially causes a decrease in the net osmotic pressure, except close to γ ) 1, where there is initially an increase. Considering eqs 16, 18, and 19, it is clear that even in the asymptotic regime the effects of adding salt can have further complications, depending on the surface charge density, since the electrolyte concentration affects S* and κ, as well as entering directly into eq 16. Effect of Chain Flexibility. The flexibility of the chain is another property that may have an impact on the osmotic pressure. Interestingly, the force curves shown in Figure 7 resemble the corresponding salt curves in the previous figure, and indeed a more flexible chain facilitates the influx of electrolyte from the bulk. In other words, a larger

Ennis et al.

Figure 7. Effect of chain flexibility on the osmotic pressure as a function of the amount of polyelectrolyte. The graphs are obtained for d ) 5, 10, 20, and 30 Å. The area per surface charge is 50 Å2, the separation is 100 Å, and cB ) 10 mM.

d-value corresponds to a higher salt content in the double layer. The mechanism behind this behavior is of entropic origin. The dense charge distribution, caused by electrostatics, implies large conformational restrictions on a polyion interacting with a weak grafting potential. Addition of salt leads to a relaxation of these constraints by an exchange of grafted ions close to the walls with free counterions, a process that is driven by the increasing chain entropy but at the expense of a lower counterion entropy and higher Gaussian energy. Thus, with a large chain flexibility one will find a larger fraction of free counterions in the vicinity of the walls, which affects the chemical potential such that the equilibrium content of salt in the double layer increases. The fact that increasing chain flexibility lowers the value of γ at which S* changes sign (i.e., the minimum in the curves in Figure 7) also agrees with the predictions of eqs 18 and 19, since increasing d or increasing κ should have a similar effect. Making the chain more flexible eventually results in formation of bridges and, as for d ) 30 Å, an attractive force between the walls. In the limit d f 0, the force curves become symmetric around γ ) 1, and the system is transformed into an ordinary double layer with an effective surface charge density given by (1 - γ)σ, as shown by eq 19. Qualitative Comparison with Experiments. Experimental investigations of the effect of adding salt to a system with adsorbed polyelectrolytes are complicated by two effects. First, it is difficult to obtain the parameter γ directly, for even when the adsorbed amount can be measured, the actual surface charge density is difficult to estimate, and the net surface charge density σ(1 - γ) has to be inferred by fitting force profiles to DLVO theory. Second, the adsorption of strong polyelectrolytes and subsequent changes to the salt concentration involve nonequilibrium effects. For example, adsorbing polyelectrolyte from solution at low ionic strength and then increasing the ionic strength to the desired level does not yield the same force measurements as a system in which the adsorption takes place at the higher ionic strength.28 In the first situation it is found that adding salt to a system that is near the charge neutralization point causes additional adsorption, an increase in the layer thickness, an increase in the magnitude of the double-layer repulsion but a decrease in the decay length, and a weaker adhesion.25 This is consistent with the picture developed above, in which increasing γ past the charge neutralization


point leads to a steep increase in double-layer repulsion. However, in order to test the effect of adding salt at fixed γ, it is necessary to do an experiment in which, after the initial adsorption of polyelectrolyte, the solution between the surfaces is replaced by one that contains only salt, so that no further adsorption can occur as the salt concentration is altered. In this situation, both the strength and the range of the double-layer repulsion decrease as salt is added, and the layer thickness decreases slightly.25 These changes are consistent with Figure 6 and the discussion there. 6. Conclusions In this paper we have explored the behavior of a very simple mean-field model for the pressure between two identical surfaces due to strong polyelectrolytes grafted on surfaces of opposite charge to the polyelectrolyte. The polyelectrolyte layer is characterized by its flexibility d and the proportion γ of the surface charge that it neutralizes. At a few points this mean-field model has been compared to Monte Carlo simulations of short grafted chains, and the results agree at least qualitatively. The behavior is also consistent with experimental force measurements on surfaces bearing adsorbed polyelectrolytes. The observed deviations between the mean-field results and the simulations are understood: ordinary electrostatic correlation effects that are important at shortrange are neglected in the mean-field model, and bridging is stronger in the Monte Carlo simulations since bridges can include several monomers, whereas the mean-field model allows only single-monomer bridges. Apart from

Langmuir, Vol. 16, No. 18, 2000 7125

solving the mean-field model numerically, which is straightforward, we have also obtained analytic results for large separations, which are useful for explaining the numerical results. In a salt-free system, for γ < 1 there is a long-range repulsion between the surfaces due to free counterions that decays as h-2 (see eq 14). If γ is close to unity and the surface charge density is large enough, then a bridging attraction can occur when the surface separation is of the order of d, the parameter controlling the flexibility of the polyelectrolyte. The range of the attraction increases with d, while the strength of the attraction decreases. In the presence of salt, the pressure at large separations has the same exponentially decaying form as for a normal double-layer interaction (eq 16), but with a modified surface charge density that depends on the properties of the grafting layer and the electrolyte. If γ or the salt concentration or the flexibility d are increased enough, then the effective surface charge may change sign, leading to a disappearance of the long-range repulsion, and a subsequent reappearance as the increase is continued. Addition of salt to an initially salt-free system leads to a reduced repulsion except near to γ ) 1, where an increased repulsion can be found. Bridging is dominant mainly near where the effective charge changes sign (which for low salt is γ ∼ 1), and the system rapidly becomes repulsive when overcompensated (γ > 1). LA9913654

Surface Interactions in the Presence of Polyelectrolytes. A Simple

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