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Interactions between Charged Surfaces Immersed in a Polyelectrolyte Solution Martin Turesson,* Torbjo¨rn A° kesson, and Jan Forsman Theoretical Chemistry, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden ReceiVed June 15, 2007. In Final Form: July 24, 2007 With grand canonical simulations invoking a configurationally weighted scheme, we have calculated interactions between charged surfaces immersed in a polyelectrolyte solution. In contrast to previous simulations of such systems, we have imposed full equilibrium conditions (i.e., we have included diffusive equilibrium with a bulk solution). This has a profound impact on the resulting interactions: even at modest surface charge densities, oppositely charged chains will, at sufficiently large separations, adsorb strongly enough to overcompensate for the nominal surface charge. This phenomenon, known as charge inversion, generates a double-layer repulsion and a free-energy barrier. Simpler canonical approaches, where the chains are assumed to neutralize the surfaces perfectly, will not capture this stabilizing barrier. The barrier height increases with the length of the polyions. Interestingly enough, the separation at which the repulsion becomes attractive is independent of chain length. The short chains here are unable to reach across from one surface to the other. We therefore conclude that the transition to an attractive regime is not provided by the formation of such “intersurface’’ bridges. With long chains and at large separations, charge inversion displays decaying oscillatory behavior (i.e., the apparent surface charge switches sign once again). This is due to polyion packing effects. We have also investigated responses to salt addition and changes in polyelectrolyte concentration. Our results are in qualitative and semiquantitative agreement with experimental findings, although it should be noted that our chains are comparatively short, and the experimental surface charge density is poorly established.
Polyelectrolytes find many applications in processes of biological and industrial interest. Polyelectrolytes can be used to stabilize colloid dispersions but can also act as coagulants. Polyelectrolyte-induced surface interactions are often sensitive to changes in experimental conditions.1 Furthermore, studies of polyelectrolyte-mediated interactions commonly suffer from nonequilibrium effects2-5 because the approach to equilibrium can be a very slow process. A manifestation of these complications is the wide range of results obtained in surface force experiments, including a number of apparent contradictions. Still, most studies report a weak repulsion at long range, with a narrow and strongly attractive regime at short separations.2,3,5,6 At low polymer concentrations or for weakly charged chains, the attraction generally becomes more long-ranged.1,4,7,8 This subject has also inspired considerable theoretical effort. van Opheusden9 adopted the Edwards-deGennes theory10,11 on a screened Coulomb model and demonstrated the existence of attractive bridging interactions between charged surfaces in the presence of oppositely charged polymers. Podgornik12,13 generalized the theory whereby the screened Coulomb approximation could be avoided. This allowed him to investigate interactions between charged surfaces in the presence of an infinitely long neutralizing polymer. Similar systems, with short chains, have (1) Claesson, P. M.; Poptoshev, E.; Blomberg, E.; Dedinaite, A. AdV. Colloid Interface Sci. 2005, 114, 173. (2) Dahlgren, M. A. G.; Waltermo, A° .; Blomberg, E.; Claesson, P. M.; Sjo¨stro¨m, L.; A° kesson, T.; Jo¨nsson, B. J. Phys. Chem. 1993, 97, 11769. (3) Dahlgren, M. A. G.; Hollenberg, H. C. M.; Claesson, P. M. Langmuir 1995, 11, 4480. (4) Biggs, S.; Proud, A. D. Langmuir 1997, 13, 7202. (5) Poptoshev, E.; Rutland, M. W.; Claesson, P. M. Langmuir 1999, 15, 7789. (6) Dahlgren, M. A. G.; Claesson, P. M.; Audebert, R. J. Colloid Interface Sci. 1994, 166, 343. (7) Sennerfors, T.; Fro¨berg, J. C.; Tiberg, F. J. Colloid Interface Sci. 2000, 228, 127. (8) Solberg, D.; Wågberg, L. Colloids Surf., A 2003, 219, 161. (9) van Opheusden, J. H. J. J. Phys. A.: Math. Gen. 1988, 21, 2739. (10) Edwards, S. F. Proc. R. Soc. London 1965, 85, 613. (11) deGennes, P. G. Rep. Prog. Phys. 1969, 32, 187. (12) Podgornik, R. J. Phys. Chem. 1991, 95, 5249. (13) Podgornik, R. J. Chem. Phys. 1993, 99, 7221.
also been approached by simulation methods and by an extension of the Poisson-Boltzmann theory for connected charges.14,15 Again, a strong but short-ranged bridging attraction has been found. When the chains are very stiff, however, the attraction there is dominated by polyion-polyion correlations, with a diminished bridging contribution.16 Effects of salt addition have also been investigated,15,17-19 with the major conclusion being that adding salt to a system with charged surfaces and neutralizing polyions will reduce the attraction because of osmotic effects. In cases where the charge of the polyion chain is comparable to those of the colloidal particles, the interactions are significantly different, with a long-ranged bridging attraction and a strong dependence on polyion chain length.13,20-23 These theoretical studies have increased our understanding of how interactions between charged particles are modulated by the presence of polyelectrolytes. Still, they all suffer from approximations on a rather fundamental level. Specifically, no study thus far has treated equilibrium interactions between large macroions in the presence of polyelectrolytes and salt without invoking mean-field approximations. The reason is that simulations including the bulk exchange of polymers are computationally demanding. In this work, we shall present results from grand canonical simulations of such interactions in slit geometry where the chemical potentials of all species (i.e., the polyelectrolyte and any additional salt) are fixed. The problem of polyion bulk (14) A° kesson, T.; Woodward, C. E.; Jo¨nsson, B. J. Chem. Phys. 1989, 91, 2461. (15) Woodward, C. E.; A° kesson, T.; Jo¨nsson, B. J. Chem. Phys. 1994, 101, 2569. (16) Turesson, M.; A° kesson, T.; Forsman, J. Langmuir 2006, 22, 5734. (17) Bo¨hmer, M. R.; Evers, O. A.; Scheutjens, J. M. H. M. Macromolecules 1990, 23, 2288. (18) Borukhov, I.; Andelmann, D.; Orland, H. J. Phys. Chem. B 1999, 103, 5042. (19) Huang, H.; Ruckenstein, E. AdV. Colloid Interface Sci. 2004, 112, 37. (20) Granfeldt, M.; Jo¨nsson, B.; Woodward, C. E. J. Phys. Chem. 1991, 95, 4819. (21) Podgornik, R.; Åkesson, T.; Jo¨nsson, B. J. Chem. Phys. 1995, 102, 9423. (22) Podgornik, R. J. Chem. Phys. 2003, 118, 11286. (23) Dzubiella, J.; Moreira, A. G.; Pincus, P. A. Macromolecules 2003, 36, 1741.
10.1021/la7017852 CCC: $37.00 © 2007 American Chemical Society Published on Web 08/15/2007
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exchange is alleviated by the inclusion of configurationally biased techniques.24-27 To maintain electroneutrality, an insertion/deletion of a polyion, containing r charged monomers, is accompanied by corresponding insertions/deletions of r monovalent ions carrying an opposite charge. Any additional monovalent salt is handled via an analogous but separate grand canonical process, albeit without biasing techniques. As we shall see, the proper inclusion of diffusive equilibrium has a fundamental impact on the surface forces. At large separations, polyions will adsorb strongly enough to overcompensate for the charge on the surfaces. This leads to long-ranged double-layer repulsion. This repulsion is absent in previously investigated simulation models in which there is no bulk exchange and where the chains are assumed to neutralize the surfaces. Effects of charge overcompensation may be even more pronounced in many experimental situations, where a rapid approach of the surfaces prevents diffusive equilibrium from being established.28 We shall use the restricted primitive model in which a uniform dielectric response of r ) 78.3 is assumed. The surfaces carry a uniform negative charge density, σ. They are flat and extend infinitely in the x and y directions. We thus define z as the axis transverse to the walls. One surface is located at z ) 0, with the other at z ) h. The system contains a polyelectrolyte and, in general, an additional monovalent salt. A polyion is composed of r monomers that are connected by rigid but orientationally flexible bonds of length b ) 7 Å. Each monomer carries a positive charge, with a sign opposite to that of the surfaces. All charges, including free ions, are centered in hard spheres of diameter d ) 4 Å. These charges also sense a soft repulsive potential, Vex, from the walls: Vex ) w(z) + w(h - z) with βw(z) ) 405 exp(-z/τ)(τ/z)3, where τ ) 1 Å. Finally, β ) 1/(kT), with k denoting Boltzmann’s constant and T denoting the temperature (T ) 298 K). The polyelectrolyte and any additional salt are allowed to equilibrate with a bulk solution. This is achieved via standard grand canonical steps for the simple salt and configurationally biased additions/deletions of the polyelectrolyte. These are performed in a stepwise manner, where electroneutrality is maintained in each step. Specifically, a monomer and an unconnected counterion are simultaneously added or deleted using the configurational-biased scheme proposed by Rosenbluth and Rosenbluth.24 A successful grand canonical step corresponds to the deletion or addition of an entire chain, together with its counterions. The required CPU time per simulated data point ranged from 2 to 7 days (about 500 million configurations). The grand canonical steps were performed with a frequency of 5%. The lateral dimension of the simulation box was usually about 200 Å, and the system typically contained 40-60 chains. Chemical potentials, corresponding to desired bulk densities, are obtained from grand canonical simulations in the bulk. The concentration of monomers and monovalent salt in the bulk are denoted Fm and Fs, respectively. The osmotic pressure, P, acting against the confining walls was calculated from the repulsive and electrostatic particlesurface interactions. The corresponding interaction free energies per unit area, gs were established by a numerical integration of the net osmotic pressure, Pn ) P - Pb, (24) Rosenbluth, M. N.; Rosenbluth., A. W. J. Chem. Phys. 1955, 23, 356. (25) Harris, J.; Rice, S. J. Chem. Phys. 1988, 88, 1298. (26) Siepmann, J. Mol. Phys. 1990, 70, 1145. (27) Turesson, M.; Forsman, J.; A° kesson, T. Phys. ReV. E, in press, 2007. (28) Horn, R. G.; Hirz, S. J.; Hadziioannou, G.; Frank, C. W.; Catala, J. M. J. Chem. Phys. 1989, 90, 6767.
Letters
gs(h) )
∫h∞ (P(h′) - Pb) dh′
(1)
where Pb is the osmotic pressure in the bulk. Bulk conditions are approached more rapidly with neutral surfaces, and in the absence of added salt, Pb was estimated from the osmotic pressure between neutral walls at h ) 300 Å. In the presence of salt, however, bulk conditions were approached rapidly enough to allow us to estimate the bulk pressure with charged surfaces (at large separations). As the surfaces are drawn apart, gs approaches twice the tension at a single surface, γs. The net surface interaction per unit area is expressed as ∆gs(h) ≡ gs(h) - 2γs. We may define an apparent surface charge density, σapp as
σapp(z) ) σ +
∫0z ∑ FR(z′)qR dz′ R
ze
h 2
(2)
where FR and qR are the density and charge of ionic species R, respectively. The sum runs over all monomers and free ions in the system. The symmetry of the system allows us to limit the integration to the left half-cell, with σapp(h/2) ) 0. Polyions may adsorb at an oppositely charged surface strongly enough to overcompensate for its net charge. This leads to an “apparent’’ surface charge density of opposite sign. We shall by σapp(max) denote the maximum value of σapp > 0. In the absence of charge inversion, σapp(max) ) 0 (at the midplane). In canonical simulations, where only neutralizing polyions are present (the zero bulk concentration limit), the effects of chain length are very small.16 In our more elaborate model incorporating proper bulk equilibrium, we find a substantial dependence on the degree of polymerization (at least for our short chains). This is highlighted in Figure 1, where we also display how the surface interactions respond to salt addition. There are several remarkable features to be noted. First, in contrast to the simple canonical model with only neutralizing polyions (Figure 2), we observe a strong, long-ranged repulsive barrier. This barrier increases with the polyion chain length. However, the position of its maximum (i.e., where the net osmotic pressure turns attractive) does not depend on the degree of polymerization. We also see how repulsive interactions always are accompanied by charge inversion, which vanishes at the separation where the net osmotic pressure turns attractive (i.e., at the top of the freeenergy barrier). The higher free-energy barrier for longer chains is a consequence of their stronger adsorption, which increases the strength of the “charge-inverted’’ surfaces. By comparing graphs a and b in Figure 1, we see that these findings are qualitatively conserved when a simple monovalent salt is added. As expected, the interactions become more short-ranged, with a concomitant decrease in the free-energy barriers. Note that surface interaction with 14-mers displays a weakly attractive regime at very long range. This is connected to a second charge inversion, as will be discussed below. In a recent SFA study, Maurdev et al.29 measured the interactions between mica surfaces in the presence of relatively short (r ) 170) and positively charged chains. At low concentrations of monovalent salt (∼0.1 mM), repulsive forces were detected at separations greater than 50 Å. By fitting DLVO theory to experimental data at large separations, they concluded that the surfaces effectively had a reversed sign. The barrier had its maximum height at 50 Å and was measured to be about 4 mN m-1 in units of force/radius. By multiplying ∆gs in Figure 1a by a factor 2π, we relate our free energies per unit area to the force (29) Maurdev, G.; Meagher, L.; Ennis, J.; Gee, L. M. Macromolecules 2001, 34, 4151-4158.
Letters
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Figure 2. Comparing interactions between charged surfaces in the presence of oppositely charged polyions as obtained with different models. In the canonical model, the surfaces are neutralized by the polyions, and there are no free simple ions (i.e., no bulk). The term “implicit simple ions” signifies that the monomers interact via a screened Coulomb (SC) potential in which the Debye screening length is obtained from the bulk concentration of simple ions. The inset demonstrates the decaying oscillatory behavior of the apparent surface charge density.
Figure 1. Variation of the interaction free energy (opaque symbols, left ordinate axis) and maximum apparent surface charge density (filled symbols, right ordinate axis) with surface separation. Note that in the absence of surface charge inversion, σapp(max) is zero. A typical van der Waals interaction, vdW, with a Hamaker constant of 10-20 J is displayed as a reference. The bulk monomer concentration is 10 mM, and the nominal surface charge density is 1 negative charge per 200 Å2. Symbols are simulated data points, and solid lines are splines to guide the eye. (a) 0 and (b) 100 mM salt.
per radius between two curved surfaces (the Derjaguin approximation). We find a 2.5 mN m-1 strong barrier, also positioned at 50 Å. This is in good agreement with the experiments, considering our comparatively short chain. From Figure 1a, we expect the barrier to increase with chain length. This is a consequence of the cooperative nature of polyelectrolyte adsorption, which generates a stronger charge inversion when the chains are longer. However, given that the experimental surface charge density is unknown, the agreement may be fortuitous. An adhesive force of about 4-42 mN m-1 (depending on pH) was observed by Maurdev et al. at short separations. We obtain values of about 10 mN m-1, increasing slowly with chain length (not shown). Interestingly, we observe the onset of attractive forces at the same separation for all three chain lengths. At a separation of h ) 50 Å, the average end-to-end distance for a hexamer is roughly 20 Å whereas the fully stretched chain is 35 Å. Hence, these chains are unable to form bridges from one surface to the other, and we must seek an alternative mechanism for the onset of attraction. First, we note that nonadsorbed short chains still may form bridges, albeit not between the minima of the adsorption potentials. Furthermore, by analyzing pressure contributions at the midplane, we have verified that polyionpolyion correlations provide an important contribution to the observed attraction.
A simplified model of our polyelectrolyte solution is obtained if we effectively integrate out simple ions by imposing screened Coulomb interactions between the monomers. The Debye screening length, κ-1, is then obtained from the concentration of all simple ions in the bulk polyelectrolyte solution, including any added monovalent ions (salt). This approach is remarkably accurate in the absence of additional simple salt, as illustrated in Figure 2. Only results for 14-mers are displayed, but the agreement is similar for the other chain lengths. When a simple monovalent salt is added, the screened Coulomb approach is less accurate (not shown). Specifically, the screening effects are then overestimated, and the predicted free-energy barriers are significantly lower than those obtained from explicit ion simulations. Results from a corresponding canonical approach, where the polyions are assumed to neutralize the surface charge (no free simple ions), are also included in Figure 2. The importance of including proper adsorption equilibrium is highlighted, and the oversimplified canonical approach fails to detect any free-energy barrier. The inset of Figure 2 shows how the apparent surface charge density varies with the distance from the left surface at a large separation. Interestingly, we observe a double charge inversion (i.e., the apparent surface charge reverses back to its original negative sign) before it approaches zero at the midplane. Hence, it seems to display decaying oscillatory behavior. As expected, this has a corresponding impact on the surface force, which displays a attraction at very long range. In fact, oscillatory interactions have indeed been measured in (almost) salt-free polyelectrolyte solutions30,31 between free-standing films. This technique allows measurements of weak repulsive interactions. These effects are weaker for shorter chains on account of the reduced adsorption. With hexamers, the double charge inversion is barely detectable at very large separations and is too weak to produce a significant attractive surface interaction. The screened Coulomb model does capture this oscillatory behavior in terms of the monomer density profile. There are also indications of an (30) v. Klitzing, R.; Espert, A.; Asnacios, A.; Hellweg, T.; Colin, A.; Langevin, D. Colloids Surf., A 1999, 149, 131-140. (31) Theodoly, O.; Tan, J. S.; Ober, R.; Williams, C. E.; Bergeron, V. Langmuir 2001, 17, 4910-4918.
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Figure 3. Responses of surface interactions to changes in the nominal surface charge density. Note that the legends provide the inVerse surface charge density.
Figure 4. Surface interactions at various polyelectrolyte concentrations.
attractive regime at long range, but the noise prevents firm conclusions. We expect the interactions to become stronger at higher nominal surface charge densities. This is corroborated by the simulation results presented in Figure 3. The repulsive barrier, as well as the attractive minimum at short separations, becomes weaker and more long-ranged at lower surface charge densities. Finally, we have investigated how the surface interactions respond to the addition of a polyelectrolyte salt at a given concentration of monovalent salt. In accordance with experimental results,5 the charge inversion increases with increasing polyion concentration. We see, in Figure 4, that it is primarily the range that is affected, being reduced as the polyelectrolyte concentration is increased. The barrier height is primarily dictated by chain length and the concentration of simple salt. The behavior and mechanistic picture that emerge are thus partially different from findings in previous studies on canonical
Letters
models in which the number of polyions in the slit is restricted so as to counterbalance the surface charge. In these latter cases, one finds no free-energy barrier. Instead, there is a short-ranged attraction that is dominated by bridging. In our case, where we allow proper diffusive equilibrium with a bulk solution, there is a double-layer repulsion at long range that is caused by strong chain adsorption and concomitant charge inversion. At short range, this charge inversion disappears, and the interaction becomes attractive. Interestingly enough, for the investigated chain lengths, the separation at which this transition occurs is independent of the degree of polymerization. This is true even for chains that are unable to adsorb simultaneously at both surfaces, thereby connecting them with a bridge. Bridges across the midplane are still formed but only between nonadsorbed chains (i.e., the bridges do not reach from one surface to the other). The surface charge inversion and concomitant repulsive barrier that we find in all cases appear to contradict the fact that polyelectrolytes sometimes can be used to flocculate dispersions. However, experimental studies have shown that aggregation of a dispersion occurs only in a rather narrow concentration window as a polyelectrolyte is added.32-36 The optimum dosage, where the flocculation rate is highest, usually corresponds to perfect charge neutralization (i.e., zero electrophoretic mobility). This generally corresponds to a very low polyion concentration, in agreement with our findings. At higher concentration, charge inversion occurs, and the aggregation rate decreases. Note that we, in our model system, assume the presence of an infinite bulk (i.e., the bulk solution always contains a sufficient number of polyions to neutralize the surfaces). This should mimic a typical SFA or AFM experiment, but the situation is often different at low dosage in flocculation experiments. Our predicted second charge inversion is probably less relevant to flocculation experiments because they occur only at large separations, where the Derjaguin approximation is less valid. The packing effects that we observe between flat and infinite surfaces may not be directly transferable to colloid-sized spherical particles when the separation is large. Finally, it should also be noted that, especially in practical applications, nonequilibrium effects are substantial, and the flocculation rate may depend on the order of mixing, shearing effects, and so forth.37 Acknowledgment. We thank Christophe Labbez for many fruitful discussions. LA7017852 (32) Dixon, J. K.; LaMer, V. K.; Li, C.; Messinger, S.; Linford, H. B. J. Colloid Interface Sci. 1967, 23, 465. (33) Vincent, B. AdV. Colloid Interface Sci. 1974, 4, 193. (34) Zhang, J.; Buffle, J. J. Colloid Interface Sci. 1995, 174, 500. (35) Walker, H. W.; Grant, S. B. Colloids Surf., A 1996, 119, 229. (36) Feretti, R.; Zhang, J.; Buffle, J. Colloids Surf., A 1997, 121, 203. (37) Gregory, J. Colloids Surf. 1988, 31, 231.
