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Simple Correlation-Corrected Theory of Systems Described by Screened Coulomb Interactions Jan Forsman Theoretical Chemistry, Chemical Centre, P.O. Box 124, S-221 00 Lund, Sweden ReceiVed October 31, 2006. In Final Form: December 14, 2006 We present a simple correlation-corrected density functional treatment of dispersions containing macroions, where these are assumed to interact via screened Coulomb potentials, as given by Debye-Hu¨ckel theory. A straightforward mean-field description even fails to qualitatively capture important correlation effects displayed by such systems. However, if an effective, correlation-corrected potential is adopted at short range, then the predictions are in qualitative and semiquantitative agreement with simulated results. The correlation corrections are evaluated in a manner that is completely analogous to those recently presented in correlation-corrected Poisson-Boltzmann theory (Forsman, J. J. Phys. Chem. B 2004, 108, 9236). The accuracy of the theory is evaluated by comparison with simulation data on systems displaying correlation-generated packing effects and stratification forces.
1. Introduction Liquid dispersions often contain macroions such as charged colloid particles, micelles, or proteins. Mutual charge interactions between such aggregates are commonly approximated by screened Coulomb potentials using Debye-Hu¨ckel arguments. The most famous example of this approach is the DLVO theory,1,2 which has proven to be extremely useful and often remarkably accurate. Nevertheless, a straightforward mean-field description of such a model system will not capture macroion-macroion correlations, which are crucial to many relevant phenomena such as oscillatory stratification forces. In a recent study,3 it was shown that the Poisson-Boltzmann theory of simple ionic solutions can be considerably improved by the use of an effective correlation-corrected potential between ions of like charge at short range. In other words, the effective potential was chosen so as to approximate correlation effects. This amounts to a correction of the overestimated repulsion between ions of like charge that results from a straightforward mean-field (Poisson-Boltzmann) treatment (i.e., the effective potential is weaker than the pure Coulombic charge-charge interaction at close range). Using these ideas, a correlationcorrected Poisson-Boltzmann theory (cPB) was constructed. It is simple and versatile and can rather accurately account for effects from ion correlations,4,5 as has been demonstrated for a wide range of systems.3,6 Although there are other theories that are even more accurate,5,7-25 the cPB approach has the advantage (1) Derjaguin, B. V.; Landau, L. Acta Phys. Chim. URSS 1941, 14, 633-662. (2) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (3) Forsman, J. J. Phys. Chem. B 2004, 108, 9236. (4) Guldbrand, L.; Jo¨nsson, B.; Wennerstro¨m, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221. (5) Kjellander, R. J. Chem Soc., Faraday Trans. 2 1984, 80, 1323. (6) Forsman, J. Langmuir 2006, 22, 2975. (7) Patey, G. N. J. Chem. Phys. 1980, 72, 5763. (8) Kjellander, R.; Marcˇelja, S. J. Chem. Phys. 1985, 82, 2122. (9) Torrie, G. M.; Kusalik, P. G.; Patey, G. N. J. Chem. Phys. 1989, 91, 6367. (10) Kjellander, R.; Sarman, S. Mol. Phys. 1990, 70, 215. (11) Outhwaite, C. W.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 2 1983, 79, 707. (12) Outhwaite, C. W.; Bhuiyan, L. B. Mol. Phys. 1991, 74, 367. (13) Lamperski, S.; Outhwaite, C. W. Langmuir 2002, 18, 3423. (14) Kjellander, R.; Mitchell, D. J. Chem. Phys. Lett. 1992, 200, 76-82. (15) Kjellander, R.; Mitchell, D. J. J. Chem. Phys. 1994, 101, 603-626. (16) Kjellander, R.; Mitchell, D. J. Mol. Phys. 1997, 91, 173-188. (17) Nordholm, S. Chem. Phys. Lett. 1984, 105, 302. (18) Nordholm, S. Chem. Phys. Lett. 1991, 105, 302.
of being extremely simple, with calculations running very fast because the mean-field character is retained. In some cases, notably for bulk salt solutions, it is possible to derive analytical expressions.3 Furthermore, the cPB focuses on improving the most relevant flaw of the PB method, which even has some pedagogical merits. Ion-ion correlation attractions are known to exist at a zero salt level in the presence of charged surfaces, where the dissolved counterions are multivalent (the Swedish double layer). In this case, it is natural to model the mobile ions as pointlike, without hard cores. Most density functional approaches,22-27 however, rely on the existence of hard cores, without which they reduce to the Poisson-Boltzmann theory. Thus, they completely fail to account for correlation attractions in systems with pointlike multivalent counterions. “Hidden” in these theories is the fact that these hard cores effectively act as Coulomb holes, which largely account for the ion-ion correlations. The presence of these, possibly together with some further tail correction, makes the density functional approaches quite successful in many circumstances. Still, given the way in which the theories are formulated, the capture of ion-ion correlation effects appears to be almost accidental. The primary mechanism behind these correlations is unrelated to hard core repulsions. One may of course model a Coulomb hole by an effective hard core, but the radius of this should then be determined by electrostatics, not exchange repulsion. This route has been explored by Nordholm and co-workers.18-22 Our main objective in this work is to demonstrate that a treatment, completely analogous to the cPB, of particles interacting via screened Coulomb interactions also leads to a versatile and accurate theory. Not unimportantly, this is achieved with minimum effort in terms of theoretical complexity and computational effort. We shall focus on the influence that the (19) Nordholm, S.; Penfold, R.; Jo¨nsson, B.; Woodward, C. E. J. Chem. Phys. 1991, 95, 2048. (20) Nordholm, S.; Penfold, R. J. Chem. Phys. 1992, 96, 3102. (21) Abbas, Z.; Gunnarsson, M.; Ahlberg, E.; Nordholm, S. J. Colloid Interface Sci. 2001, 243, 11. (22) Nordholm, S. Aust. J. Chem. 1984, 37, 1. (23) Boyle, E. J.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1987, 86, 2309. (24) Mier-y-Teran, L.; Suh, S.; White, H. S.; Davis, H. T. J. Chem. Phys. 1990, 92, 5087. (25) Tang, Z.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1992, 97, 494. (26) Patra, C. N.; Ghosh, S. K. J. Chem. Phys. 1994, 101, 5219. (27) Patra, C. N.; Ghosh, S. K. J. Chem. Phys. 2002, 117, 8938.
10.1021/la063179l CCC: $37.00 © 2007 American Chemical Society Published on Web 04/07/2007
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presence of charged colloid-sized particles (micelles, proteins, and the like) has on the interaction between large (macro- or mesoscopic) surfaces in solution. At low salt concentrations, we expect to observe oscillatory surface forces as a result of correlations (packing) between the macroions. This phenomenon, often denoted stratification, has attracted a lot of interest in recent years28-42 and is also known to occur in polyelectrolyte solutions. Hence, our goal is not only to validate a theory; we believe the results are interesting in their own right. We do not focus on the relevant question concerning the validity of the model system itself, but we expect that macroion-macroion interactions often can be reasonably well modeled by screened Coulomb potentials as long as the macroion surface density is low and the screening counterions are monovalent. This is supported by a structural comparison with a more elaborate primitive model system in which the counterions are explicitly included. This comparison is presented in the Appendix. At high concentrations, bringing the macroions close together, the model is likely to be less valid, at least if the macroion charges are anisotropically distributed such as in most proteins. We shall therefore focus on cases where the bulk macroion concentration is low. Our proposed theory is, together with relevant model system definitions, presented in the next section. This is followed by comparisons between theoretical predictions and corresponding simulation results. The last section contains a few summarizing comments.
2. Theory 2.1. Model System. Our model system consist of dispersed macroions that interact via a screened Coulomb potential, φSC,
βφSC(r) ) Q2
lB exp(-κr) r
(1)
where r ) |r - r′| is the distance between the macroions at positions r and r′. Q is the macroion valency, and lB is the Bjerrum length
lB )
βe2 4π0r
(2)
with 0 denoting the dielectric permittivity of vacuum. Assuming an aqueous solvent, the relative permittivity, r, is set equal to 78.3. Furthermore, e is the elementary charge, and β ) 1/kBT denotes the inverse thermal energy. All calculations and simulations were performed at room temperature, 298 K. (28) Richetti, P.; Kekicheff, P. Phys. ReV. Lett. 1992, 68, 1951-1954. (29) Sober, D. L.; Walz, J. Y. Langmuir 1995, 11, 2352-2356. (30) Bergeron, V.; Langevin, D.; Asnacios, A. Langmuir 1996, 12, 15501556. (31) Milling, A. J. J. Phys. Chem. 1996, 100, 8986-89993. (32) Asnacios, A.; Espert, A.; Colin, A.; Langevin, D. Phys. ReV. Lett. 1997, 78, 4974-4977. (33) Carignano, M. A.; Dan, N. Langmuir 1998, 14, 3475-3478. (34) von Klitzing, R.; Espert, A.; Asnacios, A.; Hellweg, T.; Colin, A.; Langevin, D. Colloids Surf. 1999, 149, 131-140. (35) Kolaric, B.; Jaeger, W.; v. Klitzing, R. J. Phys. Chem. B 2000, 104, 5096-5101. (36) von Klitzing, R.; Kolaric, B. Prog. Colloid Polym. Sci. 2003, 122, 122. (37) Theodoly, O.; Tan, J. S.; Ober, R.; Williams, C. E.; Bergeron, V. Langmuir 2001, 17, 4910-4918. (38) von Klitzing, R.; Kolaric, B.; Jaeger, W.; Brandt, A. Phys. Chem. Chem. Phys. 2002, 4, 1907. (39) Jo¨nsson, B.; Broukhno, A.; Forsman, J.; Åkesson, T. Langmuir 2003, 19, 9914. (40) Trokhymchuk, A.; Henderson, D.; Nikolov, A.; Wasan, D. T. J. Phys. Chem. B 2003, 107, 3927. (41) Piech, M.; Walz, J. Y. J. Phys. Chem. B 2004, 108, 9177. (42) Tulpar, A.; van Tassel, P.; Walz, J. Y. Langmuir 2006, 22, 2876.
The inverse Debye-Hu¨ckel screening length, κ, is given by
κ2 ) 4πlB
∑i ciqi2
(3)
where the summation runs over all salt particles and counterions of concentration ci and valency qi. To keep the model system as simple as possible, we do not include any hard cores. Furthermore, because the entire screened Coulomb approach may break down (even qualitatively) in the presence of multivalent counterions, we have limited ourselves to cases where all implicit screening ions are monovalent. Given the success with which the DLVO theory is able to describe interactions between macroions (with monovalent counterions), we expect our screened Coulomb model to capture the essential physics of the system, although there may be quantitative deviations from an explicit all-ion description. A structural comparison between simulations where all ions are explicitly included and the corresponding screened Coulomb version is provided in the Appendix. Our comparisons between theory and simulations will focus on interactions between large surfaces in the presence of macroions. We shall adopt the commonly used slit geometry. The two confining surfaces are located at z ) 0 and h. They are infinitely large, flat, and perpendicular to the z direction. The surfaces exert a soft repulsion on the confined macroions, βwrep(x) ) (Dw/x)4, where x is the macroion-surface distance and Dw ) 40 Å. This rather long-ranged, soft repulsion is present in all studied cases in this work. It serves as a very rough measure of the effective repulsion that results in the “real” system, at a simple inert wall, as a result of the loss of counterion interactions imposed by the presence of such a surface. This effect is not directly captured by a screened Coulomb model system in which the counterions enter only implicitly. In addition to the soft repulsion, the walls may carry a uniform surface charge density, σ, with a sign opposite to that of the macroions. In those cases, the macroions will sense an electrostatic attraction, βwatt(z) ) -Q exp[(-κx)/(κlGC)], where lGC ) e/(2πlB|σ|) is the GuyChapman length. The total surface-macroion interaction, Vex(z, h), can thus be written Vex(z, h) ) wrep(z) + wrep(h - z) + wattr(z) + wattr(h - z), where the latter two terms naturally vanish when the surfaces are neutral. Charged surfaces will furthermore repel each other, making a contribution of Pww ) 2πlBσ2 exp(-κh) to the total interaction pressure acting between the surfaces. 2.2. Correlation-Corrected Density Functional Theory. Armed with the definitions given above, we can formulate a mean-field free-energy functional, F, for the system
∫ F(r) ln[F(r)] - 1) dr + β ∫ ∫ F(r) F(r′) φSC(|r - r′|) dr dr′ + 2 β∫ Vex(r) F(r) dr
βF [F(r)] )
(4)
where F is the density of the macroions and Vex is an external field. This defines the ordinary mean-field functional for pointlike charged particles interacting via a repulsive screened Coulomb potential. We note that the formulation above will significantly overestimate the interparticle repulsion. The reason is the meanfield assumption of a uniform radial distribution function, g(r), which is a poor approximation at close range. A common way to correct for this is to invoke an estimate of g(r) explicitly, with the Ornstein-Zernike relation as a reference. A drawback is that the resulting formulation becomes considerably more complex,
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with a concomitant increase in the required computational effort. Borrowing ideas from cPB theory, we propose a simpler, though more approximate, approach wherein the mean-field character of the theory is retained. We include the correlations in an implicit way by reducing the mean-field interparticle repulsion at separations below some threshold value Rc, where correlations are significant. This amounts to the use of an effective potential, φeff(r), which differs from φSC(r) at close range
φeff(r) )
[
φSC(r), r > Rc φcorr(r), r e Rc
(5)
where φcorr(r) < φSC(r). The value of Rc should reflect the distance below which g(r) drops significantly (i.e., a typical separation between neighboring macroions in the system). It will be specified below. The effective potential in the cPB formulation is chosen in a physically reasonable but otherwise rather ad hoc manner. Still, it has subsequently been confirmed that the choice results in an accurate theory. This confirmation relies on numerous comparisons with simulated data in systems with weak as well as strong electrostatic coupling and with salt concentrations ranging from zero up to molar levels.3,6 Encouraged by this success, we shall here adopt the same equation for the correlation part, φcorr(r), of the effective potential:
( )
dφSC φcorr(r) ) dr′
(r - Rc) + φSC(Rc)
r′)Rc
(6)
This choice is particularly appealing in the sense that the effective potential and its derivative are continuous across the transition separation Rc. Nevertheless, as with the cPB theory, the choice relies heavily on explicit demonstrations of accuracy, under various circumstances. These will be provided in the next section. In our slit geometry, the mean-field character of our theory allows us to integrate φeff(r) in the direction (x, y) parallel to the surfaces to obtain
[
βφeff(|z - z′|) )
[
-2πlBQ2 exp (-κRc)
R2c - |z - z′|2 (1 + κRc) + 2Rc κ (|z - z′|3 - R3c ) + κ-1 3Rc
2πlBQ2 exp(-κ|z - z′) κ
|z - z′| e Rc
]
Rc )
xπe |Qσ |
where σ, as usual, is the surface charge density. In this manner, Rc is defined from the area per ion when the counterions are condensed at the surface to the extent that they exactly neutralize the surface charge. Defining this area to be circular, Rc is the radius of that circle. The transition distance is thus determined by the proximity between the macroions in the vicinity of the charged surfaces, where the density is highest and correlation effects are most important. In the absence of charged surfaces, we adopt the simple arguments by van der Waals. Denoting the excluded volume per macroions by V0, we write
V0 )
1 4π 3 R 2 3 c
(7)
where the z direction is normal to the surfaces. The transition distance is also defined in the same manner as in cPB theory. Specifically, in systems containing macroscopic surfaces that carry a substantial concentration of charges of sign opposite to that of the macroions, there will be a clustering of macroions at the fluid-surface interfaces. Hence, the average interparticle separation will, in the vicinity of these surfaces, be relatively small. This implies a short transition distance. Far away from such surfaces, the nearest-neighbor separation is larger, suggesting that the transition distance in principle should also increase. Such an anisotropic treatment of the interparticle interactions is, however, not without problems. Furthermore, cPB results thus far indicate that an isotropic treatment, where the typical nearest-neighbor separation in the entire system governs the (fixed) value of Rc, is sufficient in most cases. Hence, by analogy to the cPB formula, the transition distance is, in the presence of oppositely charged surfaces, given by
(9)
where van der Waals would describe Rc as the distance of closest approach. If we now assume a cubically close-packed arrangement (i.e., V0 ) 1/Fb), then we get
Rc )
( ) 3 2πFb
1/3
(10)
where Fb is the bulk density of macroions. This relation differs somewhat from the one adopted in the cPB work. The difference is small, but we believe the present formulation is better motivated, being based on well-established arguments from van der Waals theory. In situations where there are very weakly charged surfaces present, the transition distance is of course still given by the nearest-neighbor separation:
Rc ) min
[x ( ) ] eQ 3 | |, π σ 2πFb
1/3
(11)
The grand potential, Ω, is obtained from Ω[F(r)] )F [F(r)] - µ ∫ F(r) dr, where µ is the bulk macroion chemical potential. The functional is simplified by integration over the (x, y) plane parallel to the surfaces. The density profile, F(z), that minimizes Ω is given by
F(z) ) exp(β(µ - Vex(z) |z - z′| > Rc
(8)
∫0h F(z′) φ
eff
(|z-z′|) dz′) (12)
which can be solved by simple Picard iterations. The ideal part of the chemical potential can be written as ln[FbVnorm], where Vnorm is an uninteresting normalization volume. Note also that βφeff has the unit of square length. The osmotic pressure, P(h), acting perpendicularly to the surfaces, is given by P(h) ) -∂Ωeq/ ∂h, where Ωeq is the equilibrium grand potential. At large separations, the osmotic pressure will approach the bulk value, Pb, and net surface interactions are conveniently measured by Pn(h) ≡ P(h) - Pb. Predictions by the correlation-corrected theory will be denoted by cSC, and corresponding ones without any such corrections (i.e., where φSC is used instead of φeff) are dubbed SC. 2.3. Simulations. The simulations were performed in a standard manner, with the usual minimum image convention and a grand canonical formulation to ensure a constant (bulk) chemical potential at different surface separations. A spherical cutoff was adopted, truncating all interactions except the soft wall repulsion at a distance given by half the lateral (x, y) extension of the simulation box. To treat all charge-charge interactions consistently, the macroion-wall and wall-wall interactions were also truncated, as described in ref 39. Bulk simulations were performed separately to determine the chemical potential of the
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Figure 1. Interactions between neutral surfaces immersed in a dispersion containing macroions, with Q ) 60, at a concentration of 0.33 mM. The screening is due to counterions only (i.e., no salt has been added). The inset is an enlargement demonstrating the force oscillations obtained by correlation-corrected screened Coulomb density functional (cSC) calculations and Metropolis Monte Carlo simulations (MC), respectively. The corresponding uncorrected meanfield density functional theory, SC, fails to reproduce such correlationinduced behaviors. No salt.
Figure 2. Comparing simulated macroion density profiles, with predictions by cSC and CS theories. Neutral surfaces, no salt.
chosen bulk state. Tests were conducted to ensure that system size effects were small.
3. Results 3.1. Neutral Surfaces. 3.1.1. Concentration Dependence at Constant Screening (No Salt). We shall first consider the interaction between neutral surfaces in the presence of charged macroions and the corresponding implicit counterions. We postpone, for the time being, effects of additional salt and fix the counterion density at 20 mM, which in our screened Coulomb approach is described by a Debye length of about 30 Å. In Figure 1, we provide SC and cSC predictions as well as simulated data of the surface interactions in such systems, where each macroion carries 60 unit charges. The cSC is obviously able to reproduce the simulated oscillatory stratification forces quite well. The uncorrected CS version, however, fails to describe any oscillatory behavior at all if we disregard the complete depletion at close range. This latter phenomenon is due to ionsurface rather than ion-ion correlations. A similar conclusion holds for typical density distribution profiles across the slit, as shown in Figure 2. These graphs also illustrate the close connection between oscillatory forces, packing effects, and ion-ion correlations. In fact, correlations between like-charged ions may be viewed as a packing effect, although the repulsive cores of course are much softer than in the more well-established cases of molecular packing governed by exchange repulsion.
Figure 3. Predicted and simulated responses of surface interactions to a change in macroion charge under the constraint of constant screening (counterion concentration). Neutral surfaces, no salt. Q values: (a) 20, (b) 40, and (c) 60.
Stratification forces in similar systems were recently simulated by Jo¨nsson et al.39 They observed much more pronounced density oscillations, with a concomitantly stronger amplitude in typical surface forces. The reason for this discrepancy is that they regarded the screening length as a freely adjustable parameter (i.e., it was not subject to constraints imposed by Debye-Hu¨ckel theory). In other words, the interparticle repulsion was generally more long-ranged than Debye-Hu¨ckel screening from counterions would allow. Figure 3 demonstrates how the surface forces respond to changes in the macroionic charge under the constraint of constant counterion concentration (screening length). Again, we observe satisfactory predictions by the cSC theory. Corresponding CS predictions are not included because they naturally fail to capture the observed oscillations, even qualitatively. We see that as the macroion charge is increased the amplitude of the oscillations is almost unaffected but the period is significantly increased. This reflects the increased distance between neighboring macroions. 3.1.2. Concentration Dependence at Constant Macroion Charge (No Salt). Experimentally, it has been found that the
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Figure 6. Interactions between equally charged surfaces (e/σ ) -400 Å2) in the presence of macroions, with Q ) 40. No salt is added, and the macroion concentration is 0.5 mM.
Figure 4. Increasing the concentration of macroions, with Q ) 40. Neutral surfaces, no salt. (a) cSC predictions and (b) simulation data.
Figure 7. Response of surface charges to variations of the surface charge density. The micellar valency is Q ) 20, and the bulk macroion concentration is Fb ) 1 mM.
Figure 5. Effects of adding salt with neutral surfaces.
period of the oscillations depend on the macroion concentration.41,42 This is expected if we regard the phenomenon to be governed by ion correlation packing. In our salt-free system, there is the additional complication that the effective range of the potential also varies with concentration. Still, in Figure 4 we see that the period indeed increases as the concentration of a given type of macroion is lowered. We see that the height of the primary force maximum varies non-monotonically with macroion concentration. In contrast, the depth of the primary minimum increases monotonically with concentration, at least in the investigated regime. Not surprisingly, these behaviors are captured by the cSC theory, and the quantitative agreement with simulation results is satisfactory, although there are some noticeable discrepancies at the highest concentrations. 3.1.3. Adding MonoValent Salt. Adding salt to these suspensions will effectively reduce the range of the electrostatic interactions, Hence, we expect electrostatic packing effects to be less significant in the presence of salt. This is confirmed in Figure 5. What may come as a surprise, however, is the extreme sensitivity that the surface forces display upon the addition of salt. At a level of 10 mM monovalent salt, the force oscillations nearly vanish, at
Figure 8. Effects of adding salt in the presence of equally charged surfaces (e/σ ) -400 Å2). Fb ) 0.5 mM, and Q ) 40.
least when plotted on the same scale as for those obtained in the absence of salt. The addition does result in a 3-fold increase in the total monovalent ion concentration in the system, but the response is nevertheless remarkably strong. 3.2. Charged Surfaces. We now proceed to situations where the walls carry a surface charge, with a sign opposite to that of the macroions. This has a dramatic effect on the surface forces, as illustrated in Figure 6. If we compare these curves with corresponding ones for uncharged surfaces, (i.e., Figure 3b), then we see that the secondary force maximum in the present case far exceeds even the primary one obtained when the surfaces are neutral. The ordinary mean-field SC approach fails to reproduce any attractive regime at all, but the cSC performs quite well. If we increase the surface charge density, then the repulsive and attractive regimes grow in strength, as shown in Figure 7.
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Figure 9. Comparing explicit and implicit (screened Coulomb) counterion models. The midplane macroion concentration is in both cases rougly 1.06 mM. The counterion density, in the explicit case, has been divided by Q ) 20 in order to be visible on the same scale. The separation is h ) 500 Å in the explicit case and h ) 520 Å in the screened Coulomb case. (See the text.) The inset highlights the central portion of the slit. In the explicit model, the counterions are excluded within a distance of RM ) 20 Å from the charge of a macroion.
Forsman
cases can be described in terms of simple packing considerations, taking into account that the Coulombic repulsion is soft. Hence, the way in which correlations between multivalent ions affect the interaction between two surfaces is dominated by some characteristic packing. If the surfaces are charged (opposite to the multivalent ions), then this characteristic packing occurs in the immediate vicinity of the surfaces. The packing is different in the bulk or far away from the surfaces. This is, however, of secondary importance because the surface interactions are strong and mainly governed by the adsorbed layers. If the surfaces are neutral, however, the packing in the bulk is the only relevant parameter. In the presence of macroions, this leads to weak and oscillatory surface forces, at least when the salt concentration is low. As mentioned previously, a possible further development of the cPB and cSC theories would be to allow spatial variation of the transition distance. This could improve the description of charged surface interactions at long range, where the interactions are weak and the bulk packing is important. This part is relatively poorly described with the present formula, containing a single transition distance. A more accurate approach to the whole subject is naturally to use the restricted primitive model in full detail (i.e., to include all ions explicitly). We have included an example and a structural comparison with the corresponding screened Coulomb model in the Appendix. With explicit counterions, surface force simulations are considerably more problematic, whereas cPB calculations are a tempting (though more approximate) alternative. We are currently investigating this option. Acknowledgment. I acknowledge financial support from the Swedish Research Council and the Linneaus program “Organizing Molecular Matter”.
Appendix Figure 10. Comparing simulated macroion density distributions as obtained with pointlike and hard core adjusted screened Coulomb (SC) interactions. In the latter case, we have set RM ) 20 Å (cf. Figure 9).
The cSC somewhat underestimates the primary force peak at the highest surface charge density, but the overall agreement is satisfactory. We saw earlier that the net interaction forces were sensitive to the addition of salt when the surfaces are neutral. Such a strong dependence is not observed when the surfaces are (sufficiently) charged, as demonstrated in Figure 8. This is not too surprising given that the magnitude of the oscillations is dramatically stronger in the salt-free case, when the surfaces are charged. This will allow correlation effects (packing) to stay relevant at a higher degree of screening.
4. Summary The simple ideas behind the cPB approach can be directly transferred to formulate a correlation-corrected theory, cSC, of systems containing macroions that interact via screened Coulomb potentials. The cSC is very easily implemented, and the calculations run quickly. Explicit comparisons with simulated results, for a large range of system parameters, have confirmed that cSC predictions are accurate. Given this and the simplicity of the cSC formulation, we believe that it will become a useful theory for descriptions of dispersions containing various kinds of macroions. Why, then, do the simple cPB and cSC formulations work so well? The reason is probably that ion correlation effects in most
Rough Evaluation of the Model. We have already discussed some of the drawbacks associated with our model, where the macroions appear as point charges interacting via a screened Coulomb potential. In this Appendix, we will nevertheless make a few simple structural comparisons that we believe will lend some support to our simplified model system. We will first present simulated slit density distributions, as obtained using the screened Coulomb adopted in the present work as well as with a more elaborate model in which the counterions are included explicitly. We have limited this comparison to a salt-free macroionic dispersion, with Q ) 20, confined between neutral, softly repulsive walls. In the presence of explicit counterions, we need an additional parameter, namely, the distance of closest approach, RM, between the macroion and its counterions. We have set RM ) 20 Å, which a reasonable choice for typical macroions. No other hard cores were present. The explicit counterion simulations were performed in a standard manner, as described in ref 4. The macroion-wall potentials were identical to those used in the screened Coulomb model, but for the counterions, we used Dw ) 5 Å. Macroions will have a tendency to avoid surfaces in the explicit model because these distort the counterion “cloud”. This effect is lost with the screened Coulomb ansatz. However, in essence this just means that the surface interactions in the screened model simply will be shifted to slightly smaller separations. This is highlighted in Figure 9, where macroion density distributions are compared, as obtained with explicit and screened models, respectively. The separation is 520 Å in the latter case rather than the 500 Å used in the explicit case. Given this minor separation adjustment, the overall agreement between the models is quite satisfactory, in particular
Simple Correlation-Corrected Theory
in the central potion of the slit (see inset), which governs the oscillatory behavior that we have observed. The density peaks that are closest to either surface are clearly exaggerated in the screened Coulomb model, but one would expect this to primarily influence the height of the first force peak at short separations. Structural similarities are insufficient to guarantee that the same surface forces are produced, but the agreement is nevertheless encouraging and provides some merit to the model that we have used in this work. Another possible concern with our model is the use of pointlike macroions. In reality, these naturally have a size (i.e., exclude counterions). According to standard Debye-Hu¨ckel theory, we should then adjust the macroion-macroion screened Coulomb potential by a factor of exp(κRM)/(1 + κRM). To avoid unnecessary parameters in our evaluation of the cSC theory, we have excluded
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such hard core adjustments. Furthermore, the amplitude adjustment is of course identical in the simulations and in the cSC theory, respectively. Still, it is of interest at least to make a rough estimate of the effect of such a screened Coulomb potential adjustment. A simple structural comparison is provided in Figure 10 using the same model system as in Figure 9. We see that the hard core potential adjustment plays only a minor role and, at least in this case, its inclusion actually increases the structural deviations from corresponding results with the more elaborate explicit counterion model. Finally, it should be noted that the hard core-hard core interactions are completely irrelevant in the present systems. The Coulomb repulsion prevents the occurrence of contact configurations. LA063179L
