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Depletion and Structural Forces in Confined Polyelectrolyte Solutions Bo Jo¨nsson,* A. Broukhno, J. Forsman, and T. Åkesson Theoretical Chemistry, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden Received May 16, 2003. In Final Form: August 22, 2003 Monte Carlo simulations and density functional calculations have been performed for charged macromolecules confined to planar slits. The force between the confining walls has been evaluated as a function of separation, while keeping the chemical potential of the macromolecules constant. Highly charged spherical particles and flexible polyelectrolyte chains in confinement give rise to depletion and structural oscillatory forces as a function of surface separation. The sign and magnitude of the surface charge of the confining walls have no dramatic effect on the qualitative behavior of the confined liquid. With neutral or oppositely charged surfaces, an accumulation of charged macroions is seen in the slit driven by the repulsive interaction between the macroions, while equally charged surfaces give rise to a pure depletion. The net charge, the range of interaction, and the particle density affect the details of the force curve. For spherical macroions, the period of the oscillations scales approximately as the bulk aggregate concentration, cbulk-1/3. Confined polyelectrolyte chains share some of these properties, but they partly display a different behavior. One clear difference is that the polyelectrolyte net charge, that is, the degree of polymerization, has no effect on the osmotic pressure. This is an indication that polyelectrolyte chains pack not as spheres but rather as cylindrical objects. Another difference is that the effective repulsive interaction between polyelectrolyte chains can be more long ranged and oscillatory forces can appear more readily than for a corresponding solution of equally charged spherical macroions.
Introduction A confined, simple fluid exerts a force on the confining surfaces. At liquid densities, this force typically shows an oscillatory behavior as a function of surface separation. This has been demonstrated both in theoretical calculations1-3 and in surface force experiments.4,5 The oscillations come as a consequence of packing of the liquid in the slit, and at reduced densities the oscillations disappear and a monotonic force-distance profile remains. Recent experimental studies of confined polyelectrolytes show, rather unexpectedly, also oscillatory force curves.6-12 Both linear and branched polyelectrolytes11 as well as spherical micelles13,14 show qualitatively the same behavior. The macromolecular concentration has in all these studies been quite low with an average solutesolute separation of 10-100 Å or more.10 The type of confinement, be it a free-standing soap film or solid silica surfaces, does not seem to matter. The measured pressures are in the range 50-2000 Pa, and the magnitude depends on macroion charge, ionic strength, and polymer density10 but does not seem to be crucially dependent on the (1) Snook, I.; van Megen, W. J. Chem. Phys. 1980, 72, 2907. (2) Karlstro¨m, G. Chem. Scr. 1985, 25, 89. (3) Henderson, D.; Lozada-Cassou, M. J. Colloid Interface Sci. 1986, 114, 180. (4) Christenson, H. K.; Gruen, W. R. D.; Horn, R. G.; Israelachvili, J. N. J. Chem. Phys. 1987, 87, 1834. (5) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (6) Bergeron, V.; Langevin, D.; Asnacios, A. Langmuir 1996, 12, 1550. (7) Milling, A. J. J. Phys. Chem. 1996, 100, 8986. (8) Asnacios, A.; Espert, A.; Colin, A.; Langevin, D. Phys. Rev. Lett. 1997, 78, 4974. (9) Klitzing, R. v.; Espert, A.; Asnacios, A.; Hellweg, T.; Colin, A.; Langevin, D. Colloids Surf. 1999, 149, 131. (10) Kolaric, B.; Jaeger, W.; Klitzing, R. v. J. Phys. Chem. B 2000, 104, 5096. (11) Klitzing, R. v.; Kolaric, B. Prog. Colloid Polym. Sci. 2002, 00, 00. (12) Klitzing, R. v.; Kolaric, B.; Jaeger, W.; Brandt, A. Phys. Chem. Chem. Phys. 2002, 4, 1907. (13) Richetti, P.; Kekicheff, P. Phys. Rev. Lett. 1992, 68, 1951. (14) Sober, D. L.; Walz, J. Y. Langmuir 1995, 11, 2352.
character of the surface, that is, whether it is charged or not.6,15,16 Oscillatory forces have been observed in micellar solutions as mentioned above, but they have also been reported to occur in more complex surfactant systems, that is, in bicontinuous microemulsions and in lamellar liquid crystalline systems.17,18 In these latter studies, the oscillations are caused not by single molecules or macroions but by semimacroscopic phases. A few theoretical studies of this problem have been published. Yethiraj19 applied an approximate integral equation technique to calculate polyelectrolyte induced pressure. Carignano and Dan20 used Monte Carlo (MC) simulations in order to investigate the density distribution of confined polyelectrolytes but did not report any intersurface forces. The experiments are performed under such conditions that the macromolecule chemical potential is constant, that is, there is an equilibrium between the confined volume and the bulk. This means that in simulations, one has to invoke a grand canonical Monte Carlo technique in order to mirror the experimental conditions. The grand canonical method works well for simple particles but is less applicable for polymer molecules. Hence, we have used an expanded ensemble method21 in order to maintain a constant chemical potential of the polyelectrolyte independent of the confinement and equal to the bulk value. Simulations of polymer molecules are usually very time-consuming and limited to rather short chains and a small number of polymers. To investigate a confined polymer under condi(15) Theodoly, O.; Tan, J. S.; Ober, R.; Williams, C. E.; Bergeron, V. Langmuir 2001, 17, 4910. (16) Klitzing, R. v. Tenside, Surfactants, Deterg. 2000, 37, 338. (17) Petrov, P.; Olsson, U.; Christenson, H.; Miklavic, S. J.; Wennerstro¨m, H. Langmuir 1994, 10, 988. (18) Petrov, P.; Miklavic, S. J.; Olsson, U.; Wennerstro¨m, H. Langmuir 1995, 11, 3928. (19) Yethiraj, A. J. Chem. Phys. 1999, 111, 1797. (20) Carignano, M. A.; Dan, N. Langmuir 1998, 14, 3475. (21) Broukhno, A. V.; Jo¨nsson, B.; A˙ kesson, T.; Vorontsov-Velyaminov, P. N. J. Chem. Phys. 2000, 113, 5493.
10.1021/la034850e CCC: $25.00 © 2003 American Chemical Society Published on Web 10/11/2003
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Figure 1. Schematic pictures of the model systems. (a) Model 1: Point particles with a net charge Qe are confined between two planar hard walls impenetrable to the particles; the walls are shown as thick lines. In some simulations, two charged surfaces are introduced, here denoted with thin lines and displaced a distance d relative to the confining walls. The particles interact via a screened Coulomb potential, eq 1. (b) Model 2: The monomers of the confined polyelectrolyte chains interact via a screened Coulomb potential. Model 3 is obtained if model 2 is augmented with counterions and all particles are allowed to interact via a pure Coulomb interaction, eq 8.
tions more akin to experimental ones, we have also invoked a density functional theory developed by Woodward.22 Models and Methods Three different models of confined macromolecules have been considered. In the first, model 1 (see Figure 1a), the macromolecules are assumed to be point charges interacting with a screened Coulomb potential,
lB exp(-κrij) βu(rij) ) Q2 rij
(1)
where rij ) |ri - rj| is the distance between the macroions at positions ri and rj, Q is the macroion charge, and lB is the Bjerrum length,
βe2 lB ) 4π0r
∑i ciqi2
(3)
Q ([exp(-κz) - exp(-κR)] + κlGC [exp(-κ(h - z)) - exp(-κR)]) (4)
where z and h - z are the distances from the charged (22) Woodward, C. E. J. Chem. Phys. 1991, 94, 3183.
kfri,i+12 lBri,i+12 ) 2 2r 3
(5)
(2)
where the summation runs over all salt particles and counterions of concentration ci and valency qi. There is no need to include a hard core, since the Coulomb repulsion will anyway prevent the particles from approaching too close. With charged surfaces, one also has to include the electrostatic wall-macroion and wall-wall interactions. These interactions are simple to integrate up using the screened Coulomb potential, and for example, the wallmacroion interaction reads
βu(z) )
βubond(ri,i+1) )
min
0 is the dielectric permittivity of a vacuum, and the relative permittivity, r, is set equal to 78. e is the elementary charge, and β ) 1/kBT, where kB is the Boltzmann constant and T is the temperature equal to 300 K. The inverse Debye-Hu¨ckel screening length is given by
κ2 ) 4πlB
surfaces. The first parenthesis on the right-hand side is zero if z > R, and the second is zero if (h - z) > R. R is the cutoff distance in the simulations, and lGC ) e/2πlBs is the Gouy-Chapman length characterizing the surface charge density s. The wall-wall interaction is also calculated according to eq 4 with Q replaced with the appropriate amount of charge coming from that part of the wall that is inside the cutoff sphere. Model 2 treats the macromolecules as flexible polymers (Figure 1b), whose monomeric units interact with a screened Coulomb potential, eq 1, with Q ) 1. The bonds are modeled as harmonic springs, and the bonding potential between two neighboring monomers reads
where rmin is the energy minimum for the dimer. rmin has been put equal to 3 Å in most of the simulations, which leads to an average monomer-monomer separation of approximately 5 Å, with a rather weak dependence on chain length, salt concentration, and the confining geometry. These two models are solved using different Monte Carlo simulation techniques. In model 1, we have extended the usual Metropolis method23 to the grand canonical ensemble with particle insertion and deletions, which is a standard technique described in detail in several textbooks.24,25 The size of the system was chosen to ensure that the repulsive interaction has decayed to a sufficiently low value, σ
(8)
The density functional theory is surprisingly accurate for many polymer systems26,27 and can be described as semiquantitative. Figure 2 shows a comparison between density functional results and force-distance curves obtained from expanded ensemble simulations. The comparison is made for a screened Coulomb chain and gives an indication of the accuracy of the approximate theory. Details of the implementation of the theory can be found in the Appendix. Results and Discussion Point Particles with Screened Coulomb Interaction. Nonadsorbing polymers escape a narrow slit, and
(26) Woodward, C. E.; Yethiraj, A. J. Chem. Phys. 1994, 100, 3181. (27) Forsman, J.; Woodward, C. E. J. Chem. Phys. 2003, 119, 1889.
Figure 2. Polyelectrolyte chains confined between two neutral surfaces and in equilibrium with a bulk solution. The monomers interact according to the screened Coulomb potential, eq 1, with Q ) 1. The polymer length is 20, the bulk monomer concentration is 50 mM, rmin ) 4 Å, and κ-1 ) 18 Å. The expanded ensemble pressure is shown as a solid line, and the dashed curve is obtained from the density functional theory.
a depletion attraction appears.28-30 The mechanism behind the depletion is that the confining walls put constraints on the configurational entropy of the polymers. For a compact macromolecule like a charged micelle, there is no configurational penalty in a confined geometry. But if the confining surfaces carry the same charge as the macroions, there will be a repulsive micelle-surface interaction forcing the micelles out of the slit, which ultimately could lead to a depletion attraction. The interesting observation made by Richetti and Kekicheff13 was that prior to this depletion zone, the force between the surfaces is oscillatory. The result has been verified by Sober and Walz14 for a slightly different geometry and at significantly lower micelle concentrations. Piech and Walz have also reported oscillatory forces between silica surfaces in the presence of nanoparticles of both silica and polystyrene.31 We have tried to mimic the conditions in some of these experiments. The screened Coulomb potential is derived under the assumption that the electrostatic potential is low and that the Boltzmann factor can be linearized. This is generally not the case with ionic micelles. There exist many attempts to retain the simple form of the screened Coulomb potential, while improving its validity by rescaling the charge and/or the screening length.32,33 We have, however, decided to use the nominal charge and screening parameters quoted in the original experimental reports with the tacit assumption that this will not affect the qualitative behavior of the system. The experimental results are reported as the force divided by the radius of the crossed cylinders, and the simulated counterparts have been obtained using the Derjaguin approximation.34 The positions of the first minimum and maximum in Figure 3a are in excellent agreement with the data of Richetti and Kekicheff. The strength of the force is, however, overestimated by almost an order of magnitude. As will be obvious later, this magnitude could easily be reduced by adjusting the micellar charge without affecting the positions of maxima and minima. The density variation within the slit shows an oscillatory behavior, where the number of maxima and minima decreases at shorter separations. The cetyltrimethylammonium bromide (CTAB) system studied by Richetti and (28) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (29) Vrij, A. Pure Appl. Chem. 1976, 48, 471. (30) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: New York, 1983. (31) Piech, M.; Walz, J. Y. J. Colloid Interface Sci. 2002, 253, 117. (32) Woodward, C. E.; Jo¨nsson, B. J. Phys. Chem. 1988, 92, 2000. (33) Kjellander, R.; Mitchell, D. J. J. Chem. Phys. 1994, 101, 603. (34) Derjaguin, B. Kolloid-Z. 1934, 69, 155.
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Figure 3. Grand canonical simulation of CTAB micelles between two charged planar surfaces. The micelle charge Q ) 90 and lGC ) 2 Å, corresponding to a surface charge density of 0.178 C/m2. Note that the charged surfaces are displaced a distance d ) 23.5 Å, relative to the confining surfaces. (a) The force, obtained from the Derjaguin approximation (ref 34), as a function of separation. The volume fraction of micelles and the screening length are 0.009 and 49 Å for the solid curve and 0.019 and 36 Å for the dashed curve, respectively. (b) Micellar concentration profiles for three surface separations: h ) 500 Å (solid line), 400 Å (dashed line), and 300 Å (dotted-dashed line). The volume fraction and screening length are 0.019 and 36 Å.
Figure 4. Grand canonical simulation results for model 1 with different surface charge densities and d ) 23.5 Å; Q ) 16, κ-1 ) 18 Å, and cbulk ) 3.2 mM. (a) Net pressure as a function of separation for different surfaces: circles, neutral surfaces; squares, surfaces oppositely charged to the macroions, lGC ) -2 Å; diamonds, surfaces with the same charge as the macroions, lGC ) 2 Å. (b) The average macroion concentration as a function of slit width, with notation as in panel a.
Kekicheff is strongly coupled, and the density profiles show an almost solidlike structure; see Figure 3b. This highly ordered structure is probably an overestimate due to the use of stoichiometric parameters in the screened Coulomb potential. Nonetheless, the figure clearly shows how micelles, layer by layer, are squeezed out when the separation decreases. Thus, a dilute solution of particles interacting with a soft repulsion, like the screened Coulomb potential, gives rise at a confining surface to a structure similar to that of a hard-sphere liquid.2 This is true as long as the range of the repulsive interaction is larger than or of the same order as the average macroionmacroion separation. In the simulations, we find strong oscillations in the force-distance curve if the interaction between a pair of macroions at their average separation is of the order of or larger than kBT, that is,
βu(rij ) cbulk-1/3) ∼ 1
(9)
The mica surfaces used in the surface force apparatus are negatively charged, and the CTA molecules form a bilayer thereon. In the simulations, we have assumed that the surface charge is reversed due to bilayer formation. As a matter of fact, this is not a crucial issue, since macroions confined between neutral surfaces and surfaces with the same as well as opposite charge to the macroions all display oscillatory force profiles; see Figure 4a. The force profiles at short separations differ, but they are surprisingly similar at intermediate and large separations. In fact, the three curves in Figure 4a can be made to coincide by a small shift of the separation scale. The force curve with
oppositely charged surfaces and macroions extends further out than the other two. The reason is that a layer of macroions is formed close to the surfaces, which reverses the sign of the surface charge and at the same time displaces the force profile. Several experimental studies with flexible polyelectrolytes in free-standing films confirm these findings. For example, Klitzing16 has measured oscillatory forces with negative polyelectrolytes and both positive and neutral surfaces. Theodoly et al.15 have studied polyelectrolytes confined between surfaces with the same charge as the macroions as well as neutral surfaces, and they find similar types of oscillatory forces in the two cases. The average density of charged macromolecules in the slit will of course be different in the three cases. With equally charged surfaces, there is a depletion of macroions, while with neutral or oppositely charged surfaces an accumulation is seen; Figure 4b. The accumulation near neutral surfaces is driven by the particles’ desire to avoid the mutual repulsive interaction. With oppositely charged surfaces, this mechanism is further strengthened by the attraction to the surface. Experimentally, one can adjust the interactions in the system in several ways. Changing the density is one possibility, but one can also modify the net charge on the particles and independently decrease the range of the electrostatic interaction by adding salt. In a salt-free system, the Debye-Hu¨ckel screening length is given by the concentration of counterions only,35,36 which means that an increase of Q at constant particle density automatically decreases κ-1. In the simulations, however, we
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Figure 5. Grand canonical simulation results for model 1 with a macromolecular charge of Q ) 16 and neutral walls. Net pressure as a function of slit separation for different bulk densities and fixed κ-1 ) 18 Å: solid line, cbulk ) 3.2 mM; dashed line, cbulk ) 6.4 mM; dotted-dashed line, cbulk ) 12.6 mM.
are free to vary both Q, κ-1, and cbulk independently, to explore the importance of the various parameters. That is, the screening length need not be related to the density and macroion charge in the way described by eq 3. In the following graphs, we have systematically varied the macroion density, the macroion charge, and the screening length. Figure 5 shows the pressure profile for three macromolecule densities. The amplitude of the oscillations increases with density, while the period decreases. From Figure 5, we can estimate that the period of the oscillations scales approximately as cbulk-0.38, which is fairly close to -1/3 expected from simple packing considerations. A higher particle charge, at constant bulk macroion density, not unexpectedly leads to a more structured liquid and pronounced oscillations in the pressure curve; see Figure 6a. The amplitude of the oscillations increases with the net charge, but the period remains approximately the same. This is in agreement with the atomic force microscopy (AFM) experiments by Piech and Walz31 on confined polystyrene spheres of varying size. The screening in the experiment was mainly caused by added KNO3. Experimentally, it has been seen that the addition of salt reduces the oscillations,7,10,31 and a similar result is found in the simulations when κ is varied; see Figure 6b. Piech and Walz31 have also investigated how the oscillations are affected by a change in macroion charge at constant volume fraction of macroions; that is, the number density varies. Figure 7 shows density profiles and pressure curves for such conditions. In agreement with the experimental results, we find that the structure in the force curve is diminished when the macroion charge increases and the number density decreases, while keeping the amount of charge per unit volume constant. This is a consequence of the exponential screening of the electrostatic interactions that decrease much faster than the average separation between the macroions proportional to cbulk-1/3. The increased structure at reduced macroion concentration is of course an intermediate phenomenon. When Q becomes sufficiently small and the average macroion separation is of the same order as or smaller than the screening length, that is, cbulk-1/3 < κ-1, then the oscillatory structure seen in Figure 7b will disappear. This can also be seen from the fact that eq 9 is no longer satisfied, but instead we have βu(rij < κ-1) < 1 for most experimental conditions. (35) Beresford-Smith, B. Some aspects of strongly interacting colloidal interactions. Ph.D. Thesis, Australian National University, Canberra, Australia, 1985. (36) Beresford-Smith, B.; Chan, D. Y. C.; Mitchell, D. J. J. Colloid Interface Sci. 1985, 216, 216.
Jo¨ nsson et al.
Polyelectrolyte Chains. To maintain a constant chemical potential of confined polymers leads to a significantly more demanding simulation than the case with point particles or hard spheres. Any direct insertion of chains is difficult with an extremely low acceptance and, thus, a poor convergence of the simulation. The expanded ensemble technique offers a viable alternative. It is still rather time-consuming and limited to fairly short chains. In the simulations presented, we have used chains consisting of up to 32 monomers. Figure 8a shows monomer density profiles at different slit widths. The general picture is similar to what is seen with point particles confined between equally charged surfaces, with a depletion zone close to the surfaces. The difference is that the depletion is not caused by repulsive macroion-wall interaction but is due to the constraint put on the polymer from the wall. The extension of the depletion is, at least for shorter chains, determined by a balance between chain entropy and interchain repulsion. The oscillatory density profiles are accompanied by periodically varying force curves; see Figure 8b. In model 1, with point particles interacting with a screened Coulomb potential, the period of the oscillations in the forcedistance curve seemed to be determined by simple packing of the macroions. Altering the macromolecule concentration and charge changed the force curve; see for example Figure 7b. This is not the case for a flexible chain, and two virtually identical force curves are obtained for 16- and 32-mers at constant bulk monomer concentration (Figure 8b). Several experimental investigations7,10,12,15 have found that monomer concentration, and not the degree of polymerization, is the crucial variable for the behavior of the intersurface pressure. The results in Figure 8b are in agreement with these experiments. The 8-mer curve also included in Figure 8b deviates slightly from the other two with more damped oscillations. This is reasonable since reducing the degree of polymerization eventually leads to a solution containing only monomeric units. There are two obvious differences between a spherical charged macroion and a flexible polyelectrolyte: (i) the chain is an elongated object and (ii) the range of interaction is significantly more long ranged for polyelectrolytes than for spherical particles. That the aggregate geometry plays a role for the density distribution is apparent from Figures 7a and 8a. The macroions accumulate close to the walls, while the polyelectrolytes avoid the immediate neighborhood of the walls and the first peak in the distribution function appears approximately a radius of gyration out from the surface. Changing the charge of a macroion has a clear effect on the distribution (see Figure 7a), which is in contrast to the monomer distribution functions in Figure 8a, where an increased chain length has essentially no effect on the profile. An even more distinct difference is found in the force curves. An increase in macromolecular charge, while keeping cbulkQ ) const, eventually leads to a disappearance of the structure in the pressure curve, while increasing the chain length has no effect on the pressure as seen from Figure 8b. Graphical representations of configurations from the MC simulations, Figure 9, clearly show the polyelectrolyte chains as elongated objects. The chains also tend to order at the walls. A calculation of the x-, y-, and z-components of the end-to-end distance supports the notion that the long axis of a chain is preferentially aligned parallel to the confining surfaces. The simulation snapshots in Figure 9 contain only fairly short chains; although the monomer density is high, there is no clear overlap of the chains. That the range of interaction is larger for polyelectrolyte
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Figure 6. Grand canonical simulation results for model 1 with cbulk ) 3.2 mM and neutral walls. Net pressure as a function of slit separation. (a) Variation of the macromolecular charge with κ-1 ) 18 Å: solid line, Q ) 24; dashed line, Q ) 32; dotted-dashed line, Q ) 48. (b) Variation of the screening length with fixed macroion charge, Q ) 16: solid line, κ-1 ) 18 Å; dashed line, κ-1 ) 27 Å; dotted-dashed line, κ-1 ) 36 Å.
Figure 7. Grand canonical simulation results for model 1 with varying macroion charge and number density, but a constant charge density in the bulk, i.e., cbulkQ ) 50 mM and κ-1 ) 18 Å. The confining walls are neutral. (a) Density profiles for a surface separation of h )200 Å: solid line, Q ) 32; dashed line, Q ) 64; dotted-dashed line, Q ) 128. (b) Net osmotic pressure as a function of separation, with notation as in panel a.
Figure 8. Expanded ensemble simulation results for model 2 with varying chain length, κ-1 ) 18 Å, rmin ) 3 Å, and a monomer bulk concentration of 100 mM. The walls are neutral. (a) Density profiles for two different surface separations, h ) 200 Å and h ) 100 Å: solid lines, 32-mers; dashed lines, 16-mers. (b) Net pressure as a function of slit separation: solid line, 32-mers; dashed line, 16-mers; dotted-dashed line, 8-mer.
chains than point particles with the same total charge is evident from Figure 10. The results reported in Figure 8b are obtained for fairly short chains and with monomers interacting via the screened Coulomb potential, eq 1. In the density functional calculations, we can relax both of these constraints. Compared to simulations, we can proceed to study larger separations and longer chains, while still maintaining a very good accuracy in the calculation of free energies and pressures. The density functional results in Figure 11 showing the pressure for chains interacting with explicit Coulomb interactions confirm the conclusions drawn from the screened Coulomb results. The chain length has virtually no effect on how the pressure varies as a function of surface separation, and only if the chain length is shorter than 20-40 monomers is a size dependence observable.
Note also that a reduced density leads to a more long ranged pressure; cf. the two curves in Figure 11. These results are, in both respects, in good agreement with experiments.8,12,15 In some density functional calculations, we have also varied the polymer stiffness and found that the stiffer the chain, the better it packs, giving less variation in the pressure curve. Conclusion We have investigated the pressure exerted by confined polyions, which are in equilibrium with a bulk at constant chemical potential, using Monte Carlo simulations and density functional theory. Both spherical macroions and flexible polyelectrolytes between two planar surfaces give rise to oscillatory forces as a function of separation. If the surfaces are charged or not has hardly any qualitative
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Coulomb interaction between the micelles in the simulations. The oscillatory forces are a consequence of the packing of the micelles, which can be regarded as soft repulsive spheres. Typically, oscillations will appear if the average interaction is larger than kBT. Confined polyelectrolyte chains share some of the properties demonstrated by spherical macroions. However, the chains seem to preferentially pack as approximate cylinders. As a consequence, no chain length dependence shows up and the forces are also rather insensitive to the line charge density of the polyelectrolyte (data not shown). The density, however, has a profound effect on the force curves, for both macroions and polyelectrolytes, and a reduced density leads to more long ranged but weaker interactions. Our approximate density functional theory seems to give an accurate description of the forces in the system. The theory allows us to investigate systems where the parameters are in the same range as in experiment. The results are, with respect to both range and magnitude, in good agreement with experimental results. Appendix: Density Functional Theory Figure 9. Snapshots from simulations of confined polyelectrolyte chains. h ) 150 Å, κ-1 ) 18 Å, rmin ) 3 Å, and the monomer density is 78 mM. The monomers are depicted as spheres with a diameter equal to κ-1, and the confining walls are shown in gray. The chain length is 16 (a) and 64 (b).
We shall use the density functional formulation of Woodward22 and also invoke its recent improvement by the inclusion of intramolecular stiffness.27 We represent an M-mer configuration as R ) (r1, ..., rM), where ri is the coordinate of monomer i. Adjacent monomers are connected by infinitely strong bonds, the fixed length of which is given by the bond length b; this is sometimes referred to as the “pearl-necklace model”. Hence, the bonding potential, Vb(R), connecting monomers along the chain can be written as M-1
exp[-βVb(R)] ∝
Figure 10. A comparison of the net pressures as a function of separation for spherical macroions (solid line) and flexible chains (line with symbols). The bulk chain and macroion concentration is 3.2 mM, Q ) 16, the chain length is 16, rmin ) 3 Å, and κ-1 ) 18 Å. The confining walls are neutral.
δ(|ri+1 - ri| - b) ∏ i)1
(10)
where δ(x) is the Dirac delta function. We have fixed the bond length to 10 Å in the calculations. We denote by N(R) dR the number of polymer molecules having configurations between R and R + dR. For a fluid of ideal chains, the exact expression for the free energy functional, 37 F id p , is
βF id p [N(R)] )
∫N(R)(ln[N(R)] - 1) dR + β ∫N(R)(Vb(R) + Vext(R)) dR
(11)
where Vext(R) is an external potential. The monomer density, nm(r), is given by M
nm(r) )
∫ ∑δ(r - ri)N(R) dR
(12)
i)1
Figure 11. Density functional results for model 3 with an explicit description of polyelectrolytes and ions. The bulk monomer concentration is 10 mM (thick line and filled symbols) and 2.5 mM (thin line and open symbols). The degree of polymerization is 300 (lines) and 75 (symbols); neutral walls.
effect on the force curves. Good agreement is found between simulations and experiments for micelles confined between curved mica cylinders. The differences seen between theory and experiments can be attributed to the use of a screened
The system we wish to describe also contains counterions, and there are Coulombic interactions which have to be taken into account. The predictive powers of the PoissonBoltzmann approximation and its polyelectrolyte version can be considerably improved simply by including a “correlation hole” between species of equal charge. This means that they do not interact with each other below a certain separation, which defines the diameter of the hole. This hole serves to mimic the correlations between equally charged species, by making the net repulsion between them smaller than the traditional mean-field estimate. (37) Woodward, C. E.; Jo¨nsson, B. Chem. Phys. 1991, 155, 207.
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The introduction of a correlation hole will, for instance, allow the theory to predict an attractive interaction between equally charged surfaces, in strongly coupled systems. This is in qualitative agreement with simulation results. In our polyelectrolyte system, we have simplified matters by restricting the correlation hole between monomers, σmm, and between counterions, σcc, to be identical, that is, σmm ) σcc ≡ σ. In other words, the interaction energy between two equally charged species separated a distance r is
βuRR(r) )
{
lB/r 0
r>σ reσ
si‚si+1 ) ∆zi∆zi+1 + (b2 - ∆zi2)1/2(b2 ∆zi+12)1/2 cos Φi,i+1 (17) where Φi,i+1 is the angle between the vectors si and si+1, projected onto the plane of the surfaces. The Boltzmann factor containing the bending potential can be averaged over the surface plane, to give the following result:
[(
(13)
A monomer and a counterion interact via the pure Coulomb potential, without any hole. If we summarize the various contributions to the complete free energy functional for the polylelectrolyte system and perform a Legendre transformation, we obtain the following expression for the grand potential, Ω:
∫nc(r)(ln[nc(r)] - 1) dr + βFex[N(R), nc(r)] - βµp ∫N(R) dR - βµc ∫nc(r) dr
βΩ[N(R)] ) βFid p [N(R)] +
b2
1-
∆zi+1 b
2 1/2
(18)
where we have introduced the modified Bessel function,
I0(x) )
1 2π
∫02π exp[-x cos Φ] dΦ
(19)
This function is easily evaluated from a polynomial expansion. Minimizing the free energy functional results in an equilibrium monomer density profile, which satisfies the following equation: M
∫0 ∑ i)1
M-1
∑R ∑γ ∫ ∫ nR(r)nγ(r′)uRγ(|r -
2 1/2
∆zi b
nm(z) ) exp[β(µp + MΥD)]
1 βFex[N(R), nc(r)] ) 2
I0 1 -
( )) ( ( )) ]
(14) where
)] [ (
∆zi∆zi+1
Ψ(∆zi, ∆zi+1) ) exp 1 -
exp[-λ(zj)]
∏ k)1
h
M
∏ j)1
δ(z - zi)
M-2
Θ(|∆zk| - b)
Ψ(∆zl, ∆zl+1) dz1 ... dzr ∏ l)1 (20)
r′|) dr dr′ (15) where where µp and µc are the polymer and counterion bulk chemical potentials, respectively. Since we are interested in describing polymer fluids at or between surfaces, we define a slit geometry where two flat confining surfaces extend infinitely in the (x, y) separated a distance h. We let these surfaces be hard but otherwise inert. The symmetry of this slit geometry allows us to integrate out the (x, y) dependence of all quantities. To maintain electroneutrality in the slit, we need to invoke a Donnan potential, ΥD, which varies with separation. However, electroneutrality itself ensures that this potential does not contribute to the grand potential, since the energy contribution to positive charges will exactly cancel the corresponding contribution to negative charges. We shall approximate the Coulombic intrachain repulsion by introducing a bending potential, such that next nearest neighbors repel each other along a chain.27 This provides an internal stiffness of the polymer molecule. Formally, this means that a term ∫N(R)Eb(R) dR is added to the ideal free energy functional, eq 11. The inclusion of stiffness has been shown to considerably improve the density functional theory, in descriptions of athermal polymer fluids at low density.27 It is likely to be even more important in our systems, where the chains are relatively stiff, due to intramolecular repulsions. Let si denote the bond vector between monomers i and i + 1. Furthermore, ∆zi ≡ zi+1 - zi, where zi is the z coordinate of monomer i. The strength of the bending potential, Eb, is regulated by the parameter according to the following equation:
(
βEb ) 1 -
)
si‚si+1 b2
We can write the vector product as
(16)
λi(z) )
δβFex
(21)
δnm(zi)
and Θ(x) is a step function,
Θ(x) )
{
1 0
xe0 x>0
(22)
The expression for the equilibrum counterion density profile is much simpler:
[(
nc(z) ) exp β µs - ΥD -
δFex
)]
δnc(z)
(23)
The solution to eq 20 can be obtained by numerical iteration. This can be efficiently achieved via repeated matrix-vector multiplications. The value of the stiffness parameter is given by the stiffness of the polyelectrolyte chains in the bulk solution with which the confined system is in equilibrium. Specifically, is chosen so as to give the same average end-toend separation as obtained in a Metropolis Monte Carlo simulation of the bulk solution. Having specified our model system in this manner, we are left with one adjustable parameter, namely, the size of the correlation hole between charges of equal sign. A problem is that this hole must be a compromise between the correlations between equal charges near one “tagged” polyelectrolyte and the much larger correlation hole that exists between charges on or near different polyelectrolyte chains. Naturally, we cannot with perfect quantitative agreement reproduce the properties of a laterally (in the (x, y) plane) heterogeneous system with a mean-field estimate, where a smeared-out
9922
Langmuir, Vol. 19, No. 23, 2003
behavior is assumed. We nevertheless expect to correctly reproduce the qualitative features of such systems, with this kind of correlation-corrected mean-field analysis. Note that if lateral heterogeneity is of qualitative importance for the surface forces we wish to study, one would expect to see a qualitative response to changes of the nature of the confining surfaces. Such a response has not been found experimentally. At any rate, we have chosen to specify the size of the correlation hole in a very simplistic manner.
Jo¨ nsson et al.
We simulate a polyelectrolyte system with 30-mer chains in a canonical ensemble, with confining surfaces separated by h/b ) 60. The average monomer density in the slit is set equal to the bulk density of the system we wish to study. The size of the correlation hole is then simply chosen so as to give the best possible agreement with the monomer density profile that was obtained in the simulation. LA034850E
