Langmuir 2007, 23, 9713-9721
9713
Mesoscale Simulations of the Behavior of Charged Polymer Brushes under Normal Compression and Lateral Shear Forces Mohan Sirchabesan† and Suzanne Giasson*,†,‡ Faculty of Pharmacy and Department of Chemistry, UniVersite´ de Montre´ al, Montre´ al, Que´ bec, Canada H3C 3J7 ReceiVed March 30, 2007. In Final Form: June 5, 2007 Dissipative particle dynamics (DPD) was used to investigate the behavior of two opposing end-grafted charged polymer brushes in aqueous media under normal compression and lateral shear. The effect of polymer molecular weight, degree of ionization, grafting density, ionic strength, and compression on the polymer conformation and the resulting shear force between the opposing polymer layers were investigated. The simulations were carried out for the poly(tert-butyl methacrylate)-block-poly(sodium sulfonate glycidyl methacrylate) copolymer, referred as PtBMAb-PGMAS, end-attached to a hydrophobic surface for comparison with previous experimental data. Mutual interpenetration of the opposing end-grafted chains upon compression is negligible for highly charged polymer brushes for compression ratios ranging from 2.5 to 0.25. Under electrostatic screening effects or for weakly charged polymer brushes, a significant mutual interpenetration was measured. The variation of interpenetration thickness with separation distance, grafting density, and polymer size follows the same scaling law as the one observed for two opposing grafted neutral brushes in good solvent. However, compression between two opposing charged brushes results in less interpenetration relative to neutral brushes when considering equivalent grafting density and molecular weight. The friction coefficient between two opposing polymer-coated surfaces sliding past each other is shown to be directly correlated with the interpenetration thickness and more specifically to the number of polymer segments within the interpenetration layer.
Introduction Numerous research activities on end-grafted charged polymers, or polyelectrolytes, have been carried out in the past few years because of their important role in controlling material surface properties (adhesion, lubricity, biocompatibility, and colloidal stability)1-3 which critically depend on the conformation of the polymer chains. Grafted polymer chains can adopt different conformations (i.e., pancake, mushroom, and brush)4-6 depending on the polymer molecular weight, degree of ionization of the chain, grafting density, and environmental conditions (i.e., temperature, pressure, solvent, ionic strength, and pH). The brush conformation offers very interesting properties such as controlled wettability, autophobicity, lubricity, and steric hindrance.1,7,8 Brush conformation with end-grafted neutral polymers occurs when the distance between adjacent grafting sites s is less than the unperturbed coil size, whereas brush conformation with a charged polymer can be obtained at much smaller grafting densities 1/s2 due to the fact that electrostatic interactions within and between grafted chains promote chain stretching. The variation of the thickness of polymer brushes upon changes in environmental conditions has been extensively investigated both theoretically and experimentally.9-17 The charged polymer brush * To whom correspondence should be addressed. E-mail: suzanne.
[email protected]. † Faculty of Pharmacy. ‡ Department of Chemistry. (1) Claesson, P. M.; Poptoshev, E.; Blomberg, E.; Dedinaite, A. AdV. Colloid Interface Sci. 2005, 114, 173. (2) Guenoun, P.; Argillier, J.-F.; Tirrell, M. C. R. Acad. Sci., Ser. IV: Phys., Astrophys. 2000, 1 (9), 1163. (3) Zhao, B.; Brittain, W. J. Prog. Polym. Sci. 2000, 25 (5), 677. (4) Degennes, P. G. Macromolecules 1980, 13 (5), 1069. (5) Zhulina, E.; Balazs, A. C. Macromolecules 1996, 29 (7), 2667. (6) Zhulina, E. B.; Borisov, O. V. Macromolecules 1996, 29 (7), 2618. (7) Claesson, P. M.; Dedinaite, A.; Rojas, O. J. AdV. Colloid Interface Sci. 2003, 104 (1-3), 53. (8) Raviv, U.; Giasson, S.; Kampf, N.; Gohy, J. F.; Jerome, R.; Klein, J. Nature 2003, 425 (6954), 163.
thickness is governed by an energy balance between the stretching of the chains and the osmotic contribution of the counterions. Some theoretical approaches using self-consistent mean field theory, scaling laws, and molecular dynamics have been used to investigate the monomer distribution, or the density profile, throughout charged brushes.18-21 According to these approaches and under salt-free conditions, the charged brush thickness h is expected to vary linearly with the degree of polymerization N when all polyelectrolyte counterions are located inside the brush and to vary as N3s-2 when some of these counterions extend outside the brush.22 The monomer density profile along the brush is expected to follow a Gaussian law in salt-free solutions for large grafting densities,23 whereas a parabolic profile independent of the grafting density is predicted when salt or ions are added in the solution as observed with neutral polymer brushes.19,24 This is still an open issue.25 The brush thickness h is expected (9) Auroy, P.; Auvray, L.; Leger, L. J. Phys.: Condens. Matter 1990, 2, SA317. (10) Auroy, P.; Auvray, L.; Leger, L. Macromolecules 1991, 24 (9), 2523. (11) Auroy, P.; Auvray, L.; Leger, L. Phys. ReV. Lett. 1991, 66 (6), 719. (12) Auroy, P.; Mir, Y.; Auvray, L. Phys. ReV. Lett. 1992, 69 (1), 93. (13) Baranowski, R.; Whitmore, M. D. J. Chem. Phys. 1995, 103 (6), 2343. (14) Baranowski, R.; Whitmore, M. D. J. Chem. Phys. 1998, 108 (23), 9885. (15) Cosgrove, T. J. Chem. Soc., Faraday Trans. 1990, 86 (9), 1323. (16) Cosgrove, T.; Heath, T.; Vanlent, B.; Leermakers, F.; Scheutjens, J. Macromolecules 1987, 20 (7), 1692. (17) Cosgrove, T.; Heath, T. G.; Phipps, J. S.; Richardson, R. M. Macromolecules 1991, 24 (1), 94. (18) Borisov, O. V.; Leermakers, F. A. M.; Fleer, G. J.; Zhulina, E. B. J. Chem. Phys. 2001, 114 (17), 7700. (19) Israels, R.; Leermakers, F. A. M.; Fleer, G. J.; Zhulina, E. B. Macromolecules 1994, 27 (12), 3249. (20) Zhulina, E. B.; Borisov, O. V. J. Chem. Phys. 1997, 107 (15), 5952. (21) Zhulina, E. B.; Wolterink, J. K.; Borisov, O. V. Macromolecules 2000, 33 (13), 4945. (22) Pincus, P. Macromolecules 1991, 24 (10), 2912. (23) Fleer, G.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B.; Polymers at Interfaces; Chapman & Hall: New York, 1993. (24) Zhulina, E. B.; Borisov, O. V.; Birshtein, T. M. J. Phys. II 1992, 2 (1), 63. (25) Muller, F.; Romet-Lemonne, G.; Delsanti, M.; Mays, J. W.; Daillant, J.; Guenoun, P. J. Phys.: Condens. Matter 2005, 17 (45), S3355.
10.1021/la7009226 CCC: $37.00 © 2007 American Chemical Society Published on Web 08/14/2007
9714 Langmuir, Vol. 23, No. 19, 2007
to decrease with increasing Cs but only as a relatively weak power law, that is, h ∝ NCs -1/3s-2/3.22,26 The interactions between two surfaces bearing charged brushes as a function of separation distance, that is, interaction profiles, have also been extensively studied experimentally and theoretically by several groups.1,7,8,26-35 The interaction profiles between charged brushes are characterized by stronger and longer-ranged repulsive forces relative to neutral brushes when considering equivalent grafting density and molecular weight. The interaction energy at small separation distances, corresponding to the physical contact and slight compression between the two opposing charged brushes, is mainly governed by osmotic pressure associated with the polyelectrolyte counterions. At larger separation distances, that is, when two opposing brushes are not in contact, the interactions depend on the effective potential of the polyelectrolyte brushes or the relative position of polyelectrolyte conterions. Long-range repulsive electrostatic forces can be measured if counterions extend outside the brush,26 whereas only short-range steric forces are observed if the counterions are located inside the brush.36 In both cases, the behavior of charged brushes is much less sensitive to the Debye screening effect than that of two simply charged surfaces. The capacity of an end-grafted charged polymer to prevent particles from aggregation as well as to reduce friction between two sliding surfaces has been investigated.8,29,37,38 Charged polymer brushes have shown the ability to significantly reduce friction between two surfaces in aqueous medium at a relatively high applied load,8 whereas a non-negligible friction or shear force is usually obtained between neutral brushes in good solvent, which is associated with an important increase in the effective viscosity of the compressed film together with mutual interpenetration of the opposing neutral polymer layers.30,32,39,40 The very good lubricity observed with charged brushes was attributed to the counterion-induced brush swelling and the fluid hydration sheaths around the charged segments which are both effective in overcoming factors leading to large friction force.8 A recent study using molecular dynamic simulations has shown that two opposing polyelectrolyte brushes avoid mutual interpenetration upon compression by folding in on themselves.27 In the absence of mutual interpenetration, the shear force between two opposing charged brushes is expected to result from shearing hydrated ions and/or hydrated charged polymer segments which mainly rely on the low effective viscosity of water confined in thin films. This friction mechanism would explain the very low friction coefficient previously measured experimentally.8 However, to (26) Abraham, T.; Giasson, S.; Gohy, J. F.; Jerome, R. Langmuir 2000, 16 (9), 4286. (27) Hehmeyer, O. J.; Stevens, M. J. J. Chem. Phys. 2005, 122 (13), 134909. (28) Kampf, N.; Gohy, J. F.; Jerome, R.; Klein, J. J. Polym. Sci., Part B: Polym. Phys. 2005, 43 (2), 193. (29) Kampf, N.; Raviv, U.; Klein, J. Macromolecules 2004, 37 (3), 1134. (30) Klein, J. Pure Appl. Chem. 1992, 64 (11), 1577. (31) Miklavic, S. J.; Marcelja, S. J. Phys. Chem. 1988, 92 (23), 6718. (32) Neelov, I. M.; Borisov, O. V.; Binder, K. Macromol. Theory Simul. 1998, 7 (1), 141. (33) Raviv, U.; Tadmor, R.; Klein, J. J. Phys. Chem. B 2001, 105 (34), 8125. (34) Tsarkova, L. A.; Protsenko, P. V.; Klein, J. Kolloidn. Zh. 2004, 66 (1), 84. (35) Abe, T. H.; Higashi, N.; Niwa, M.; Kurihara, K. Langmuir 1999, 15 (22), 7725. (36) Balastre, M.; Li, F.; Schorr, P.; Yang, J. C.; Mays, J. W.; Tirrell, M. V. Macromolecules 2002, 35 (25), 9480. (37) Benz, M.; Chen, N. H.; Israelachvili, J. J. Biomed. Mater. Res., Part A 2004, 71A (1), 6. (38) Muller, M.; Lee, S.; Spikes, H. A.; Spencer, N. D. Tribol. Lett. 2003, 15 (4), 395. (39) Neelov, I. M.; Borisov, O. V.; Binder, K. J. Chem. Phys. 1998, 108 (16), 6973. (40) Schorr, P. A.; Kwan, T. C. B.; Kilbey, S. M.; Shaqfeh, E. S. G.; Tirrell, M. Macromolecules 2003, 36 (2), 389.
Sirchabesan and Giasson
our knowledge, reliable correlations between parameters controlling the interpenetration and the friction have not yet been established. In this study, we used a coarse-grained simulation approach to investigate the effects of the polymer conformation, compression, degree of ionization, and distribution of the charges on the interpenetration thickness and frictional properties of charged polymers end-grafted onto surfaces. Methodology Using coarse-grained simulations, we investigated the behavior of a diblock copolymer, poly(tert-butyl methacrylate)-b-poly(sodium sulfonate glycidyl methacrylate) or PtBMA-b-PGMAS, that has been previously studied experimentally using a surface forces apparatus and ellipsometry.26,41 The copolymer is composed of a short hydrophobic anchoring block, PtBMA, and a relatively long hydrophilic and ionizable block, PGMAS. Our previous studies have shown that PtBMA-b-PGMAS can form brushes on hydrophobic surfaces using selective self-adsorption in aqueous media.26,41 The hydrophobic PtBMA blocks adsorb on hydrophobic surfaces, while the nonadsorbing charged PGMAS blocks can dangle in the aqueous solution to form a brush conformation. The degree of polymerization of the anchoring block NPtBMA was fixed to 26 (Mw ) 3700), which corresponds to the previous experimental value.26,41 The degree of polymerization of the charged chains, the distance between the adjacent grafting sites s, the degree of ionization, the location of the charges along the chains, and the concentration of the added monovalent ions were used as variable parameters. The solvent used was water, and a monovalent salt, sodium chloride (NaCl), was used to investigate the Debye screening effect on the copolymer behavior. No boundary constraint was used so that the nonadsorbing parts of the grafted chains were free to explore all directions including the adjacent polymer-free aqueous reservoir. Two system sizes were simulated: 225 grafted chains in a 15 × 15 array representing relatively large interacting areas (102-103 times the polymer size) and 9 chains in a 3 × 3 array representing small areas ( rc) 0
}
χij + 78 0.231
(4)
where χ is the Flory-Huggins parameter defined as46,47
χij ) (∆Eij/RT)V
(5)
with V being the bead volume, ∆Eij being the cohesive energy per unit volume, R being the gas constant, and T being the absolute temperature. The cohesive energy was calculated using molecular dynamics stimulations (amorphous cell module in Material Studio). The interaction parameters aij used for our simulations are given in Table 1. The dissipative force FijD representing the viscosity of the fluid phase is governed by a friction coefficient γ according to the following equation:
FijD ) -γw(rij) (rˆijVij)rˆij (43) Groot, R. D. J. Chem. Phys. 2003, 118 (24), 11265. (44) Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 107 (11), 4423. (45) Groot, R. D.; Madden, T. J. J. Chem. Phys. 1998, 108 (20), 8713. (46) Case, F. H.; Honeycutt, J. D. Trends Polym. Sci. 1994, 2 (8), 259. (47) Espanol, Warren. Europhys. Lett. 1995, 30, 191. (48) Groot, R. D.; Madden, T. J. J. Chem. Phys. 1999, 110 (19), 9739. (49) Groot, R. D.; Rabone, K. L. Biophys. J. 2001, 81 (2), 725. (50) Yip, S. Handbook of Materials Modeling; Springer: Berlin, 2005. (51) Groot, R. D. Langmuir 2000, 16 (19), 7493.
water PGMAS PtBMA surface
water 78 83 121 118
PGMAS 83 78 81 81
(6)
PtBMA 121 81 78 65
surface 118 81 65 78
bi - b vj|. where w(rij) is a short-range weight function and Vij ) |v As outlined by Espanol, this function could be a Mexican hat, but for no specific reason.47 The random force FijR representing the Brownian motion of the polymer is expressed as
FijR ) σw1/2(rij)ζijrˆij
(7)
where ζij(t) is a delta-correlated stochastic variable with zero mean and σ is the noise amplitude describing the degree of freedom eliminated by the coarse-graining approach. The dissipative and random forces act as a heat source and sink, respectively, allowing angular momentum to be conserved. Espanol and Warren47 have shown that to reach the steady-state regime, corresponding to the Gibbs canonical ensemble, the following relations must be satisfied:
(
σij ) 0.75FkBTrCχij0.26 1 -
(3)
where rˆij is the unit vector joining beads i and j. The amplitude of conservative force aij depends on the level of coarse graining, and it can be mapped to the Flory-Huggins parameter as demonstrated by Groot and Warren.45 In our study, the number of water molecules per bead was set to 3 to reach the optimal degree of coarse graining which corresponds to the balance between the fastness of the simulation and the reliable physical representation of the system. The value of aij for water was set to 78 as reported in literature.46-48 The other values of aij were determined using the following equation:49-51
aij )
Table 1. Interaction Parameters aij between the Different Beads Used for the Simulations
)
2.36 χij
1.5
γij ) σij2/2kBT
(8) (9)
where T is the absolute temperature, kB is the Boltzmann constant, and χ is the Flory-Huggins parameter described by eq 5. Espanol and Warren have shown that the form of the weight function47
w(rij) )
{( ) 1-
0
rij rc
2
(rij e rc) (rij > rc)
}
(10)
allows the thermodynamic equilibrium to be maintained. The function w(rij) balances the heat generated by the random forces and the energy dissipated byFijD. The spring force FijS represents the flexibility and internal degrees of freedom of the polymer chains.53 Each polymer bead, composed of a constant number of repeat units, is assumed to be connected to an adjacent bead through a spring. A simple Hooke’s law is used to represent the force on a polymer bead i bounded to polymer bead j, FijS, with a spring of stiffness C:
FijS )
∑j Crij
(11)
The electrostatic force FijE between charged beads i and j was analyzed as reported in Groot’s study.43 According to this study, the charged beads are not treated as point charges. The electrostatic field is solved by smearing the charges over a lattice grid whose size is determined by a balance between the fast implementation and the correct representation of the electrostatic field. In our study, the grid size was set to rc. For each charged bead, a charge proportional to 1 - r/Re is assigned to every grid node within a radius Re in such a way that the sum of all charged nodes is equal to the charge of the bead. The electric field is solved according to the following equation:43 (52) Irfachsyad, D.; Tildesley, D.; Malfreyt, P. Phys. Chem. Chem. Phys. 2002, 4 (13), 3008. (53) Martin, J. I.; Wang, Z. G.; Zuckerman, D.; Bruinsma, R.; Pincus, P. J. Phys. II 1997, 7 (8), 1111.
9716 Langmuir, Vol. 23, No. 19, 2007
∇
(
Sirchabesan and Giasson
)
∇ψ ) -Fje,nΓ r0 n
(12)
where , r, and 0 are the values of the dielectric permittivity of the medium, the vacuum, and the water, respectively, at 25 °C, Fje,n is the averaged charge density, Γ is the coupling constant, and ∇ψn is the electric field gradient at lattice node n. The electrostatic force FijE is then given by43
((
∑n ∇ψn
FijE ) -qi
1 - |rn - rb|/Re
∑n
1 - |rn - rb|/Re
)
(13)
where qi is the number of unit charges of a charged bead, the sum runs over all the lattice nodes within the smearing radius Re, rn is the position of lattice node n, and rb is the position of the charged bead. The polyelectrolyte counterions are represented as charges with no mass and are localized around the beads according to the Poisson-Boltzmann equation. When using suitable values for the relative magnitudes of the different forces, the simulated system evolves into a steady-state regime corresponding to the Gibbs canonical ensemble. The integration of the equation of motion generates a trajectory from which thermodynamic observables (e.g., density fields, order parameters, correlation functions, and stress tensors) are determined. The Verlet algorithm modified by Groot43 was used to solve eq 1. The programs were written using FORTRAN 77. In our study, the equilibrium polymer conformations are determined from the relative equilibrium positions of the beads rij. The thickness of a single end-grafted polymer chain is defined as the distance between the farthest bead from the surface and the surface in the direction perpendicular to the surface Z. The thickness of the polymer layer is defined as the arithmetic mean of all single polymer chain thicknesses. Interpenetration thickness is defined as the distance over which two opposing polymer layers undergo mutual interpenetration. It is arbitrarily determined as the distance between the two mean positions of all free chain ends of the two opposing surfaces. The simulations of the shear behavior between two opposing polyelectrolyte brushes were carried out according the DPD model developed by Irfachsyad et al. for neutral brush-coated surfaces assuming a laminar Couette flow.52 The friction coefficient is defined as the ratio between the off-diagonal pressure tensor PXZ in the direction of the shear and the normal pressure tensor PZZ:
µ)
-PXZ PZZ
(14)
The pressure tensor is calculated using the Irving-Kirkwood equation:52
1 PRβ ) ( (VR)(Vβ)H(zi)) V i,R rR,ijrβ,ij 1 z - zi zj - z 1 θ (Fij) θ LXLY i 70 nm, no significant interaction was measured between adjacent chains but the chains adopt a relatively stretched conformation due to the electrostatic repulsion within the chains (Figure 1a). As the grafting distance decreases, an overlap of the repulsive interactions between adjacent chains arises and increases with decreasing s. To minimize the free energy of interaction, the dangling parts of the chains stretch away from the surface and form a brushlike conformation, as illustrated in Figure 1b and c. The critical grafting distance (CDG) is arbitrarily defined as the minimum grafting distance for which the overlap of repulsive interaction fields between adjacent chains is sufficient to move the beads over a mean distance of 0.1 nm away from the surface in the Z-direction (i.e., perpendicular to the surface). The value of CGD determined for PtBMA26-b-PGMAS97 (70% charged) is 62 nm, and it corresponds to 4 times the unperturbed chain length L determined using a surface forces apparatus.26 As a reference, repulsive interactions between adjacent grafted neutral polymer chains in good solvent begin to overlap at s equivalent to the unperturbed coil size, ∼aN3/5.4 As previously mentioned, charged polymer brushes can be obtained for relatively large grafting distances because of the presence of long-range electrostatic interactions between adjacent chains promoting chain stretching. Variations
BehaVior of Charged Polymer Brushes
Langmuir, Vol. 23, No. 19, 2007 9717
Figure 2. Variation of brush thickness for 9 end-grafted PtBMA26b-PGMAS97 chains (Mw(PGMAS) ) 23 000 and Mw(PtBMA) ) 3700) as a function of grafting distance s for different added salt concentrations Cs: (+) 0.0001 M, ([) 0.01 M, (9) 0.1 M, (2) 1 M, (×) 1.5 M, and (O) 2 M. Inset: Distribution of the added counterions around the copolymer chains for s ) 3.32 nm and Cs) 0.0001 M.
Figure 3. Variation of scaling product hs2/3 for 9 end-grafted PtBMA26-b-PGMAS97 chains as a function of salt concentration for different numbers of PGMAS monomers N: ([) 97, (9) 40, (2) 122, (×) 162, (/) 203, (b) 244, and (+) 304. The slope is ∼1/3 as predicted from eq 17. Table 4. Comparison between the Simulation and Experimental Data for PtBMA26-b-PGMAS97 (Mn(PGMAS) ) 23 900 and Mn(PtBMA) ) 3700)
Figure 1. Characteristic conformations adopted by 9 PtBMA26b-PGMAS97 (Mw(PGMAS) ) 23 000 and Mw(PtBMA) ) 3700) chains end-grafted onto a hydrophobic surface with no added salt and for different grafting distances s: (a) 70 nm, (b) 30 nm, and (c) 3.32 nm.
brush thickness (nm) grafting distance, s (nm)
added salt concentration, Cs (M)
simulation resultsa
experimental resultsb
10 6.1 3.32
0.01 0.1 1
15.7 ( 0.1 12.2 ( 0.1 7.4 ( 0.1
16.8 13.9 7.3
Table 3. Variation of the Critical Grafting Distance CGD with the Concentration of Added Salt (Cs) and the Molecular Weight of PGMAS (Mw(PGMAS))a a
CGD (nm)b salt concentration, Cs (M) Mw(PGMAS)
0.01
0.1
1
1.5
2
10 000 23 900 30 000 40 000 50 000 60 000 75 000
31 62 77 98 120 150 180
22 45 54 68 80 100 120
15 30 38 49 60 70 90
14 28 36 46 50 70 80
14 27 35 44 50 60 80
a The molecular weight of the PtBMA block is 3700. b The CGD is determined with a precision of (1 nm for Mw(PGMAS) ranging from 10 000 to 40 000 and (10 nm for Mw(PGMAS) greater than 40 000.
in the CGD with molecular weight and salt concentration are presented in Table 3. A linear increase in the CGD with PGMAS molecular weight is observed, as the contour length of highly charged polymers is expected to increase linearly with molecular weight.22,24,54 Moreover, the CGD decreases with increasing salt concentration as expected from the Debye screening effect.
Data from Figure 3. b Data from Abraham et al.26
The effects of the added salt concentration Cs and the grafting distance s on the brush thickness are shown in Figure 2. The brush thickness decreases with increasing grafting distance as the chains undergo structural changes from a brush to a flat conformation. The effect of added salt on the change in brush thickness is significant (>1%) for salt concentrations greater than ∼10-4 M. Our results show that, for salt concentrations less than 10-4 M, added counterions do not penetrate into the brush and extend beyond the brush surface as schematically represented by the inset of Figure 3. For salt concentrations greater than 10-4 M, the added counterions penetrate into the brushes and screen the electrostatic interactions between adjacent chains and along the chains, thereby causing the brush thickness h to decrease (Figure 3). Our simulations are in very good agreement with previous experimental studies26 (Table 4), and they follow the simple scaling model developed by Pincus:22 (54) Argillier, J. F.; Tirrell, M. Theor. Chim. Acta 1992, 82 (5), 343.
9718 Langmuir, Vol. 23, No. 19, 2007
hs2/3 ∝
number of repeat units [Cs]1/3
Sirchabesan and Giasson
(17)
as shown in Figure 3. Compression Studies. Simulations of the density profiles across the gap between two opposing surfaces bearing 9 and 225 grafted PtBMA26-b-PGMAS97chains with a degree of ionization of 70% and s ) 3.32 nm in the absence of added salt and at different compression ratios are presented in Figures 4-6. A grafting distance of 3.32 nm was used for comparison with neutral brushes arising at a grafting distance of ∼4 nm (aN3/5) for equivalent polymer molecular weights. The monomer density corresponds to the number of monomers per cubic nanometer, Z represents the axis perpendicular to the polymer-coated surfaces, D is the separation distance between the two opposing polymercoated surfaces, and L is the unperturbed brush thickness determined at a single wall (Figure 2). Figure 4a illustrates the density profiles between two surfaces bearing 9 grafted PtBMA26b-PGMAS97 chains at different separation distances corresponding to compression ratios D/2L ranging from 2.5 to 0.25. The total area covered by the polymer corresponds to 0.81 nm2. For D/2L > 2.5, no significant interaction was determined between the opposing chains and the resulting density profiles (for 0 < Z/D < 0.5 and 0.5 < Z/D < 1) exhibit a plateau as that observed for unperturbed neutral polymer brushes.16 In this regime, the unperturbed polymer chains adopt a brush conformation as illustrated in Figure 4b. As the compression ratio decreases, or as surface separation decreases, the plateau disappears progressively and the density profiles exhibit a maximum (Figure 5a). For small values of D/2L, the repulsive interactions between the opposing chains become significant and the free chain ends (dangling parts of the chains) tend to move out of the gap to minimize excluded volume effects. The maximum in the density profiles is associated with the lateral motion of the chains as depicted in Figure 4c. The monomer density at Z/D ) 0.5 (gap center) calculated over a distance of 1 nm is less than 10-4/nm3 for all compression ratios, which is negligible compared to the monomer density along the brush that is 10-1/nm3. For comparison, simulated density profiles between two opposing surfaces bearing 9 noncharged PtBMA26-b-PGMAS97 copolymer chains with a similar grafting distance, that is, s ) 3.32 nm, and for different compression ratios are illustrated in Figure 5. Negligible interpenetration between the opposing chains is observed for compression ratios larger than 1.75. However, our simulations show interpenetration of opposing chains for D/2L < 1, as also reported in the literature.53,55 The inherent charges of the polyelectrolyte chains give rise to additional repulsive forces between adjacent and opposing chains, relative to neutral brushes, which prevent a close proximity of the opposing charged chains. Figure 6 illustrates the density profiles between two surfaces bearing 225 grafted PtBMA26-b-PGMAS97 chains at different separation distances and under the same conditions as those used for simulations with 9 chains, that is, a degree of ionization of 70% and s ) 3.32 nm. The monomer density at the gap center is negligible ( 0.75) and increases as the separation distance decreases. However, the maximum monomer density at the gap center remains smaller (0.03/nm3) than the one observed with neutral chains (0.07/nm3) when considering equivalent compression ratios of D/2L ) 0.25. Moreover, the progressive increase in monomer density with a decrease in the compression ratio near the surface (55) Whitmore, M. D.; Baranowski, R. Macromol. Theory Simul. 2005, 14 (2), 75.
Figure 4. (a) Density profiles between two opposing surfaces bearing 9 end-grafted PtBMA26-b-PGMAS97 chains for different compression ratios D/2L: ([) 2.5, (9) 1.5, (2) 1, (×) 0.75, (/) 0.5, and (O) 0.25. Degree of ionization ) 70% and s ) 3.32 nm (no added salt); Z is the position along the axis perpendicular to the two surfaces bearing the polymers; and D is the separation distance between the two opposing surfaces bearing the polymers. (b) Conformation of the polymer chains at D/2L ) 2.5. (c) Conformation of the polymer chains at D/2L ) 0.25.
(Figure 6a) indicates that the free chain ends of the central chains fold in on themselves while the dangling parts of the side chains (close to the reservoir) tend to move toward the reservoir (Figure 6b). The density profiles agree relatively well with those from molecular dynamics simulations carried out for charged sodium poly(styrene sulfonate) using periodic boundary conditions.27 Significant interpenetration thickness is arbitrarily defined as a length longer than the monomer size that is 0.25 nm. The maximum interpenetration thickness with 225 chains is 0.28 nm, corresponding to 0.1% of the unperturbed chain length, and it occurs at the smallest D/2L ratio of 0.25 and for s ) 3.32 nm. The maximum interpenetration observed for 9 chains is 0.002
BehaVior of Charged Polymer Brushes
Figure 5. (a) Density profiles between two opposing surfaces bearing 9 end-grafted neutral PtBMA26-b-PGMAS97 chains for different compression ratios D/2L: ([) 1.75, (9) 1, (2) 0.75, (×) 0.6, (/) 0.5, and (O) 0.25. Degree of ionization ) 0% and s ) 3.32 nm (no added salt). (b) Conformation of the grafted neutral chains at D/2L ) 0.25.
nm under the same conditions of grafting density and compression ratio. Our simulations show that the extent of mutual interpenetration under relatively high applied load between the opposing chains depends on the number of interacting chains. For a relatively large number of chains (i.e., 225 chains), the opposing chains are constrained to slightly interpenetrate, whereas, for only few interacting chains (9 chains), all dangling parts of the chains can move laterally toward the polymer-free reservoir to avoid any significant interpenetration under applied load. As charged brushes are sensitive to the electrostatic screening effect, we simulated the compression behavior of two opposing surfaces bearing 9 and 225 end-grafted charged brushes (70% ionized) in the presence of added monovalent ions (Na+ and Cl-) in solution. Under the electrostatic screening effect (i.e., for salt concentrations greater than 10-4 M), a significant interpenetration between the opposing polymer chains was measured upon compression. To establish scaling relations between t and the variable parameters (s, N, D, and Cs), we carried out simulations for PGMAS molecular weights ranging from 10 to 50K, for s ranging from 3 to 20 nm, for Cs ranging from 10-5 to 1 M, and for different compression ratios (Figure 7). The variation of the interpenetration thickness as a function of each parameter was assessed independently. For conditions where the
Langmuir, Vol. 23, No. 19, 2007 9719
Figure 6. (a) Density profiles between two opposing surfaces bearing 225 end-grafted PtBMA26-b-PGMAS97 chains for different compression ratios D/2L: ([) 2.5, (9) 1.5, (2) 1, (×) 0.75, (/) 0.5, and (O) 0.25. Degree of ionization ) 70% and s ) 3.32 nm (no added salt). (b) Conformation of the end-grafted polymers at D/2L ) 0.5.
Figure 7. Correlation between interpenetration thickness t, number of PGMAS monomers N, concentration of added salt Cs, grafting distance s, and separation distance D for two opposing surfaces bearing 9 and 225 end-grafted chains. Degree of ionization of the PGMAS blocks ) 70%.
measured interpenetration thickness is significant (t > 0.25 nm), t is found to scale as
t ∝ N2/3Cs0.98s-2/3D-1/3
(18)
9720 Langmuir, Vol. 23, No. 19, 2007
Figure 8. Interpenetration thickness between two opposing surfaces bearing 9 end-grafted PtBMA26-b-PGMAS97 chains as a function of the degree of ionization of the PGMAS block for s ) 3.32 nm and D/2L ) 0.25. The charged brushes are distributed progressively on every monomer starting from the attached end (]) and from the free end (0) and homogeneously distributed along the chain (2).
where N is the number of repeat units of the hydrophilic block. For salt concentrations less than 10-4 M, the interpenetration thickness is independent of salt concentration and scales as t ∝ N2/3s-2/3D-1/3. The interpenetration thickness exhibits the same dependence on separation distance as the one theoretically predicted for neutral polymer brushes.55-57 Our simulations also showed that the long-range electrostatic interactions between chains are screened by the added counterions over a distance equivalent to the interpenetration thickness, since the contribution of the electrostatic force to the total force acting on the polymer beads within the interpenetration layer is negligible. This implies that interpenetration between the opposing charged chains occurs between electrostatically screened segments. However, the magnitude of t is significantly smaller than the value expected for neutral brushes when considering equivalent compression ratio, molecular weight, and grafting density (see Figure 8). This is explained by the fact that, within the charged brushes, the excluded volume effect is augmented by the additional osmotic pressure exerted by the mobile counterions. Therefore, a applied load can be supported by charged brushes with less interpenetration than that for neutral brushes. The effect of the degree of ionization and the distribution of charges along the polymer chain on the interpenetration thickness between two opposing grafted brushes is illustrated in Figure 8. These simulations were carried out for 9 chains with a grafting distance s ) 3.32 nm and at a small compression ratio D/2L ) 0.25 where the interpenetration thickness is at maximum. The mutual interpenetration between opposing chains is significantly sensitive to the degree of ionization and to the distribution of the charges along the chains. When the charges are distributed on every adjacent repeat unit along the chains from the grafted ends (close to the surface) to the free ends or homogeneously distributed throughout the chains, the interpenetration thickness between the opposing chains is negligible for a degree of ionization greater than 60%. When the charges are distributed from the free to the grafted ends of the chains, a degree of ionization of 40% is sufficient to prevent significant interpenetration. Below 40%, independently of the location of the charges along the chain, the electrostatic forces are not sufficient to prevent mutual interpenetration and the polymer chains behave as neutral polymers. (56) Klein, J. Annu. ReV. Mater. Sci. 1996, 26, 581. (57) Wijmans, C. M.; Zhulina, E. B.; Fleer, G. J. Macromolecules 1994, 27 (12), 3238.
Sirchabesan and Giasson
Figure 9. Friction coefficient as a function of interpenetration thickness between two opposing surfaces bearing 9 end-grafted charged ([) and 9 neutral (9) PtBMA26-b-PGMAS97 chains. s ) 3.32 nm and shear velocity V ) 400 nm/s. Inset: variation of the friction coefficient with N, Cs, s, and D.
Shearing. Simulations of the shear behavior between two opposing surfaces bearing 9 and 225 grafted PtBMA-b-PGMAS chains undergoing lateral sliding motion were carried out for different PGMAS molecular weights (10 000-50 000), grafting distances s (3-20 nm), salt concentrations Cs (0.0001-1 M), compression ratios D/2L (0.25-0.8), degrees of ionization (0 and 70%), and a constant shear velocity V of 400 nm/s which was used in our previous experimental study.8 At steady state, the sliding or traveled distance was larger than the separation distance between the two surfaces D. The scaling relation between the friction coefficient and the variable parameters s, Cs, N, and D is similar to the one obtained for the interpenetration thickness, that is, eq 18 (Figure 9). This similarity indicates that frictional dissipation occurs predominantly within the interpenetration layer. This behavior has also been theoretically predicted for neutral polymer brushes.56,58 Moreover, our simulations clearly show that the friction coefficient between electrostatically screened charged brushes is smaller than the one observed with neutral brushes when considering equivalent grafting density, molecular weight, and interpenetration thickness. At a given applied load, the interpenetration thickness between charged brushes is small compared to that of neutral brushes, due to the presence of additional electrostatic repulsions, so that the extent of the sheared interfacial region between the opposing charged brushes is smaller as well. Therefore, for a given interpenetration thickness, the applied load between charged brushes is greater than that for neutral brushes, thereby reducing the friction coefficient compared to that of neutral brushes. Since friction mainly arises from shearing the polymer segments within the interpenetration layer, the shear force can be normalized by the number of beads within the interpenetration layer. Figure 10 reports the values of friction or shear force as a function of the number of beads within the sheared interpenetration layer for neutral, charged, and electrostatically screened polymer segments. It is important to recall that the highly charged beads (70%) are electrostatically screened within the interpenetration layer, since significant interpenetration was measured only under screening effects. A significant mutual interpenetration between weakly charged brushes (40%) was measured with no added salt, that is, without the screening effect. Moreover, a given number of beads within the interpenetration layer is obtained under a (58) Kreer, T.; Muser, M. H.; Binder, K.; Klein, J. Langmuir 2001, 17 (25), 7804.
BehaVior of Charged Polymer Brushes
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charged, electrostatically screened, and neutral grafted polymer segments in water are equivalent. Nevertheless, our simulations support the notion that the main parameter controlling friction dissipation between the polymer brushes is the mutual interpenetration which largely depends on the charges along the chains and the presence of added ions.
Conclusion
Figure 10. Shear force as a function of the number of beads within the interpenetration layer between two opposing surfaces bearing 9 end-grafted PtBMA26-b-PGMAS97 chains for different degrees of ionization of the PGMAS block: (]) 70%, (×) 40%, and (0) 0%. The charges are homogenously distributed along the PGMAS block. s ) 3.32 nm and V ) 400 nm/s.
relatively high applied load between charged brushes, relative to neutral brushes, when considering all other parameters equivalent. The simulations show that the shear force per sheared bead is equivalent for charged, uncharged, and electrostatically screened polymer segments regardless of the presence of added salt. There is a slight difference in the shear force between charged and neutral brushes for large numbers of interpenetrating beads, but this cannot explain the significant difference in the frictional drag between charged and uncharged brushes previously reported.8 Indeed, we previously proposed that the polyelectrolyte counterions inducing polyethylene (PE)-brush swelling, together with the fluid hydration sheaths around the charged segments, were effective in overcoming the factors leading to larger frictional dissipation for all other types of polymeric lubricants studied.8 However, our simulations show that the friction forces between
The behavior of two opposing end-grafted polymer chains undergoing compression and shear motions were simulated using dissipative particle dynamics. Between two highly charged polymer brushes (degree of ionization greater than 60%) and in the absence of added ions, mutual interpenetration of the opposing chains upon compression is negligible for compression ratios up to 0.25, corresponding to a separation distance between the two surfaces of 50% of the unperturbed chain length. However, significant mutual interpenetration is predicted under the electrostatic screening effect, and the scaling relation between the interpenetration thickness t, the grafting distance s, the number of monomers N, and the separation distance between the two surfaces D is similar to that for neutral polymer brushes. The friction coefficient was found to be directly correlated to the extent of the interpenetration layer between the opposing polymer brushes and more specifically to the number of sheared polymer segments within the interpenetration layer. The presence of electrostatic interactions and added counterions within the interpenetrated chains was not shown to significantly affect the friction force between sheared polymer beads. Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council (NSERC). The authors thank Armand Soldera for fruitful discussions and for a critical analysis of the simulation approach and results. LA7009226
