Article pubs.acs.org/Langmuir
Lateral Interactions in Brush Layers of Bottle-Brush Polymers Erik Wernersson* and Per Linse Physical Chemistry, Department of Chemistry, Lund University, P.O. Box 124, S-221 00 Lund, Sweden ABSTRACT: We used isotension-ensemble Monte Carlo simulations to study the properties of brush layers of bottle-brush polymers under lateral compression. The polymers were represented by a freely jointed hard-bead model with one side chain grafted to each bead of the main chain, and we considered variations in side-chain length and bead size. Brush properties, including brush height and surface pressure, were analyzed in the context of a generalized box model. The surface pressure was found to have a steeper dependence on the grafting density than predicted by classical theories of polymer brushes. This discrepancy could be traced to the equation of state of the polymer fluid composing the brush, which was found to be more reminiscent of the concentrated regime than of the semidilute conditions normally expected in polymer brushes. The conformational properties of individual polymer molecules were found to be insensitive to lateral compression; in particular, the side-chain end-to-end distance remained essentially constant.
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INTRODUCTION Polymer brushes composed of surface-grafted branched polymers have attracted recent experimental interest on the basis that such polymers can provide a larger steric exclusion volume per grafting point than nonbranched polymers.1,2 Short branched polymers have been found to be effective as antifouling agents and for other purposes.3,4 However, many of the traditional tools of polymer physics are inapplicable in describing such polymers. For example, an existing scaling theory for branched polymers5 requires that the polymers are long and the branching points are far apart, which is not the typical situation for polymer antifoulants. Therefore, direct numerical computation methods, normally associated with small molecules, are preferred for the examination of short, branched polymers. A recent study using the Scheutjens−Fleer self-consistent field (SCF) theory showed that the presence of side chains in branched polymers with bottle-brush architecture increased the excluded volume per main-chain segment.6 Moreover, the SCF results were, in important respects, reproduced by a mean-field box model6 in the spirit of Alexander−de Gennes theory7,8 but further simplified through a mean-field assumption. In this box model, the side chains contribute to the segment density but not to the stretching elasticity. Under comparable conditions, i.e., the same main-chain contour length and number of segments per unit area, branched polymers thus are predicted to give rise to thicker brushes with a lower segment concentration than linear polymers do. A previous simulation study of the compression isotherms of bottle-brush polymers has shown notable deviations from mean-field theory.9 Here, we investigate the properties of brush layers formed by short, surface-grafted bottle-brush polymers with a high density of side chains of varying length. Our central question is how do brush properties such as surface pressure and brush height and polymer properties such as lateral width © 2014 American Chemical Society
and side-chain end-to-end separation depend on the side-chain length and bead size under various degree of lateral compression? We employ the isotension-ensemble Metropolis Monte Carlo simulation method to calculate the compression isotherms and structural properties of the brush layer. Our results are discussed in the framework of a generalized box model.
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MODEL AND METHOD
We considered a freely jointed bead model of surface-grafted bottlebrush polymers. The bottle-brush polymers are composed of a main chain with Nm = 24 beads plus a grafting bead constrained to a hard surface at z = 0. Each main-chain bead possesses one side chain containing Ns = 3, 7, or 15 beads. The total number of beads in each bottle-brush polymer is NT = Nm(1 + Ns) + 1. The beads are represented by hard spheres and are connected by rigid bonds. We consider bead diameters σ/d = 0.64, 0.8, and 1.0, with d denoting the bond length. We employed isotension-ensemble Monte Carlo (MC) simulations to compute compression isotherms, i.e., the surface pressure, Π, as a function of the surface area per polymer, S. This method is analogous to NPT ensemble MC, wherein the pressure is specified and the system volume is allowed to vary, but in isotension MC, Π is specified and S varies.10,11 This was accomplished by making moves that change the surface area, in which the system was expanded or compressed through an affine deformation of the xy plane. The individual polymer molecules were treated as rigid bodies. If the deformation resulted in bead overlap, then the move was rejected. Otherwise, it was accepted with the probability
p = min(1, e−ΔA Π / kT )
(1)
where ΔA is the change in surface area. Note that this acceptance rule implies that Π is the excess surface pressure, i.e., the surface pressure Received: July 22, 2014 Revised: August 26, 2014 Published: September 10, 2014 11117
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due to interactions, and corresponds to the situation where polymer molecules are grafted to the surface and there is no contribution to the surface pressure from the lateral motion of the polymers. In addition to area change moves, we employed whole polymer translation moves in the x and y directions and pivot rotations to sample the configuration space of the brush layer. The latter were accomplished by randomly selecting a bead and rotating a part of the molecule around a random axis passing through the center of that bead. For main-chain and grafting beads, the main-chain beads of the end not attached to the surface and all beads in side chains attached to these beads were moved. For side-chain beads, the beads of the chain ends not attached to the main chain were moved. When the last bead of a side chain was selected, the whole side chain was rotated around the main-chain bead to which it was grafted. In this way, all internal degrees of freedom of the polymer molecules were sampled. Moves were rejected if they resulted in bead overlap and accepted otherwise. A series of simulations with surface pressures Π/(kBT/d2) ranging from 0.003 to 0.3 with the pressures logarithmically separated were performed. Periodic boundary conditions were applied in the x and y directions. The system size was selected such that the simulation box contained ca. 5000 beads. Properties at infinite dilution were obtained by performing simulations of a system consisting of a single grafted polymer. The results of the simulations were interpreted in terms of a generalized box model (GBM). The free energy of this model, F, is defined by
F(H , S) = F el(H ) + F osm(V )
W=2
=0
S0 = πW0 2
∂F osm ∂S
= HP H
(2)
Figure 1. Height, H0 (circles), and width, W0 (crosses), of surfacegrafted bottle-brush polymers as a function of the side-chain length, Ns, at infinite dilution for bead radii σ/d = 0.64 (red), 0.8 (green), and 1.0 (blue).
number of side chains for each bead radius. Increased excluded volume of side chains causes the main chain to stretch. Both H0 and W0 increase sublinearly with Ns, and the curves shift upward with increasing σ. The increase in the height with Ns is faster than that of the width, suggesting that the side chains cause the main chain to align along the surface normal to some (small) extent. Figure 2 shows ρ(z) and ρs(R) and the corresponding values of H and W at three specified surface pressures, Π. The distribution ρ(z) develops from a distorted bell shape for large S appearing at low pressure toward a steplike profile for small S appearing at high surface pressure, a change which is associated with an increase in H. As the pressure increases, the distribution ρs(R) becomes more concentrated around the grafting point. Consequently, W becomes smaller. In Figure 3a, the brush height H is shown as a function of S/ S0. The relative variation of height over the whole range in S/S0 is small, on the order of tens of percent, and the height approaches that of an isolated polymer for large S/S0. While the values of H differ between different polymers for the same S/S0, the shape of the curves is notably similar. This is clearly visible in Figure 3b, where PS is shown as a function of H − H0, i.e., the difference in H compared to the low compression limiting value H0. At small surface pressure leading to small brush
(4)
(5)
Our simulation model is linked to the GBM by computing key quantities from simulated equilibrium averages. The thickness of the brush, H, was taken as H=2
∫ dzzρ(z) ∫ dzρ(z)
(8)
RESULTS AND DISCUSSION Structural Properties. In this subsection, we provide structural properties of the grafted bottle-brush polymers under lateral compression, both in the context of the GBM and in terms of properties of individual polymers in the brush. The GBM is not applicable in the infinite dilution limit, but the properties of polymers in this state are nevertheless of interest for reference. The heights, H0, and widths, W0, of isolated polymers are shown in Figure 1 as a function of the
which can be interpreted as a force balance between the main-chain elasticity and the (three-dimensional) osmotic pressure in the brush, for which we have introduced the symbol P. The surface pressure of the brush, Π, is related to P through
Π=−
s
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Insertion of the free-energy function of the GBM into eq 3 leads to dF el dF osm = −S = SP dH dV
2 1/2
where subscript 0 denotes that the quantities are evaluated in the limit of low surface density. Because the GBM contains no explicit information about the lateral extent of individual polymers, W plays no role in that model.
(3)
S
(7)
where R = (x + y ) and ρ (R) is the distribution of beads belonging to one polymer around its grafting point expressed as the number of beads per unit area. The intrinsic area of a polymer, S0, is taken to be 2
where Fel is an elastic free energy associated with stretching the polymer main chain and Fosm is the free energy associated with the interactions within the brush layer, H is the brush height, S is the surface area per polymer, and V = SH is the volume per polymer. Central to the GBM is that (i) Fel depends only on H and (ii) Fosm depends on product SH, not on S or H separately. We expect Fel to be determined mainly by the main-chain properties. Many existing theories of polymer brushes have a form consistent with the GBM, most notably, the Alexander−de Gennes theory,7,8 and each of these theories is characterized by some combination of Fel and Fosm expressions. Here, we do not postulate any particular expression for either term but merely employ the GBM theory, as defined by eq 2, as a framework for analyzing and interpreting our simulation data. Thermodynamic equilibrium requires that the brush height corresponds to a free-energy minimum
∂F ∂H
∫ dRR2ρs (R ) ∫ dRRρs (R )
(6)
where ρ(z) is the bead density profile and the integrals are taken over the whole space. From Π and H, the osmotic pressure can be computed using eq 5. The lateral extension of the polymer, W, was evaluated though 11118
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extension. Also, the positive deviation for the large main-chain extension is expected,12 in light of the fact that the extension starts to approach the main-chain contour length of 24d for the larger Ns at low S/S0 obtained by high lateral compression. In Figure 4a, the width W of the polymer evaluated through eq 7 is shown as a function of S/S0. The width shows the
Figure 2. (a) Concentration profiles, ρ(z), and (b) single-polymer segment distributions, ρs(R), for polymer with Ns = 7 and σ = 0.8d at surface pressures of Π/(kBT/d2) = 0.003 (red), 0.03 (green), and 0.3 (blue), which correspond to surface areas per polymer of S/d2 = 295, 119, and 52, respectively. The values of H and W are also given (vertical lines) in (a) and (b), respectively.
Figure 4. (a) Lateral width of a polymer W and (b) side-chain end-toend distance rsee as a function of S/S0 for Ns = 3 (red symbols), 7 (green symbols), and 15 (blue symbols) and σ/d = 0.64 (circles), 0.8 (squares), and 1.0 (crosses).
opposite behavior compared to that of H in that it decreases by a comparable amount over the same range in S/S0. The sidechain end-to-end distance, rsee, shown in Figure 4b, changes very little over the same range in S/S0. This independence of the side-chain extension on density has been observed before. It indicates that the lateral compression of polymers causes a stretching of the main chain and is therefore intimately related to the change in brush height. The extension of the main chain affects the average density in the brush, unlike the stretching or compression of side chains.6 Compression Isotherms. Compression isotherms for the bottle-brush polymer brushes are shown in Figure 5. In Figure 5a, the compression isotherms are expressed in terms of the surface pressure, Π, and the reduced surface area per polymer, S/S0. The surface pressure nearly follows a power law Π ∝ S−ν with ν ≈ 3, which is significantly faster than that in a mean-field theory with ν = 5/3 or in the Alexander−de Gennes theory7,8,13 with ν = 11/6. For a similar model of bottle-brush polymers, an apparent scaling with ν ≈ 2.7 was previously obtained.9 Here, we observe that ν varies only slightly with Ns and σ. In the vicinity of S/S0 = 1, the surface pressures for the different values of σ are superimposed, while they differ for small S/S0. This suggests that in layers of low grafting density the bead size affects the isotherms mainly through the swelling of individual polymers. In Figure 5b, the osmotic pressure as defined in the GBM is presented in terms of the reduced pressure PV0, where V0 =
Figure 3. (a) Brush height H as a function of the reduced surface area per polymer, S/S0, and (b) PS as a function of H − H0 with H0 denoting the height at infinite dilution for Ns = 3 (red symbols), 7 (green symbols), and 15 (blue symbols) and σ/d = 0.64 (circles), 0.8 (squares), and 1.0 (crosses).
extension, the data falls on a linear master curve, while for large extensions the data for polymers with larger Ns progressively deviate upward. This dependence is broadly consistent with the GBM. The linear behavior of PS for small extension corresponds, according to Figure 3b, to the backbone behaving as a harmonic spring. This is the expected behavior for small 11119
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The properties are determined mainly by the properties of the individual beads rather than their connectivity. Thus, the semidilute regime, which is the domain of both Alexander−de Gennes and mean-field theory, is bypassed, and the steeper isotherms associated with the concentrated regime are found instead. The literature shows that this behavior is not unique to bottle-brush polymers;16,17 isotherms that are steeper than expected from theory have been found in simulations of brushes of linear polymer where the chains were kept short, presumably for feasibility reasons. The considerably reduced pressure in the brush compared to that in the hard-sphere fluid may have practical consequences. The brush equation of state determines the free-energy cost due to lateral interactions of introducing an additional polymer. Namely, the chemical potential of a polymer in the brush is given by μ(S) = μ0 +
S
∫∞ Π(S′) dS′
(9)
where μ is the chemical potential in the infinitely dilute standard state. A smaller surface pressure anywhere in the range in S′ from ∞ to S gives rise to a smaller μ, with μ0 being equal. The smaller but more rapidly increasing pressure in the polymer fluid compared to the disconnected hard-sphere fluid means that the free-energy cost due to lateral polymer− polymer interactions is smaller than that which mean-field theory would suggest. For brushes formed by polymer adsorption from solution, this allows a higher segment density for the same adsorption strength. Thus, the steep isotherms seen here for bottle-brush polymers suggest that this type of polymer is especially suitable for creating brush layers by adsorption. Effectively, much of the free-energy cost of creating a dense segment fluid is paid at the time of polymer synthesis and assembling them into a brush layer, which becomes correspondingly cheaper. 0
Figure 5. (a) Surface pressure, Π, as a function of the reduced surface area per polymer, S/S0, and (b) the reduced (three-dimensional) osmotic pressure PV0 as a function of the bead volume fraction, Vb/V0, for Ns = 3 (red symbols), 7 (green symbols), and 15 (blue symbols) and σ/d = 0.64 (circles), 0.8 (squares), and 1.0 (crosses). In (a), power laws Π ∝ S−ν at indicted values ν are given (dashed lines), and in (b) the Carnahan−Starling equation of state (full curve) and the second (short-dashed line) and third (long-dashed line) virial terms are given.
4πσ3/3 is the bead volume, as a function of the reduced volume per bead Vb/V0, where Vb = HS/NT. The isotherms for the different values of Ns appear to converge at high compression (low Vb/V0). Moreover, for σ = d, the three curves for different values of Ns cluster around the line corresponding to the third virial term of the hard-sphere equation of state. For σ ≠ d, the dependence is similar but the curves are shifted. For high densities appearing at high compression, the bead equation of state thus appears to be close to P ∝ Vb−3. Similar findings have been reported for bulk solutions of highly branched polymers.14 Presumably, the apparent absence of the second virial term is due to the screening of excluded volume interactions. Connecting hard spheres to form polymers causes their respective excluded volumes to overlap and increases the free volume in the system compared to that in the monomer fluid. This is true for linear and branched polymers alike, but a much higher local bead density can be obtained with branched polymers. The highly branched polymers considered here might be conceptualized as globules of concentrated polymer solution, where the excluded volume of each bead corresponds to a density far higher than the system average. Even when the average concentration of beads is low, the local environment of each bead resembles a concentrated solution. Note that recent theoretical calculations suggest that the concentrated regime may be encountered at relatively modest volume fractions.15 Comparing the representations of the isotherms in Figure 5a,b gives the impression that the system passes from the “mushroom” regime, characterized by weakly overlapping polymers, directly into a brush in the concentrated regime.
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CONCLUSIONS The results presented here show that the conceptual framework of the box model that forms the basis of the classical theories of polymer brushes can be applied to short, branched polymers. They also suggest that the failure of common theories for polymer brushes, that predict too slow of a variation in surface pressure with grafting density in this regime, is due to the fact that brushes of short, branched polymers are not well described by theories pertaining to the semidilute regime because the conditions in the brush are more similar to those of concentrated solutions. Furthermore, the present results suggest a rationale for why it appears advantageous to use short, branched polymers for antifouling purposes. The local concentration of monomers in the individual branched polymers is higher, and less additional work must be performed to create brush layers of high monomer density.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Financial support by the Swedish Research Council (VR) through the Linnaeus grant Organizing Molecular Matter 11120
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(OMM) center of excellence (239-2009-6794) and through individual grants to P.L. (2010-2253-78321-47) is gratefully acknowledged.
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REFERENCES
(1) Peleshanko, S.; Tsukruk, V. V. The Architectures and Surface Behavior of Highly Branched Molecules. Prog. Polym. Sci. 2008, 33, 523−580. (2) Gunkel, G.; Weinhart, M.; Becher, T.; Haag, R.; Huck, W. T. S. Effect of Polymer Brush Architecture on Antibiofouling Properties. Biomacromolecules 2011, 12, 4169−4172. (3) Blaszykowski, C.; Sheikh, S.; Thompson, M. Surface Chemistry to Minimize Fouling from Blood-Based Fluids. Chem. Soc. Rev. 2012, 41, 5599−5612. (4) Chen, Y.-W.; Chang, Y.; Lee, R.-H.; Li, W.-T.; Chinnathambi, A.; Alharbi, S. A.; Hsuie G.-H. Adjustable Bioadhesive Control of PEGylated Hyperbranch Brushes on Polystyrene Microplate Interface for the Improved Sensitivity of Human Blood Typing Langmuir 2014, 30, 9139−9146. (5) Zhulina, E. B.; Vigis, T. A. Scaling Theory of Planar Brushes Formed by Branched Polymers. Macromolecules 1995, 28, 1008−1015. (6) de Vos, W. M.; Leermakers, F. A. M.; Lindhoud, S.; Stuart, W. P. Modeling the Structure and Antifouling Properties of a Polymer Brush of Grafted Comb-Polymers. Macromolecules 2011, 44, 2334−2342. (7) Alexander, S. Adsorption of Chain Molecules with a Polar Head a Scaling Description. J. Phys. (Paris) 1977, 38, 983−987. (8) de Gennes, P. G. Conformations of Polymers Attached to an Interface. Macromolecules 1980, 13, 1069−1075. (9) Wernersson, E.; Linse, P. Spreading and Brush Formation by End-Grafted Bottle-Brush Polymers with Adsorbing Side Chains. Langmuir 2013, 29, 10455−10462. (10) Svensson, B.; Woodward, C. Simulations in Planar Slits at Constant Chemical Potential. J. Chem. Phys. 1994, 100, 4575−4581. (11) Turesson, M.; Woodward, C. E.; Åkesson, T.; Forsman, J. Simulations of Surface Forces in Polyelectrolyte Solutions. J. Phys. Chem. B 2008, 112, 5116−5125. (12) Lai, P.-Y.; Halperin, A. Polymer Brush at High Coverage. Macromolecules 1991, 24, 4981−4982. (13) Naji, A.; Seidel, C.; Netz, R. R. Theoretical Approaches to Neutral and Charged Polymer Brushes. Adv. Polym. Sci. 2006, 198, 149−183. (14) Wang, T.-Y.; Fang, C.-M.; Sheng, Y.-J.; Tsao, H.-K. Osmotic Pressure and Virial Coefficients of Star and Comb Polymer Solutions: Dissipative Particle Dynamics. J. Chem. Phys. 2009, 130, 124904. (15) Suematsu, K. Concentration Dependence of Excluded Volume Effects. Colloid Polym. Sci. 2012, 290, 481−490. (16) Grest, G. S. Grafted Polymer Brushes: A Constant Surface Pressure Molecular Dynamics Simulation. Macromolecules 1994, 27, 418−426. (17) Dimitrov, D. I.; Milchev, A.; Binder, K. Polymer Brushes in Solvents of Variable Quality: Molecular Dynamics Simulations Using Explicit Solvent. J. Chem. Phys. 2007, 127, 084905.
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