Article pubs.acs.org/Langmuir
Spreading and Brush Formation by End-Grafted Bottle-Brush Polymers with Adsorbing Side Chains Erik Wernersson* and Per Linse Physical Chemistry, Department of Chemistry, Lund University, Box 124, SE-221 00 Lund, Sweden ABSTRACT: We investigate structural and thermodynamic properties of surfacegrafted layers of model “bottle-brush” polymers by Monte Carlo simulation. The polymers consist of a longer main chain densely grafted with shorter side chains, of which the latter have some degree of affinity to the surface. Our focus is on the effect of the side-chain surface affinity on the brush properties, which we study in terms of compression isotherms spanning a broad range of grafting densities. For low grafting densities, side-chain adsorption causes the polymers to spread on the surface. As the grafting density is increased, the layer goes through a “pancake-to-brush” transition to form a brush with the main chains aligned perpendicular to the surface. We find that side-chain adsorption is decisive for the structure of dilute layers and in the transition region but has little influence on the properties of dense brushes. The close relation between compression and adsorption isotherms is discussed, and the implications of side-chain adsorption for the ability of the polymer to form a dense brush are investigated. This analysis suggests that side-chain surface affinity alone will not give rise to “brush of bottle-brushes” layers by adsorption of polymers from solution, in agreement with recent experimental results.
■
INTRODUCTION Dense layers of end-attached polymers, commonly known as polymer brushes,1 can be used to induce steric repulsion between surfaces and other objects. This makes such structures useful for promoting colloidal stability2 and lubrication3 and for preventing biofouling.4 Polymer brushes can be created on a surface either by in situ synthesis of the polymer (“grafting from”) or from ready-made chains, which can be covalently (“grafting to”) or noncovalently (“adsorption”) attached. It is difficult, however, to achieve dense brushes via the latter approach because polymer chains are sterically repelled by the partially formed brush. Also, the availability of sites for adsorption or grafting may be limiting for the brush density. Polymer brushes can be formed from branched as well as linear polymers.5 As highly branched polymers offer the potential for attaining a high segment density per grafting or adsorption site, such are likely to offer an advantage over linear polymers in situations where the density of sites is the limiting factor for the achievable brush density. Synthesis of brush layers of “bottle-brush” polymers by grafting from various surfaces has been described,6−8 where the polymer is composed of a highly and regularly branched main chain with nonbranched side chains. Such layers have also been created by deposition of complex coacervate core micelles containing a diblock copolymer with a polyelectrolyte block and a bottle-brush block.9 The antifouling properties have been tested in two instances and found favorable.8,9 Brushes of bottle-brush polymers have been considered theoretically using scaling theory,10 density functional theory,11 and numerical self-consistent field (SCF) theory.12 Some simulation works on brush layers of bottle-brush polyelectrolytes, where the side-chains have charged groups, have recently been reported.13−16 In refs 11 and 12, the antifouling properties of the model brushes were investigated. The © 2013 American Chemical Society
calculations reported in ref 11 indicate that brushes of branched polymers, including bottle-brush polymers, are more effective than linear polymers in excluding small spherical particles from the surface region for the same number of polymer molecules per surface area. Reference 12 reports, however, that brushes of bottle-brush polymers with a larger proportion of beads in the side chains are less effective in repelling a sphere for the same number of polymer segments per surface area. Mean-field theory calculations have been reported for the adsorption of bottle-brush copolymers, composed of cationic and grafted beads, on a negatively charged surface. A diblock architecture, where the grafted beads form a separate bottle-brush block and the charged beads a polyelectrolyte block, was predicted to give rise to a “brush of bottle-brushes” structure.17,18 Recently, a closely similar system has been investigated experimentally, confirming the qualitative expectations from theory.19 In this work, we perform coarse-grained simulations to investigate the situation where bottle-brush polymers are each strongly attached by one of the ends of the main chain and where the side chains have some degree of affinity to the surface. The analogous situation for linear polymers, namely strongly end-attached, weakly adsorbing chains, have been thoroughly investigated using theory20−22 as well as experiments.23,24 The typical behavior is that the polymers spread on the surface for low grafting densities and form a brush for high grafting densities, similar to that formed by nonadsorbing chains. The transition between these two regimes is commonly referred to as the “pancake-to-brush” transition, and is well understood for linear polymers. The structural consequences of side-chain adsorption in a lattice model of a single end-attached Received: June 10, 2013 Revised: July 22, 2013 Published: August 7, 2013 10455
dx.doi.org/10.1021/la4021959 | Langmuir 2013, 29, 10455−10462
Langmuir
Article
bottle-brush polymer have been studied in detail.25 To the best of our knowledge, however, the behavior of branched, adsorbing polymers at finite grafting densities has not previously been studied by simulation. A simple bead−spring model of an end-grafted bottle-brush polymer is employed. The properties of this model are evaluated using Metropolis Monte Carlo (MC) simulation. We first characterize the model system by considering the spreading due to side-chain adsorption of a single bottle-brush polymer on a surface. Then, we investigate the effects of finite grafting density via compression isotherms for surface-grafted but laterally mobile bottle-brush polymers. From the compression isotherms we estimate the excess chemical potential as a function of the grafting density, which serves as a measure of lateral interactions between polymers. The paper concludes with a discussion on the close relation between compression isotherms and adsorption isotherms and their experimental relevance.
■
where z is the bead−surface distance. The potential is given in a form such that εw corresponds to the depth of the potential minimum; uw crosses zero at z = (2/15)1/6σw ≈ 0.715σw. σw was taken equal to σ in all case. To model purely repulsive interactions, we used a truncated and shifted Lennard-Jones 9−3 potential ⎧ ⎛2 σ 9 σw 3 3 w ⎪ εw ⎜ + ⎪ 9 − z3 u w (z) = ⎨ 10 ⎝ 15 z ⎪ ⎪0 ⎩
We performed MC simulations for a system composed of one or several bottle-brush polymers grafted to an idealized surface. The polymers were represented by a bead−spring model, in which the main chain consists of 24 beads, each grafted with a side chain consisting of 8 beads. A schematic illustration of the polymer connectivity and a snapshot of a single polymer are shown in Figure 1.
Figure 1. Schematic illustration showing the connectivity of the polymer (panel a) and a snapshot of a single polymer (panel b). Color indicates the designation as main-chain (red), side-chain (blue), and grafting (green) beads. The bead size in terms of the Lennard-Jones σ parameter, see eq 1, is indicated in panel a.
All bead−bead nonbonded interactions were given by the LennardJones potential
(1)
where r denotes the bead−bead distance, with ε = 0.14 kBT. Below, lengths are given in units of σ. The second virial coefficient of the bead fluid is positive, ∼0.8 σ3, which corresponds to good solvent conditions. A 5 σ cutoff was applied to all bead−bead interactions. The springs were harmonic, i.e., associated with the interaction energy 2
us(r ) = ks(r − r0)
γ = (P − P )h
(2)
⎛2 σ 9 σw 3 ⎞ 3 w ⎟ εw ⎜ − 10 ⎝ 15 z9 z3 ⎠
(4)
(5)
where h is the box length in the direction perpendicular to the surface. In the grafted systems, the bulk pressure P is zero and the surface tension is γ = −P||h. Below, the lateral pressure will be given as the surface pressure π = −γ. The compression isotherms, in terms of the surface area per polymer molecule, Sm, were obtained by carrying out isotension ensemble simulations for a series of surface pressures ranging between ∼3 × 10−3 and 3 kBT/σ2 and spaced approximately equally on a
for each bond. The equilibrium distance r0 was set equal to σ and the force constant ks to 37.5 kBT/σ2. The bead−surface interaction was of the Lennard-Jones 9−3 type
u w (z) =
if z > (2/5)1/6 σw .
Two bead-surface interaction types were considered: (i) all beads interacting via truncated and shifted Lennard-Jones 9−3 potential, eq 4, with εw = 1.5 kBT and (ii) main-chain beads subject to the same truncated and shifted Lennard-Jones 9−3 potential and side-chain beads interacting with the Lennard-Jones 9−3 potential, eq 3, with εw between 0.5 and 3.5 kBT. These cases are referred to as nonadsorbing side chains and adsorbing side chains. The grafting to the surface was effected by adding a “grafting bead” at the end of the backbone. This bead was constrained to move within the plane located at z = zg = σ but was otherwise identical to the other beads. The length of the simulation box in the z-direction, perpendicular to the grafted surface, was sufficient that no bead would ever encounter the far edge of the box. Periodic boundary conditions were applied in the x and y directions. Single bead and whole polymer translation MC moves as well as pivot rotations of parts of both the main chains and side chains were used. Grafting-bead and whole-polymer translation moves were made only in the x and y directions to preserve the grafting. The lateral mobility of the polymers ensured that the equilibrium distribution with respect to polymer positions was sampled. For pivot rotations, the nongrafted end of the chain was rotated, so main-chain pivot moves did not displace the grafting bead and side-chain pivot moves did not displace the main chain. For main-chain pivot moves, the side chains grafted to moved beads were moved as well. All simulations were carried out using the MOLSIM program package. Spreading of a Single Polymer. The free energy of spreading a single polymer, ΔAspread, i.e., the change in free energy associated with change from nonadsorbing to adsorbing side-chain−surface interaction, was calculated by thermodynamic integration in the canonical ensemble.26 Adsorbing side chains with εw = 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 kBT were considered. The free energy increment between nonadsorbing side chains and adsorbing side chains with εw = 0.5 kBT as well as that between adjacent values of εw for adsorbing side chains was calculated using the trapezoid rule with nine equally spaced intervening points, for a total of eleven points in the quadrature. The length of each simulation was 106 MC cycles, and each was preceded by an equilibration run of 105 cycles. Isotension Simulations of Compression Isotherms. We computed compression isotherms for a system containing 24 polymers. For this purpose, we employed MC simulations in the isotension ensemble.27,28 This ensemble is closely analogous to the isothermal−isobaric ensemble,26 except that the system is permitted to deform in two dimensions rather than three. Thus, in addition to the MC moves detailed above, area-change moves were also made. These were accomplished by rescaling the x and y coordinates of each bead. The acceptance rule was analogous to that for isothermal−isobaric MC.27 In the isotension ensemble, the isotropic pressure P of the isothermal−isobaric ensemble is replaced by the lateral pressure P||. The surface tension is
MODEL AND METHOD
⎛ σ 12 σ6 ⎞ unb(r ) = 4ε⎜ 12 − 6 ⎟ ⎝r r ⎠
10 ⎞ ⎟ if z ≤ (2/5)1/6 σw 3 ⎠
(3) 10456
dx.doi.org/10.1021/la4021959 | Langmuir 2013, 29, 10455−10462
Langmuir
Article
logarithmic scale. The system was equilibrated starting from an equilibrium configuration for an adjacent pressure, except for the “starting pressure” π = 1.2 × 10−2 kBT/σ2, where a random starting configuration was used. The length of the equilibration run was between 105 and 5 × 105 MC cycles, and the length of the production run was 105 cycles. Each run was carried out in five statistically independent replicas.
■
RESULTS AND DISCUSSION Spreading of a Single Polymer. In this section, the spreading of a single polymer on a surface is studied to obtain the properties of the model in the dilute limit. The dependence of the bead concentration profile and the polymer extension on εw is shown in Figure 2. Here, the bead number density profiles,
Figure 3. Free energy of spreading of a single bottle-brush polymer (squares) as well as its entropic (diamonds) and enthalpic (circles) components, for adsorbing side chains for varying εw. The free energy of spreading is the difference in free energy between polymer with nonadsorbing and with adsorbing side chains, see text.
of ΔAspread increases in a superlinear fashion, to eventually approach a linear trend for εw > 2.5 kBT. The lowest of the values of εw considered in the isotension MC calculations, 1.5 kBT, is at the lower end of the range of εw that gives rise to spreading of the side chains on the surface. As can be seen from Figure 2, there is a considerable population of beads that are not close to the potential minimum. For εw = 3.0 kBT, on the other hand, a majority of the beads are in direct contact with the surface. These values of εw may therefore be regarded as representative of what might be called “marginal adsorption” and “moderate adsorption” regimes, respectively. Note, especially, that the free energy of spreading differs by more than a factor of 5 between εw = 1.5 and 3.0 kBT. Layers of Finite Grafting Density. The compression isotherms obtained from the isotension MC simulations are shown in Figure 4. For nonadsorbing side chains the
Figure 2. Bead concentration profiles, normalized by the number of beads per surface area, Γ, for nonadsorbing side chains and adsorbing side chains with varying εw, as indicated by the curve labels. The insert shows a measure of the of the bead mean distance from the surface for adsorbing side chains, see text. The arrow indicates the value for nonadsorbing side chains.
ρ(z), are normalized by the number of beads per surface area, Γ, to remove the trivial surface area dependence. In this representation, the density profiles correspond to the distribution of bead z positions. No distinction is made between side- and main-chain beads, but grafting beads are excluded. For purely repulsive interactions, the density profile has a distorted bell-shape, characteristic of an end-tethered, but otherwise mildly perturbed, bottle-brush polymer. The first change that becomes apparent as attractive interactions are introduced is that the maximum of the bead distribution shifts toward the surface with increasing εw. This is a consequence of the polymer aligning the backbone parallel to the surface to maximize bead-surface interaction. For further increase in εw, there is a gradual change from a broad toward a sharply spiked distribution, of which the latter is indicative of the polymer spreading on the surface. This change is reflected also in the dependence of the bead−surface mean distance h = 2⟨z − zg⟩ on εw, shown in Figure 2. The factor two is included to ensure comparability to the analogous measure of the thickness of polymer brushes. Except for the region near εw = 0, the bead mean-distance decreases roughly exponentially with increasing εw. The free energy change associated with side-chain adsorption, as well as its enthalpic and entropic components, is shown in Figure 3. As should be expected, the entropy decreases as the polymer spreads on the surface to maximize the attractive interaction. For weak adsorption, εw ≤ kBT, the change in free energy is small. As εw is increased, the magnitude
Figure 4. Compression isotherms for grafted layers of bottle-brush polymers with nonadsorbing side chains (red circles), and adsorbing side chains with εw = 1.5 kBT (green squares) and 3.0 kBT (blue diamonds). The full line is the ideal contribution to the surface pressure, πid = kBT/Sm. The dashed line corresponds to the scaling law π ∝ Sm−8/3. Error bars give the standard deviation of Sm for a set of five statistically independent simulations.
compression isotherm is remarkably simple; a single apparent scaling relation captures the behavior of the isotherm between the dense-brush (small Sm) and dilute (large Sm) regimes. The surface pressure appears to scale with Sm as a power law with an exponent close to −8/3 for high compression, significantly different the mean-field value of −5/3.29 For polymers of a 10457
dx.doi.org/10.1021/la4021959 | Langmuir 2013, 29, 10455−10462
Langmuir
Article
density (large Sm) to an almost rectangular shape at high grafting density (small Sm). The height of the brush increases from about 14 σ to about 22 σ as the brush is compressed over the whole range in Sm. For adsorbing side chains, the brush height is always smaller, and especially so for large Sm, where the brush height is almost independent of Sm. This reflects the spreading of the polymer on the surface. Again, this behavior is also seen in Figure 2. The onset of appreciable Sm-dependence occurs for Sm smaller than about 200 σ2, a value close to that which would correspond to a close-packed bead monolayer. For smaller Sm, the brush height increases with decreasing Sm at a similar rate for adsorbing side chains with both values of εw. Concurrent with the increase in brush height, the shape of the density profiles appears to approach that for nonadsorbing side chains. This corresponds to a gradual reversal of the spreading of the polymer due to the lateral compression, to form a brush similar to that obtained for nonadsorbing side chains for small Sm . Figure 6 shows representative configurations from two simulations for strongly adsorbing side chains. Panel a
length comparable to the main chains of the polymers considered here, mean-field theory scaling would not necessarily be expected; similar non-mean-field scaling behavior has been observed previously for short, linear polymers at high grafting density.30,31 However, no theoretical explanation has been furnished. We are surprised to find a non-mean-field apparent scaling law over such a large range in the grafting density; there is no clear indication of a gradual transition between a “mushroom” and a “brush” regime in the pressure isotherms. Work is currently under way to elucidate the physical origins of this apparent scaling law. At lower compression, the pressure approaches the scaling law corresponding to a two-dimensional ideal gas, as it should for laterally mobile polymers. For adsorbing side chains, the isotherms converge to that for the nonadsorbing case at high compression. At intermediate compression, the dependence of the pressure on Sm shows a plateau. As will be shown below, and as has been observed for adsorbing linear polymers,22,32 this shape of the compression isotherm reflects the transition from a flat to a brush-like adsorbed layer. Notably, the plateau occurs for a similar range of Sm for εw = 1.5 and 3.0 kBT. The structural transition between flat and brush-like regimes is not a phase transitions, but the possibility of a surface phase transition in some region of parameter space cannot be ruled out. In Figure 5, bead density profiles and the corresponding brush heights are shown. For nonadsorbing side chains, the shape of the density profile goes from a skewed bell-shape, similar to that for the single polymer, Figure 2, at low grafting
Figure 6. Snapshot from two of the simulations for strongly adsorbing side chains, εw = 3.0 kBT, displaying main-chain (red), side-chain (blue), and grafting (green) beads. Panel a corresponds to Sm ≈ 130 σ2 and panel b to Sm ≈ 70 σ2, cf. panels e and f of Figure 5.
corresponds to an Sm just above that at the onset of the transition to a brush-like structure and panel b to an Sm just below it. The snapshots illustrate the origin of the qualitative change in the density profiles corresponding to the onset of exponential dependence of brush height on Sm: a subpopulation of polymers with the back-bones aligned perpendicular to the surface appears. To quantify the conformational state of the polymer for different surface pressures, we calculated the radius of gyration Figure 5. Bead density profiles, in terms of volume fraction with the bead volume taken to be Vb = (πσ3)/6, for all values of Sm considered (panels a, c, and e) and brush height, h = 2⟨z − zg⟩, as a function of Sm (panels b, d, and f) for bottle brush polymer with nonadsorbing side chains (panels a and b) and for adsorbing side chains with εw = 1.5 kBT (panels c and d) and εw = 3.0 kBT (panels e and f). As both the brush thickness and density increases with decreasing Sm, the curves are nested in the order of increasing Sm, as indicated by the arrow in panel a. The color-coding is matched between adjacent panels: a and b, c and d, and e and f.
⟨(rg)2 ⟩ =
1 N
∑ (xi − xcog)2 + (yi − ycog )2 + (zi − zcog)2 i
(6)
as well as the radius of gyration of the projection of the chains on the xy plane ⟨(rgxy)2 ⟩ = 10458
1 N
∑ (xi − xcog)2 + (yi − ycog )2 i
(7)
dx.doi.org/10.1021/la4021959 | Langmuir 2013, 29, 10455−10462
Langmuir
Article
where xcog (ycog, zcog) is the x (y and z) coordinate of the chain center of geometry and the summation is over all the N beads in a chain. The main chain and the side chains were considered separately. As we are studying lateral compression, rxy g is the quantity of main interest. In Figure 7, the distributions of main-chain rxy g is shown for the range of Sm together with the averages ⟨rxy g ⟩ and ⟨rg⟩ as
entalpically favored spread state in the transition region. The divergence between ⟨rxy g ⟩ and ⟨rg⟩ is concurrent with the appearance of perpendicularly aligned polymers, see Figures 5 and 6. The variation in side-chain radius of gyration is shown in Figure 8. For nonadsorbing and weakly adsorbing side chains,
Figure 8. Side-chain ⟨rxy g ⟩ (circles) and ⟨rg⟩ (squares) for all values of Sm considered, for bottle brush polymer with nonadsorbing side chains (panel a) and for adsorbing side chains with εw = 1.5 kBT (panel b) and εw = 3.0 kBT (panel c). The color-coding is the same as in Figure 7.
Figure 7. Distributions of main-chain rxy g for all values of Sm considered (panels a, c, and e) and the averages ⟨rxy g ⟩ (circles) and ⟨rg⟩ (squares) as a function of Sm (panels b, d, and f), for bottle-brush polymer with nonadsorbing side chains (panels a and b) and for adsorbing side chains with εw = 1.5 kBT (panels c and d) and εw = 3.0 kBT (panels e and f). The the distributions shift toward larger values of rxy g with larger Sm, the arrows in panel a shows the order of increasing Sm to highlight this. The color-coding is matched between adjacent panels: a and b, c and d, and e and f.
both ⟨rg⟩ and ⟨rxy g ⟩ are remarkably insensitive to Sm, changing by 10% or less over the whole interval. As extension or compression of the side chains, unlike the main chains, do not affect the overall bead density in the dense brush regime, there is no comparable driving force to that which causes the main chain to extend with lateral compression.12 For adsorbing side chains the change is visibly larger, reflecting the detachment of side chains from the surface upon compression. Relation Between Compression and Adsorption Isotherms. In this section, we extract information about interpolymer repulsion from the compression isotherms and discuss its significance for the extent to which brush layers can be formed by adsorption. We envision the situation of bottlebrush polymer strongly adsorbed by its end with less strongly adsorbing, or nonadsorbing, side chains. This situation can be realized, for instance, by grafting a block of adsorbing polymer to the end of the bottle-brush polymer. For this situation, the polymer−polymer interaction can be expected to be similar between the grafted and adsorbed brush. As the grafted system is quasi-2D while adsorption must be modeled in 3D, the correspondence between compression and adsorption isotherms can only ever be approximate; extraneous assumptions must be made to enable treatment of the equilibrium between surface-adsorbed and bulk polymer. The adsorption process can be conceptually divided into three fictitious subprocesses: (i) adsorption of anchor block for noninteracting polymer, (ii) adsorption of side chains for
functions of Sm . For nonadsorbing side chains, the rgxy distribution becomes narrower and the mean is shifted to smaller values as Sm decreases. ⟨rxy g ⟩ follows a trend that is opposite to that of ⟨rg⟩, which implies alignment in the z direction; as the main chains are stretched due to compression, they also align perpendicular the surface. For adsorbing chains, ⟨rxy g ⟩ and ⟨rg⟩ virtually coincides for low compressions. This indicates almost complete alignment parallel to the surface, and is consistent with the small height of the brush under these conditions, see Figure 5. As the brush layer is compressed, ⟨rxy g⟩ initially decreases together with ⟨rg⟩. For a certain density, somewhat below 100 σ2, the distribution of rxy g turns bimodal and ⟨rxy ⟩ and ⟨r ⟩ starts diverging from each other, signifying g g that the chains orient perpendicular to the surface; the bimodal distribution in panel e of Figure 7 corresponds to the system depicted in panel b of Figure 6. Presumably, the bimodal distribution arises because complete unspreading of a polymer is required to allow sufficient conformational freedom to realize a significant entropy gain. Partially unspread states are likely to be unfavorable because they are associated with a loss of favorable interactions with the surface without an offsetting entropy gain. Thus, there is a conformational equilibrium between the entropically favored unspread state and the 10459
dx.doi.org/10.1021/la4021959 | Langmuir 2013, 29, 10455−10462
Langmuir
Article
noninteracting polymer, and (iii) turning on the polymer− polymer interactions. The associated change in free energy can be written as ΔAads = ΔAanchor + ΔAspread + μex
(8)
where ΔA is the free energy of adsorption and the terms on the right-hand side correspond to the three subprocesses. Here, μ ex refers to the excess chemical potential of polymers in the adsorbed layer. This, rather than the free energy per polymer, is the appropriate quantity as the adsorption free energy pertains to the process of adding one additional polymer to the layer. ΔAanchor is taken to include both ideal and excess contributions and depends on the bulk concentration. This term has no counterpart in the grafted system. However, if the bulk concentration is sufficiently low and the structure is sufficiently similar between the grafted and adsorbed layers, ΔAspread and μex may be approximated by their values for the grafted system. This corresponds to the assumption that the free-energy consequences of polymer−polymer and side-chain−surface interactions are similar between grafted and adsorbed systems. The chemical potential, μ, can be identified with the work required to add a particle to the system and can be calculated from the compression isotherms as ads
μ(π ) = μ(π 0) +
∫π
Figure 9. Sum of the free energy associated with the spreading of a single polymer, ΔAspread, and the excess chemical potential, Δμex, relative to the state corresponding to π0 = 3 × 10−3 kBT/σ2, for bottlebrush polymers with nonadsorbing side chains (red circles), and adsorbing side chains with εw = 1.5 kBT (green squares) and εw = 3.0 kBT (blue diamonds). ΔAspread (horizontal lines) does not depend on Sm and is included to relate the curves to a common standard state, an isolated polymer with nonadsorbing side chains.
without a cationic block. The experiments were performed with silica surfaces, on which the PEO side chains adsorbs. The kinetics of formation of the polymer layer showed a quick initial adsorption of an amount corresponding to a monolayer, followed by a slow adsorption of several times this amount. Removing the cationic block made adsorption stop after the initial phase. Thus, the initial adsorption could be ascribed to the affinity of the PEO side chains to silica, and the second stage to electrostatic attraction. The thickness of the saturated adsorbed layer of the diblock polyelectrolyte was estimated to 50 nm and that of the bottle-brush polymer without adsorbing cationic block to 2 nm. This difference in layer thickness is disproportionately large compared to the difference in adsorbed amount, which leads to the conclusion that a structural transition takes place. The present results offer an explanation for why a cationic block was necessary to create a thick, brush-like adsorbed layer. As can be seen from Figure 9, the side-chain adsorption free energy is overcome by polymer−polymer interactions at Sm ≈ 100 σ2. This Sm is near the onset of the transition to a brush-like structure, as can be seen from Figures 5 and 7. Thus, side-chain adsorption has the potential to create a layer with a density close to the brush regime, but not to cause the transition to a brush-like structure. This is not accidental: In order for a brush structure to form, a majority of the side-chain beads must desorb from the surface. The bead−bead interaction must then overcome the side-chain attraction to the surface. In the absence of an additional source of attraction, this removes the driving force for further adsorption. This shows that side-chain adsorption neither help nor hinder the formation of a dense brush; only in the transition region between pancake and brush behavior does the side-chain adsorption have a significant effect on the structure of the brush-like layer.
π 0
Sm(π ′) dπ ′
(9)
where π0 is the surface pressure in the reference state. The excess chemical potential, μex = μ − μid, is the work against intermolecular interactions. It can be obtained by subtraction of the ideal contribution, which is due to changes in translational entropy, according to μid = kBT ln S0m/Sm, where S0m is the molecular surface area at surface pressure π0. If π0 is sufficiently small, the excess chemical potential thus obtained will be close to that pertaining to a noninteracting reference state. In practice, we use eq 9 to calculate Δμex(π) = μex(π) − μex(π0) by numerical integration of the compression isotherms in Figure 4, see Appendix A. Here, π0 = 3 × 10−3 kBT/σ2. We expect this reference state to be sufficiently dilute to approximate a noninteracting standard state, i.e., Δμex(π) ≈ μex(π), for the purpose of the discussion below. The sum of ΔAspread and Δμex is shown in Figure 9. As explained above, this quantity gives a measure of the change in adsorption free energy due to polymer crowding and side-chain adsorption. In a rough sense, the sum ΔAspread + Δμex can be interpreted as giving the anchor block adsorption free energy required to form a brush of a given density. For polymer with nonadsorbing side chains, where ΔAspread = 0, Δμex increases steadily and reaches a value slightly above 100 kBT for the highest compression considered. For adsorbing side chains at large Sm, the sum ΔAspread + Δμex is dominated by ΔAspread, which is independent of Sm. Furthermore, Δμex increases upon compression to reach a plateau region where Δμex varies slowly and ΔAspread + Δμex is close to zero. This region coincides with the region of polymer unspreading and the onset of exponential change in brush height with Sm, see Figure 5. Upon further compression Δμex increases at a rate similar to that for nonadsorbing side chains. The relation between compression and adsorption isotherms enables comparison between some features of the present results to the recent adsorption experiments reported in ref 19. In that work, it was found that diblock copolymers with an adsorbing cationic block in addition to a bottle-brush block adsorbed to a greater extent than bottle-brush polymers
■
SUMMARY AND CONCLUSIONS We investigated layers of surface-attached but laterally mobile bottle-brush polymers with adsorbing and, for reference, nonadsorbing side chains. For nonadsorbing side chains, the surface pressure varies with the molecular surface area approximately as a power law according to S−8/3 m , which is steeper than the one expected from mean-field theory, S−5/3 m . 10460
dx.doi.org/10.1021/la4021959 | Langmuir 2013, 29, 10455−10462
Langmuir
Article
(2) Tatsuo, S. Stabilization of colloidal dispersions by polymer adsorption; Marcel Dekker: New York, 1980. (3) Dėdinaitė, A. Biomimetic Lubrication. Soft Matter 2012, 8, 273− 284. (4) Blaszykowski, C.; Sheikh, S.; Thompson, M. Surface Chemistry to Minimize Fouling from Blood-Based Fluids. Chem. Soc. Rev. 2012, 41, 5599−5612. (5) Peleshanko, S.; Tsukruk, V. V. The Architectures and Surface Behavior of Highly Branched Molecules. Prog. Polym. Sci. 2008, 33, 523−580. (6) Lee, H. J.; Nakayama, Y.; Matsuda, T. Spatio-Resolved, Macromolecular Architectural Surface: Highly Branched Graft Polymer via Photochemically Driven Quasiliving Polymerization Technique. Macromolecules 1999, 32, 6989−6995. (7) Yu, W. H.; Kang, E. T.; Neoh, K. G. Controlled Grafting of Comb Copolymer Brushes on Poly(tetrafluoroethylene) Films by Surface-Initiated Living Radical Polymerizations. Langmuir 2005, 21, 450−456. (8) 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. (9) Brzozowska, A. M.; de Keizer, A.; Norde, W.; Detrembleur, C.; Cohen Stuart, M. A. Grafted Block complex Coacervate Core Micelles and their Effect on Protein Adsorption on Silica and Polystyrene. Colloid Polym. Sci. 2010, 288, 1081−1095. (10) Zhulina, E. B.; Vilgis, T. A. Scaling Theory of Planar Brushes Formed by Branched Polymers. Macromolecules 1995, 28, 1008−1015. (11) Xu, X.; Cao, D. Density Functional Theory for Adsorption of Colloids on the Polymer-Tethered Surfaces: Effect of Polymer Chain Architecture. J. Chem. Phys. 2009, 130, 164901. (12) de Vos, W. M.; Leermakers, F. A. M.; Lindhoud, S.; Prescott, S., W. Modeling the Structure and Antifouling Properties of a Polymer Brush of Grafted Comb-Polymers. Macromolecules 2011, 44, 2334− 2342. (13) Carrillo, J.-M. Y.; Dobrynin, A. V. Molecular Dynamics Simulations of Grafted Layers of Bottle-Brush Polyelectrolytes. Langmuir 2010, 26, 18374−18381. (14) Russano, D.; Carrillo, J.-M. Y.; Dobrynin, A. V. Interaction between Brush Layers of Bottle-Brush Polyelectrolytes: Molecular Dynamics Simulations. Langmuir 2011, 27, 11044−11051. (15) Carrillo, J.-M. Y.; Russano, D.; Dobrynin, A. V. Friction between Brush Layers of Charged and Neutral Bottle-Brush Macromolecules: Molecular Dynamics Simulations. Langmuir 2011, 27, 14599−14608. (16) Cao, Q.; Zuo, C.; Ma, Y.; Li, L.; Chen, S.; Hu, Z. Molecular Dynamics Simulations of End-Grafted Polymers with Charged Side Chains. J. Polym. Sci., Part B: Polym. Phys. 2011, 49, 882−889. (17) Linse, P.; Claesson, P. M. Modeling of Bottle-Brush Polymer Adsorption onto Mica and Silica Surfaces. Macromolecules 2009, 42, 6310−6318. (18) Linse, P.; Claesson, P. M. Modeling of Bottle-Brush Polymer Adsorption onto Mica and Silica Surfaces: Effect of Side-Chain Length. Macromolecules 2010, 43, 2076−2083. (19) Liu, X.; Dedinaite, A.; Rutland, M.; Thormann, E.; Visnevskij, C.; Makuska, R.; Claesson, P. M. Electrostatically Anchored Branched Brush Layers. Langmuir 2012, 28, 15537−15547. (20) Alexander, S. Adsorption of Chain Molecules with a Polar Head a Scaling Description. J. Phys. (Paris) 1977, 38, 983−987. (21) Ligoure, C. Polymers at Interfaces: From a Quasi Self Similar Adsorbed Layer to a Quasi Brush First Order Phase Transition. J. Phys. II Fr. 1993, 3, 1607−1617. (22) Szleifer, I. Pancake-to-Brush Transition in Block Copolymers. Europhys. Lett. 1998, 44, 721−727. (23) Ou-Yang, H. D.; Gao, Z. A Pancake-to-Brush Transition in Polymer Adsorption. J. Phys. II Fr. 1991, 1, 1376−1385. (24) Liu, G.; Cheng, H.; Yan, L.; Zhang, G. Study of the Kinetics of the Pancake-to-Brush Transition of Poly(N-isopropylacrylamide) Chains. J. Phys. Chem. B 2005, 109, 22603−22607.
For low grafting densities, side-chain adsorption forces the backbone to align parallel with the surface. As the grafting density is increased, the repulsive interpolymer interaction eventually causes side chains to desorb. Above a certain grafting density, polymers with the main chain perpendicular to the surface start to appear, and their population relative to polymers with parallel main chains progressively increases. At high densities a polymer brush is formed, regardless of the strength of the interactions between the side chains and the surface. Estimation of the free energy change associated with side-chain adsorption and compression of the polymer layer resulted in the conclusion that a similar adsorption free energy is required to form a dense brush regardless of the side-chain adsorption strength.
■
APPENDIX A For an accurate determination of the chemical potential with respect to a standard state where μex ≈ 0, it may be necessary to integrate eq 9 over many orders of magnitude in pressure. To accomplish this, it is expedient to rewrite eq 9 as μ(π ) = μ(π 0) +
∫π
π 0
Sm(π ′)π ′ d ln π ′
= μ(π 0) + kBT ln
π + π0
∫π
π 0
(Sm(π ′)π ′ − kBT ) d ln π ′
(10)
The integrand in the remaining integral is well-behaved and amenable to numerical integration with tolerable error. The second term is not identical to the ideal chemical potential, except under conditions when the adsorbed layer is in fact ideal. In what follows, the effect of the statistical uncertainty of the integrand on the integration error will be examined, as distinct from error associated with the quadrature formula. It was found numerically that the absolute error in πSm was approximately independent of π. This observation can be rationalized by noting the fact that πS = πSmN enters the Boltzmann factor associated with the isotension ensemble. The fluctuations in this quantity should thus be of the order kBT, and otherwise independent of the thermodynamic state. Thus, the absolute error in πSm calculated from a simulation of finite length should be approximately independent of πSm and, consequently, the relative error inversely proportional to the magnitude of πSm. These considerations suggest that this method is well suited for systems with a large positive deviation from ideality, πSm ≫ kBT, which is the situation studied here. While the relative error in Sm is large for low pressure, this will have only modest consequences for the relative error in μ for higher compression, as the contribution to the integral in eq 10 from the low pressure region is small.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS Financial support by the Swedish Research Council (VR) through an individual grant to P.L. (2010-4986) is gratefully acknowledged.
■
REFERENCES
(1) Zhao, B.; Brittain, W. J. Polymer Brushes: Surface-Immobilized Macromolecules. Prog. Polym. Sci. 2000, 25, 677−710. 10461
dx.doi.org/10.1021/la4021959 | Langmuir 2013, 29, 10455−10462
Langmuir
Article
(25) Hsu, H.-P.; Paul, W.; Binder, K. Structure of Bottle Brush Polymers on Surfaces: Weak versus Strong Adsorption. J. Phys. Chem. B 2011, 115, 14116−14126. (26) Frenkel, D.; Smit, B. Understanding molecular simuation; Academic Press: London, 2002. (27) Svensson, B.; Woodward, C. Simulations in Planar Slits at Constant Chemical Potential. J. Chem. Phys. 1994, 100, 4575−4581. (28) Turesson, M.; Woodward, C. E.; Åkesson, T.; Forsman, J. Simulations of Surface Forces in Polyelectrolyte Solutions. J. Phys. Chem. B 2008, 112, 5116−5125. (29) Naji, A.; Seidel, C.; Netz, R. R. Theoretical Approaches to Neutral and Charged Polymer Brushes. Adv. Polym. Sci. 2006, 198, 149−183. (30) Grest, G. S. Grafted Polymer Brushes: A Constant Surface Pressure Molecular Dynamics Simulation. Macromolecules 1994, 27, 418−426. (31) 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. (32) Currie, E. P. K.; Leermakers, F. A. M.; Cohen Stuart, M. A.; Fleer, G. J. Grafted Adsorbing Polymers: Scaling Behavior and Phase Transitions. Macromolecules 1999, 32, 487−498.
10462
dx.doi.org/10.1021/la4021959 | Langmuir 2013, 29, 10455−10462
