Article pubs.acs.org/Macromolecules
Adsorption of Oligomers and Polymers into a Polymer Brush Formed from Grafted Ring Polymers Andrey Milchev*,†,§ and Kurt Binder§ †
Institute of Physical Chemistry, Academy of Sciences, 1113 Sofia, Bulgaria Institut für Physik, Johannes Gutenberg Universität Mainz, Staudinger Weg 7, 55099 Mainz
§
ABSTRACT: The interaction of a ring polymer brush with a solution containing oligomers or free linear flexible macromolecules is studied by Monte Carlo simulation, varying the chain length of the free chains, and in selected cases also the lengths of the rings. Two grafting densities are studied, corresponding to semidilute and very concentrated conditions, and a comparison with the corresponding case of brushes formed from grafted linear chains is made. Although the ring polymer linear dimensions in the brushes show an anomalous scaling with ring length, similar to (noncatenated) ring polymer melts, the concentration profiles of oligomers and long macromolecules in ring polymer brushes differ only very little from their linear polymer brush counterparts.
substrate. By “crresponding brush” we mean that each ring polymer corresponds to two linear polymers, the number NR, NL of effective monomers in the ring (R) and chain (L) being related as NL = NR/2, and the grafting densities are σLg = 2σRg , so the total number of effective monomers would be identical. While one then finds that the monomer density profiles in both systems essentially coincide, and also the mean square gyration radii components of the polymers (in the z-direction) perpendicular to the surface are identical, ⟨Rgz2⟩L = ⟨Rgz2⟩R, there is a significant difference with respect to the mean square gyration radii components parallel to the surface (in x, ydirections). Specifically, it has been found that12−14
I. INTRODUCTION Polymer brushes are formed when macromolecules are densely grafted on nonabsorbing substrate surfaces.1−7 These systems find important applications to stabilize colloids, for the tuning of adhesion and wetting properties, as lubricants, and find use as protective coatings preventing adsorption of proteins in biological environments, etc. This last example of applications already shows that an important aspect of polymer brushes is their interaction with other (large) molecules present in the solution. This problem has been considered theoretically mostly for the ideal situation when the chemical nature of the grafted macromolecules and of the nongrafted free polymers in solution is identical,8−10 and only very recently the effect of attractive interactions between the effective monomers of the free polymers (or oligomers, respectively) and the grafted chain molecules has been considered.11 Without such an attraction, small oligomers can penetrate, at least partially, into the brush. In contrast, long chains are almost completely expelled, apart from the region near the outer edge of the brush. Therefore, such attractive interactions may lead to partial or complete absorption of the free (chain) molecules into the brush.11 Since the structure of macromolecules in brushes under good solvent conditions is the result of a delicate interplay of excluded volume repulsive forces with the entropic elasticity of the macromolecules that are more or less stretched in direction normal to the grafting surface, the free oligomers/polymers absorbed into the brush require a rearrangement of this structure on the local level. Recently, it has been found that the organization of the polymers in the brush may significantly differ from the corresponding brush formed from polymers with linear chain chemical architecture12−14 when one grafts ring polymers from dilute solution (where they are not mutually entangled) to a © 2013 American Chemical Society
⟨R gxy 2⟩L ∝ NL , ⟨R gxy 2⟩R ∝ NR 2ν⊥
(1)
where ν⊥ is distinctly less than the random walk value of 1/2, namely12−14 ν⊥ ≈ 0.4. This behavior is somehow reminiscent of the topological effects much discussed for (noncatenated) ring polymer melts, where it is known that the topological repulsion between the rings leads to their collapse into almost compact configuration,15−27 ⟨Rg2⟩R,melt ∝ NR2/3. Note also that in such ring polymer melts there occurs an extended range of lengths, NR, where one crosses from the standard Gaussian behavior of polymers in a melt (⟨Rg2⟩R,melt ∝ NR with an intermediate regime (resembling ⟨Rg2⟩R,melt ∝ NR0.8) over to the “crumpled globule”19 behavior with ⟨Rg2⟩R,melt ∝ NR2/3. Thus, it is also possible that the behavior found for ring polymer brushes12−14 is not the asymptotic scaling for NR → ∞, but rather an intermediate crossover regime as well. Received: August 9, 2013 Revised: October 4, 2013 Published: October 31, 2013 8724
dx.doi.org/10.1021/ma401678z | Macromolecules 2013, 46, 8724−8731
Macromolecules
Article
first case, and −1.85 in the second case. In the latter case, the oligomers (or free chains) are attracted into the brush, while in the first case they are expelled. Note that irrespective of these choices, overlap of monomers still is excluded, to a very good approximation. In order to guarantee noncatenation of the rings, we create the starting conformation of the brush as a regular array of strongly deformed rings taken as vertical “rods” that comprise two stretched strands of length NR/2 (i.e., half of a loop) which are tightly pressed to one another. This starting configuration is then equilibrated by MD simulation whereby these “1D”conformations attain their eventual equilibrium shape of grafted ring macromolecules, subject to the imposed grafting density σRg , and number of beads NR in the ring. The simulation box is chosen as L × L × Lz with normally both L = 32 and Lz = 32 and two grafting densities are used, σRg = 0.125 and σRg = 0.500. Thus, the largest simulation box involves a total number of N = 512 ring polymers, and with NR = 64 the brush contains in total 32764 effective monomers. At z = 0 and z = Lz, hard repulsive walls are used, confining all effective monomers (including these of the oligomers) in the system. Note that for this choice of parameters no monomers of the brush polymers may come close to the upper wall at z = Lz. Some data have also been generated for significantly longer rings, up to NR = 512, and then much more elongated boxes were used (with Lz = 128). As an example, Fig. 1 presents snapshot pictures of a polymer brush made of ring polymers with NR = 256, grafting density σRg = 0.0625, and free linear chains with chain length NL = 64, at a monomer concentration co = 0.1875, for both choices of εpo. One can clearly recognize that in the compatible case, Fig. 1b,
The present work now extends the studies of brushes interacting with free chains by attractive monomer−monomer interactions11 to brushes formed from ring polymers, as in our earlier studies.12−14 Since the rings in these brushes have more compact configurations in the xy-directions, as noted in eq 1, one might expect that there is an interesting competition between the (entropic) self-attraction of the monomers of a ring polymer, causing the compaction, and the attraction between monomers of the ring polymers and monomers of the free linear chains. After this brief introduction in section I, in section II, we briefly describe our model, while section III presents our numerical results, including a comparison with corresponding data for brushes formed from linear chains. Section IV summarizes our conclusions.
II. MODEL AND SIMULATION METHOD The present study is based on the use of a well-established coarse-grained off-lattice bead−spring model.28−31 The polymer brush consists of rings of “length” NR (NR is just the number of effective monomers in the ring), where one monomer is fixed on the plane z = 0 at a grafting site; these grafting sites form a regular square lattice, and periodic boundary conditions in x, y directions are used. The “springs” of the bead−spring model are described by the same finitely extensible nonlinear elastic FENE potential for both the rings and the linear chains, acting between neighboring bonded effective monomers ⎡ ⎛ S − S0 ⎞2 ⎤ UFENE(S) = U0 ln⎢1 − ⎜ ⎟⎥ ⎢⎣ ⎝ S0 − Smax ⎠ ⎥⎦
(2)
where S is the length of a bond connecting two neighboring effective monomers, which can vary between Smax (≡1, chosen as our unit of length) and Smin (=0.4), with a potential minimum at S0 = 0.7, half way in between the minimum and the maximum bond extension. The amplitude U0 of this potential is chosen as U0 = 20, choosing the thermal energy kBT = 1 as our energy unit. The nonbonded interaction between any pair of monomers is described by the Morse potential, αβ UM (r )/εαβ = exp[−2κ(r − rmin)] − 2 exp[−κ(r − rmin)]
(3)
which has a sharp minimum at rmin = 0.8 (the minimum is sharp because we use the parameter κ large, namely28−31 κ = 24, implying that the range of this attraction is always very short, so it can be truncated with little error already for r = 1, which is useful to make the algorithm efficient and fast). Note that neither κ nor rmin depend on the choice of the type of pair, irrespective whether we consider effective monomers of the brush polymers (p), or oligomers (or free chains, respectively), denoted by the index (o). To work safely in the good solvent regime, for both brush polymers and oligomers, we choose the energy parameters εα,β = (εpp, εoo, εpo) as follows
εpp = 0.2,
εoo = 0.1
(4)
Figure 1. Snapshots of a polymer brush made of ring polymers with NR = 256, grafting density σRg = 0.0625, and free linear chains of length NL = 64 at monomers concentration co = 0.1875 in the system. The interaction between ring monomers and linear chain monomers is (a) εpo = 0.02 and (b) εpo = 2.00. The simulation box chosen is 32 × 32 × 128. Grafted polymers are displayed in yellow and the free chains in blue.
while two choices for εpo are considered, namely εpo = 0.02 and εpo = 2.00. The reason for these choices are that they yield strongly different signs for the “compatibility parameter” χ ≡ 0.5(εpp+εoo) − εpo (this parameter would be proportional to the standard Flory−Huggins parameters32), since χ = 0.13 in the 8725
dx.doi.org/10.1021/ma401678z | Macromolecules 2013, 46, 8724−8731
Macromolecules
Article
Figure 2. Monte Carlo data for the time evolution of the absorbed amount Γ(t) against elapsed time t after the energy parameter was switched from εpo = 0.02 (repulsive case) to εpo = 2.00 (attractive case). Part a shows various chain lengths NL of the oligomers for σRg = 0.125, part b shows data for NL = 64 and different concentrations co of the monomers in the system. Note that for small co and early enough times, a power law growth of Γ(t) with time t is observed.
Figure 3. Density profile ϕp(z) of a ring polymer brush with NR = 64, plotted vs z (curve enclosing shaded area), and profiles ϕo(z) of free chains vs z, for chain lengths from NL = 1 (i.e., monomers) up to NL = 64. Four cases are shown, with two grafting densities σRg = 0.125 (upper row) and σRg = 0.5 (lower row) and two choices of εpo, εpo = 0.02 (left column) and εpo = 2.0 (right column).
randomly chosen position with displacements Δx, Δy, Δz (relative to its old position) drawn from the intervals −0.5 ≤ Δx, Δy, Δz ≤ +0.5. These trial moves are accepted or rejected according to the standard Metropolis test.33 Data are typically derived from 107−108 Monte Carlo steps (MCS) per monomer, noting that the absorbed amount (initially all free chains are put into the volume region of the box above the outer boundary of the brush) needs of the order 106 MCS to equilibrate for the smallest free species (single particles and dimers) while this time grows for the longer free chains.
some free chains are dissolved in the brush, while in the incompatible case, Fig. 1a, there is a well-defined interface between the brush monomers and those of the free polymers. Note also that the chosen monomer density in Fig. 1 still corresponds to a semidilute solution (while for grafting density σRg = 0.5, one does almost reach melt densities in the brush, see Figures 3 and 4 in the Simulation Results). The Monte Carlo algorithm consists of attempted moves whereby a monomer is chosen at random (treating monomers of the brush polymers or of the free polymers or oligomers on the same level) and one attempts to displace it to a new 8726
dx.doi.org/10.1021/ma401678z | Macromolecules 2013, 46, 8724−8731
Macromolecules
Article
Figure 4. Density profile ϕp(z) of a ring polymer brush with NR = 64 plotted vs z (curve enclosing shaded area) and profiles ϕo(z) of free linear chains with NL = 64 plotted vs z, varying their concentration. Four cases are shown, with two grafting densities σRg = 0.125 (upper row) and σRg = 0.5 (lower row), and two choices of εpo, εpo = 0.02 (left column) and εpo = 2.0 (right column). Six choices of the number concentration co of the free chains, from co = 0.0625 up to co = 0.375, are included, as indicated, by the total number of free chains No in the container.
adsorbed amount Γads with time is significantly slower than in the case of low oligomer concentration. Figure 3 shows the density profile of equilibrated ring polymer brushes, for the two choices of grafting densities studied and the choice NR = 64, varying the length NL of the free chains from monomers (NL = 1) up to NL = 64, both for the repulsive and attractive choice of the interactions. The monomer density profile for the brush shows small oscillations near the wall at z = 0, as expected when particles (effective monomers) which repel each other are rather densely packed near a hard wall (layering effect). For the smaller choice of the grafting density, the brush is still in the regime where the monomer density corresponds to a semidilute solution; The density profile then decays with z according to the common (nearly parabolic) profile.7 For the larger value σRg = 0.5, however, the monomer density inside the brush near the wall corresponds to a very concentrated solution (almost a melt), and then, for the case of repulsive interactions (εpo = 0.02), even the solubility of individual beads (NL = 1), or dimers (NL = 2), in the brush is very small, unlike the semidilute case. Note that very close to the hard grafting surface there is a slight enhancement of the monomer density. Larger oligomers (such as NL = 4,8, ...) are almost completely expelled from the brush in all cases. Evidently, outside of the brush the density of the oligomers or free polymers is essentially constant, apart from the immediate vicinity of the upper wall at z = Lz, where particles are repelled again (depletion). In the case of attractive forces (εpo = 2.0), however, the situation is completely different: both oligomers and longer free chains are absorbed into the brush. But while for smaller
III. SIMULATION RESULTS The initial stages of free chain adsorption of different size and concentration are presented in Figure 2. Here the absorbed amount, Γ(t), is defined as the fraction of all the monomers of the free chains (of oligomers) which occurs at distances z < h, where h is the height of the brush (precisely, h is defined such that 99% of the brush monomers reside at positions z < h). Thus, the slow approach to equilibrium would make it very difficult to obtain reliable data for free chains longer than those that were studied. In Figure 2a one observes a monotonous increase of Γads with time t, once the free chains are brought in contact with the brush, whereby for a given grafting density σRg = 0.125 the net amount of Γads(t) goes up with declining molecular weight of the oligomers while the adsorption kinetics steadily slows down. In Figure 2b, one may distinguish two families of curves which undergo a steady increase with time t. In both families of curves it is always the system with highest oligomer concentration co that penetrates most quickly the originally empty polymer brush. However, if one compares Figure 2b to Figure 4b (where the total number of oligomers No ∝ co−1), one would immediately notice that the two families of Figure 2b can be classified as such of “low”- and “high”-concentration whereby the former totally absorb and accommodate inside the brush whereas the latter are so numerous that they cannot be all included into the brush and remain largely present in the solvent above the brush surface. Since these oligomers exceed the absorption capacity of the brush, the increment of their 8727
dx.doi.org/10.1021/ma401678z | Macromolecules 2013, 46, 8724−8731
Macromolecules
Article
Figure 5. Monomer density profile ϕp(z) plotted vs z, at two different concentrations co = 0.0625 and co = 0.1875 of free chains, and at two grafting densities (σRg = 0.125 and σRg = 0.500), as indicated. (a) refers to the incompatible case (εpo = 0.02) and (b) to the compatible one (εpo = 2.00). The shaded profiles indicate the unperturbed brushes (no free chains in the solution).
Figure 6. Comparison of the profiles of oligomers (NL = 1) or free chains (NL = 64), penetrating into a brush formed from linear chains (NL = 32,σLg = 1) (a) for the case of repulsive interactions, εpo = 0.02, and (b) for attractive interactions, εpo = 2.0, Also given is corresponding data for a brush formed from ring polymers (NR = 64,σRg = 0.5). The monomer concentration of the penetrator was chosen in both cases as co = 0.0625. The brush density profiles are shaded.
very similar to the results for free chains interacting with a brush formed from linear chains.11 One can also ask the question, does the monomer density profile of the brush change when the linear chains get absorbed? Figure 5 shows that such changes do occur but are quite minor. In the case of repulsive forces between the effective monomers of the brush and of the free chains, the brush gets slightly compressed when the concentration of free chains in the solution increases; for the case of attractive forces, a slight swelling of the brush is observed. Qualitatively, a similar behavior for brushes formed from linear chains interacting with free chains was already reported earlier.11 Therefore, it is of interest to compare directly both systems, under condition where the brushes formed from linear chains and from rings are quantitatively comparable, i.e., NR = 2NL and σRg = σLg /2, so the numbers of monomers in both types of brushes are identical. For brushes in the absence of free chains (or oligomers), such a comparison was already given in ref 12; there it was found that the density profiles of these two types of brushes were nearly identical. Now we complement these earlier studies by comparing profiles in the presence of the free chains (or oligomers, respectively), Figure 6. While the brush profiles of rings and linear polymers are almost indistinguishable, and also the profiles for oligomers (NL = 1) do not distinguish whether they penetrate into a ring polymer brush or an ordinary
grafting density the absorption happens mostly in the brush interior, for the higher grafting density it occurs mostly in the outer region of the brush, where the brush monomer profile decays to zero. All these findings are very similar to results obtained for brushes formed by linear chains;11 the anomalous chain structure of the grafted rings12 (in comparison to grafted linear chains) hardly influences the absorption behavior (we shall discuss this aspect in more detail below). A further parameter of interest is the concentration of free chains co. Figure 4 studies the variation of this parameter, focusing on the choice NL = 64, so the length of the free chains is identical with the length of the rings. Note that the number of free chains in the simulation box varies then between No = 32 chains (for co = 0.0625) up to 192 (for co = 0.375), so we stay off from the case of the absorption of single chains into a brush (which we plan to investigate in a separate study). In the repulsive case, it is evident that enhancing the density of free chains leads to slightly deeper penetration of the free chains into the bulk. For the attractive case, it is evident that the chains at small concentration co get almost completely absorbed into the brush (for low σRg ) or adsorbed at the brush-solvent interface (high value of σRg ) while for larger co some of the free chains stay outside of the brush; thus the brush reaches a “saturation limit” of free chains, further free chains cannot be incorporated into the brush any longer. Again, the behavior is 8728
dx.doi.org/10.1021/ma401678z | Macromolecules 2013, 46, 8724−8731
Macromolecules
Article
Figure 7. (a) Comparison of the monomer density profiles ϕp(z) of the polymer brush and of absorbed oligomers, ϕo(z), with length NL = 1, or absorbed free chains with NL = 64. The monomer concentration co = 0.0625 in both cases. Here σRg = 0.125, σLg = 0.25. (b) Density profiles of absorbed free chains with NL = 64 in a brush of linear chains with σLg = 1.0, or in a ring-polymer brush with σRg = 0.5, respectively. Two concentrations of the adsorbate, co = 0.125, 0.375 are considered. The interaction energy was chosen as εpo = 2.0 in all cases.
Figure 8. (a) Variation of the absorbed amount Γ of free chains in brushes vs chain length NL, considering attractive energy εpo = 2.0 and two grafting densities both for linear chains in the brush (NL = 32 with σLg = 0.25 or 1.0, respectively) and for the equivalent grafted ring polymers (NR = 64 with σR = 0.125 and 0.5, respectively). All data are for small penetrator concentration (co = 0.0625). (b) Variation of the absorbed amount with the penetrator concentration co, for NL = 64, and otherwise the same parameters as in (a).
that is suitable for small penetrators. Here we should also like to point out the existence of a characteristic dip in Γads at the higher grafting density for N = 2, indicating that free dimers, which benefit from translational entropy only slightly less than the single oligomers with N = 1, appear as the least absorptive species. Apparently, this holds both for ring and linear brushes and is related to the fact that dimer accommodate worse than single monomers in the voids immediately at the grafting surface. Figure 9 considers the absorbed amount as a function of the energy parameter εpo, for free chains with NL = 64, for four different ring lengths NR. It is seen that in all cases Γ stays very small until about εpo ≈ 1.0, while then a steep rise of Γ is observed. Note that presumably for large εpo one can always swell the brush strongly enough so that the limit Γ → 1 is reached. Finally, Figure 10 considers the mean square gyration radii of the ring polymers as a function of ring length, both for an “empty” brush (i.e., without dissolved linear polymers), and with a fraction of absorbed free chains (at concentration co = 0.1875), for the attractive case, εpo = 2.0). We see that the polymers in the “empty” ring polymer brush show also in this case power laws similar to those that were reported before,12 namely,
polymer brush, there is a clear difference with respect to the penetration of long linear chains (NL = 64): the latter penetrate more deeply into brushes formed from linear chains, rather than into ring polymer brushes. Figure 7 gives more data, for the case of attractive interactions between the brush monomers and the monomers of the free chains (or oligomers, respectively), for two choices of the grafting density. Again one can see that the free chains mix slightly better with brushes formed from linear chains rather than ring polymers. A very interesting behavior is found for the fraction Γ of monomers of the free chains which is located inside the polymer brush (i.e., for z < h where h is the height of the brush). For small concentration co of the monomers of the free chains, one finds that oligomers and rather short polymers penetrate more easily into ring polymer brushes rather than into brushes formed from linear polymers - Figure 8a. For large co, however, the opposite trend is observed, at least for large NL (Figure 8b), as expected already from consideration of the distribution (Figures 6 and 7). Presumably for ring polymer brushes, where chain conformations are locally more compact,12 there occur somewhat larger voids which can be filled easily with oligomers and short polymers, while the structure of brushes formed from linear polymers is more homogeneous on local scales, so there is less space available 8729
dx.doi.org/10.1021/ma401678z | Macromolecules 2013, 46, 8724−8731
Macromolecules
Article
polymer brushes formed from grafted linear chains were performed (by “equivalent” we mean that the chain length NL is one-half of the ring length NR, and the grafting density is doubled, σLg = 2σRg , so that the total number of monomers is identical; in this case also the monomer density profiles of both types of brushes are almost identical). Already in previous work, it had been found that the mean square gyration radii parallel to the grafting surface for the two types of brushes scale rather differently, ⟨Rgxy2⟩L ∝ NL while ⟨Rgxy2⟩R ∝ NR0.8. The present computations indicate that the presence of free chains dissolved in the brush reduces somewhat the difference between parallel and perpendicular (⟨Rgz2⟩) mean square linear dimensions but a clear-cut power law behavior is no longer seen, when free chains are absorbed into a ring polymer brush. In this sense apparently the influence of absorbed linear polymers is at variance with what is known for melts of ring polymers where progressive dilution with linear chains removes the scaling Rg ∝ NvR with ν < 0.5. Also the absorbed amount of (relatively long) linear polymers that is taken up by ring polymer brushes is reduced, in comparison to brushes formed from linear polymers. In contrast, for oligomers the opposite trend is observed. Unfortunately, we cannot offer clear-cut theoretical arguments that could explain these results: evidently, the interplay of various entropic forces with the enthalpic effect of the chosen monomer−monomer attraction in ring polymer brushes is difficult to interpret. It is also remarkable that the monomer density profiles of linear grafted polymers and of the equivalent ring polymers brushes are almost identical, irrespective of whether linear free polymers or oligomers are present or not. We hope that the present work will stimulate analytical theoretical considerations on the effects of polymer topology on effective interactions in dense polymeric systems, and thus contribute to the understanding of entropic forces in macromolecular systems. In principle, one could design a way to test the difference in free volume between ring- and linear polymer brushes by testing the random insertion of spherical nanoparticles of different size and measuring the systematic differences in energy cost for such insertion as in our earlier work.34 However, such a study would require a lot of computational resources and thus must be left for future work. Experimentally, the free volume at polymer-solid interfaces can be measured by positron annihilation lifetime spectroscopy.35
Figure 9. Adsorbed amount Γads for linear chains of length NL = 64 penetrating into ring polymer brushes with σRg = 0.0625 plotted vs εpo for four choices of NR, as indicated. All data refer to co = 0.3125.
Figure 10. Variation of the gyration radii components Rg∥2 and Rg⊥2 for the brush polymers with ring length NR. Both an “empty” brush and a brush at adsorbate concentration co = 0.1875 and interaction εpo = 2.00 with free chains (NL = 64) are given. The grafting density is always σRg = 0.0625.
⟨R g ⊥2⟩ ∝ NR ,
⟨R g ∥2⟩ ∝ NR 0.4
(5)
the slightly smaller “effective exponents” in the fits in Figure 9 are probably due to the choice of a smaller grafting density, making it more difficult to reach the asymptotic power law. Recall that the observation of an exponent less than 1/2 for the component ⟨Rg∥2⟩1/2 parallel to the grafting surface, was discussed earlier.12 When linear chains are dissolved in the brush, no longer a simple power law applies for either component. Of course, it is likely that the asymptotic regime, where ultimately power laws apply, has not been reached. Unfortunately, due to excessive demands of computer time it has not been possible for us to investigate still larger values of NR than shown here.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: (A.M.)
[email protected]. Notes
The authors declare no competing financial interest.
■ ■
ACKNOWLEDGMENTS We are grateful to the Deutsche Forschungsgemeinschaft (DFG) for support under Grant No. BI 314/23.
IV. CONCLUSIONS In the present work a computer simulation study of a polymer brush interacting with free polymer chains (or oligomers, respectively) was presented, emphasizing the case when the brush is formed from grafted (noncatenated) ring polymers, in order to elucidate the effect of polymer topology on the absorption of oligomers or macromolecules into polymer brushes. For this purpose, also comparisons with equivalent
REFERENCES
(1) Halperin, A.; Tirrell, M.; Lodge, T. P. Adv. Polym. Sci. 1992, 100, 33. (2) Grest, G. S.; Murat; M.; in Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Binder, K., Ed.; Oxford Univ. Press: New York, 1995; p 476. (3) Klein, J. Annu. Rev. Mater. Sci. 1996, 26, 581. (4) Szleifer, I.; Cariguano, M. A. Adv. Chem. Phys. 1996, 94, 165. 8730
dx.doi.org/10.1021/ma401678z | Macromolecules 2013, 46, 8724−8731
Macromolecules
Article
(5) Advincula, R. C., Brittain, W. J., Caster, R. C., Rühe, J., Eds.; Polymer Brushes; Wiley-VCH: Weinheim, Germany, 2004. (6) Binder, K.; Kreer, T.; Milchev, A. Soft Matter 2011, 7, 7159. (7) Binder, K.; Milchev, A. J. Polym. Sci., B: Polym. Phys. 2012, 50, 1515. (8) Grest, G. S. J. Chem. Phys. 1996, 105, 5532. (9) Huang, H.; Cammers, A.; Penn, L. S. Macromolecules 2006, 39, 7064. (10) Jain, S.; Ginzburg, V. V.; Jog, P.; Weinhold, J.; Srivastava, R.; Chapman, W. G. J. Chem. Phys. 2009, 131, 044908. (11) Milchev, A.; Egorov, S. A.; Binder, K. J. Chem. Phys. 2010, 132, 184905. (12) Reith, D.; Milchev, A.; Virnau, P.; Binder, K. EPL 2011, 95, 28003. (13) Reith, D. Dissertation; Johannes Gutenberg Universität: Mainz, 2011. (14) Reith, D.; Milchev, A.; Virnau, P.; Binder, K. Macromoleclules 2012, 45, 4381. (15) Cates, M. E.; Deutsch, J. M. J. Phys. (Fr.) 1986, 47, 2121. (16) Müller, M.; Wittmer, J. P.; Cates, M. E. Phys. Rev. E 1996, 53, 5063. (17) Brown, S.; Szamel, G. J. Chem. Phys. 1998, 108, 4705. (18) Brown, S.; Szamel, G. J. Chem. Phys. 1998, 109, 6184. (19) Müller, M.; Wittmer, J.; Cates, M. Phys. Rev. E 2000, 61, 4078. (20) Brown, S.; Lenczycki, T.; Szamel, G. Phys. Rev. E 2001, 63, 3. (21) Hur, K.; Winkler, R. G.; Yoon, D. Y. Macromolecules 2006, 39, 3975. (22) Suzuki, J.; Takano, A.; Matushita, Y. J. Chem. Phys. 2008, 129, 034903. (23) Vettorel, T.; Reigh, S.; Yoon, D. Y.; Kremer, K. Macromol. Rapid Commun. 2009, 30, 345. (24) Vettorel, T.; Grosberg, A. Y.; Kremer, K. Phys. Biol. 2009, 6, 025013. (25) Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A.Yu.; Kremer, K. J. Chem. Phys. 2011, 134, 204904. (26) Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A.Yu.; Kremer, K. J. Chem. Phys. 2011, 134, 204905. (27) Sakaue, T. Phys. Rev. Lett. 2011, 106, 167802. (28) Milchev, A.; Paul, W.; Binder, K. J. Chem. Phys. 1993, 99, 4786. (29) Milchev, A.; Binder, K. Macromol. Theory Simul. 1994, 3, 915. (30) Milchev, A.; Binder, K. Macromolecules 1996, 29, 341. (31) Milchev, A.; Binder, K. J. Chem. Phys. 2001, 114, 8610. (32) Flory. P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (33) Binder, K. Rep. Prog. Phys. 1997, 60, 487. (34) Milchev, A.; Binder, K. Polymer 2008, 49, 3611. (35) Harms, S.; Rätzke, K.; Faupel, F.; Schneider, G.; Wöllner, L.; Richter, D. Macromolecules 2010, 43, 10505.
8731
dx.doi.org/10.1021/ma401678z | Macromolecules 2013, 46, 8724−8731
