Dynamics of Macromolecules Grafted in Spherical Brushes under

Article pubs.acs.org/Macromolecules

Dynamics of Macromolecules Grafted in Spherical Brushes under Good Solvent Conditions Federica Lo Verso,*,†,‡,∥ Leonid Yelash,§ and Kurt Binder† †

Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 7, D-55099 Mainz, Germany Centro de Fisica de Materiales, CSIC/UPV Pseo, Manuel de Lardizabal 5, 20018 Donostia-San Sebastian, Spain § Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, D-55099 Mainz, Germany ∥ Materials Physics Center MPC, Paseo Manuel de Lardizȧbal 5, 20018 San Sebastiȧn, Spain ‡

ABSTRACT: Spherical polymer brushes have a structure intermediate between star polymers and polymer brushes on flat substrates, and are important building blocks of polymer nanoparticles. Molecular dynamics simulations are presented for isolated spherical polymer brushes under good solvent conditions, varying the grafting density as well as the chain length, using a coarse-grained bead−spring model of flexible chains. We complement previous work on the static properties of the same model by analyzing the chain dynamics, studying the motions of monomers in relation to their position along the grafted chains, and extract suitable relaxation times. A qualitative discussion in terms of the Rouse model is given, as well as a comparison to corresponding work on planar brushes. We find that the end monomers relax faster than monomers further inside along the chain, as previously observed for planar brushes, but at variance with theoretical expectations. The relevance of our findings for experimental work is briefly discussed.

1. INTRODUCTION Spherical polymer brushes form when flexible macromolecules are grafted via a special end group to spherical nanoparticles. These molecules are a typical representative of the class of “hairy polymer nanoobjects”1 and find abiding recent interest as ingredients of polymer nanocomposites (see refs 2−4 for recent reviews). Unlike colloidal particles stabilized by polymer brushes,5,6 where the particles are in the micrometer size range and the thickness of the polymeric layer is at least an order of magnitude smaller than the colloid diameter, objects hence are intermediate between planar brushes7−9 and star polymers.10,11 In the star polymer limit a number f of chains, which may be rather large, is grafted to a core that is very small compared to the gyration radius of a grafted polymer. Spherical polymer brushes are very soft objects, and display a better solubility in polymer melts than nanoparticles without such a grafted layer, making them potentially useful building blocks for new materials.1−4,12−16 Before one can address such problems from the perspective of theoretical modeling, it is useful to study first the behavior of isolated spherical polymer brushes17−24 under various conditions, and then use this knowledge to clarify interactions between them25−34 and to obtain a systematic understanding of the structure−property relations. This is the strategy also in the present work, where we focus on the dynamics of the grafted chains in spherical polymer brushes. It is known that the dynamics of chains in dense planar brushes has very intriguing properties, different from what is known for the dynamics of free chains in solutions and melts.35 Even for short chains, that are not entangled at the relevant © XXXX American Chemical Society

monomer densities, one expects a scaling of the chain relaxation time τ with the number N of effective monomeric units as36−42 τ ∝ N3. This prediction has been interpreted in terms of the anomalous large fluctuations of the center of mass of a grafted chain,36 and described by a generalization of the well-known Rouse model of polymer dynamics35 to the case of a stretched chain in a brush, in the effective field of its neighbors.40 However, recent simulations42 did not confirm the associated prediction40 of a monotonous increase of the relaxation time of inner monomers with their position along the chain contour. The analysis of ref 42 instead found that the maximum of the relaxation time occurs for monomers well inside the brush and not at the free chain end. Moreover a larger (effective) exponent describing the chain length dependence of the relaxation time τ exceeding the theoretical value (3) was observed.42 Thus, it is interesting to ask the question whether chain dynamics exhibits similar anomalies for spherical brushes, too. We emphasize here that a deep understanding of the chain dynamics for planar polymer brushes in equilibrium is a prerequisite for clarifying the dynamics of nonequilibrium phenomena, e.g., chain expulsion from brushes,43−46 collective motions,47−54 and dynamic response to shear deformations. See ref 55 for a recent review. Thus, it is the purpose of the present work to extend the consideration of chain dynamics in brushes from the planar brush case to the case of spherical brushes, Received: March 1, 2013 Revised: May 2, 2013

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dx.doi.org/10.1021/ma400446r | Macromolecules XXXX, XXX, XXX−XXX

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applying Molecular Dynamics methods.56 Here we take a first step, addressing good solvent conditions. The outline of our paper is as follows: in section 2, we recall the model and the simulation method (we use the same model as ref 32), as well as the static properties of the spherical polymer brushes very briefly. Section 3 then describes our results on mean square displacements of individual monomers, resolved according to their position along the chain, and the associated relaxation times. Section 4 presents our conclusions, giving also an outlook to possible experiments.

the monomeric units). These radii were chosen such that a regular subset of f particles of these surfaces, which also correspond to a (coarser) geodesic subdivision, acts as anchor points for the grafted chains. The choices of f are f = 42 (Rc = 7), 92 (Rc = 7.9) and 162 (Rc = 8.35), corresponding to grafting densities σ = 0.068, 0.118 and 0.185, respectively. We remind here that in our previous studies on spherical brushes we considered specific grafting densities in order to compare our data with the results obtained for flat brushes. As already mentioned, the method we used to find the appropriate positions of the grafting sites on the colloidal sphere is a geodesic subdivision,23,32 i.e., a repeated subdivision of triangles. Once the desired number of grafting sites is reached the corresponding value of the core radius is calculated from σ. Using the standard Kremer-Grest57 bead−spring model for the chains, with N = 20, 40, 60, and 80 effective monomers per chain, the density of the effective monomers near the nanoparticle surface corresponds to a semidilute monomer concentration, with moderate stretching of the chains in radial direction. The largest system simulated thus contains 12960 effective monomers, and thus very long molecular dynamics (MD) runs are required (in view of the expected scaling of the relaxation time τ ∝ N3 in the flat brush limit36−42). As a consequence we did not attempt to study larger systems. As mentioned above, the monomers of the chains interact with a purely repulsive truncated and shifted Lennard-Jones potential, r being the interparticle distance,

2. MODEL, SIMULATION METHOD, AND STATIC PROPERTIES Following the method of ref 32 the nanoparticle which forms the core of the spherical brush is described by a hollow sphere of densely packed monomer-like units (Figure 1a). The

⎧ 4ε [(σ /r )12 − (σ /r )6 + 1/4], LJ ⎪ ⎪ LJ LJ U αβ(r ) = ⎨ for r ≤ 21/6σLJ ⎪ ⎪ 0, else ⎩

(1)

choosing also εLJ = 1 to set the scale for the energy. Equation 1 also is used as an interaction between the monomers and the (rigidly fixed) particles forming the surface of the nanosphere (Figure 1a). Bonded monomers interact also with the finitely extensible nonlinear elastic (FENE) potential57,58 VFENE(r ) = 0.5kFENE(R 0/σLJ )2 ln[1 − (r /R 0)2 ],

r < R0 (2)

while VFENE(r > R0) = ∞. As usual, the constants were fixed as kFENE = 30εLJ and R0 = 1.5σLJ. VFENE(r) also acts between the first monomers of a grafted chain and the rigidly fixed grafting site. The temperature was chosen as kBT/εLJ = 1.2, as in our previous work on brush−brush interactions.32 Thus, we do not include solvent particles explicitly. For the present study of chain dynamics under very good solvent conditions, this choice of an implicit solvent model means that we focus on the freedraining Rouse model dynamics,35 as was also considered in previous work on planar brushes.36−42 Finally, we choose the mass m of the effective monomers as m = 1, so the characteristic time unit of the MD runs is τMD = (mσLJ2/ εLJ)1/2 = 1 as well. As discussed earlier,32 this model can be simulated very efficiently using the GROMACS package.59−61 With respect to the choice of the initial configuration and the equilibration procedures, we follow our earlier work.32 Figure 1b shows a typical snapshot picture of a well-equilibrated configuration of our model for spherical polymer brushes. Figures 2 and 3 show typical data for static properties (similar data for a somewhat different model can be found also in ref 23). One can clearly recognize that the nanoparticle

Figure 1. (a) Nanoparticle with radius Rc = 8.35 showing the regular arrangement of the particles forming the surface layer. The grafting sites are a (regularly arranged) subset of these partilces in the surface layer and highlighted in red color. (b) Typical snapshot picture of a spherical polymer brush with Rc = 8.35, f = 162 and N = 40.

arrangement of these units has been obtained by the procedure of geodesic subdivision, as described earlier.23,32 Note that these particles are rigidly fixed and do not carry out any motions, but due to their interactions with the effective monomers of the grafted chains they prevent any penetration of the chains through the nanoparticle surface into the interior of the sphere. We have chosen three values of the nanoparticle radius, namely Rc = 7, 7.9 and 8.35, respectively (all lengths are measured in units of the Lennard-Jones parameter σLJ, of the truncated and shifted Lennard-Jones potential acting between B


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further out the density profile is monotonously decaying. As discussed in our earlier work,23 the data for N = 60 and N = 80 are still close to the star polymer behavior (where ρ(r) ∝ r−4/3 is expected), while for the smaller values of N the transition regime to flat brush-like behavior is entered (see panel c). ρ(r) is approximately parabolic, but rounded into a tail near the brush height h. From Figure 3, we also see that monomers occur at radial distances up to about r ≈ 2.5Rc for N = 20 and r ≈ 3.5Rc for N = 80, and the width of the distribution ρi(r) monotonously increases with the monomer index i. For the free chain end I = N + 1, the distribution extends over a width clearly comparable to the brush height. The last panel in Figure 3 shows the rescaled free end distributions for the different number of monomers per chain. We can see a narrower width for i = 21, while the differences between i = 41, 61, 81 in the scaled representation only reflect statistical uncertainties. All of these findings are quite similar to the corresponding behavior for planar brushes;42 only the depletion zone for ρN+1(r) near the grafting site is more pronounced in the spherical brush case, as expected.23

Figure 2. Radial density profile ρ(r) of all monomers in the brush plotted vs r, for the cases Rc = 7.0 (a), and 8.35 (b), including chain lengths N = 20 and N = 80 in each case, as indicated. The broken straight lines on these log−log plots indicate the star polymer power law. Panel (c) shows all the cases investigated (Rc = 7.0 (a), 7.9, and 8.35), rescaled for the appropriate radius of the core.

3. RESULTS ON TIME-DEPENDENT MEAN SQUARE DISPLACEMENTS AND RELAXATION TIMES Following studies of chain dynamics in polymer melts62,63 and planar brushes,42 we have analyzed the time-dependence of the mean square displacements of individual monomers (i) along the chain, Figures 4−6, 2 gi(t ) = ⟨[ ri (⃗ t ) − ri (0)] ⟩, ⃗

i = 2, ...,

N+1

(3)

Figure 3. Radial density profiles ρi(r) of selected monomers i = 2, 5, ..., N + 1 (note that the grafting site is labeled as i = 1), on a logarithmic scale, plotted vs r (on a linear scale), for the case Rc = 7.9, f = 92, and three choices of N: N = 20, 40, and 80 (as indicated). Bottom panel: rescaled free end distributions for N = 20, 40, 60, 80.

surface acts like a hard core, causing the characteristic density oscillations (“layering”) of the total monomer density, starting with a rather sharp peak near r ≈ Rc + σLJ, due to the monomers with index I = 2, adjacent to the rigidly fixed grafting site I = 1. A much weaker second layering peak can also be seen, but

Figure 4. Log−log plot of gi(t) plotted vs t for the case Rc = 7.0, f = 42, σ = 0.068, and chain lengths N = 20, 40, 60, and 80 as indicated and selected values of the monomer index i (as shown in the figure). Straight lines included for N = 80 and N = 60 show the theoretical slope a = 1. C


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times, t ≫ 1, where each monomer has interacted with many neighbors, and hence a Rouse-like diffusive motion occurs. Note that for times t → ∞ due to the constraint that the chain is grafted, the mean-square displacements saturate at a static fluctuation, which is easily estimated as 2 gi(∞) = ⟨[ ri (⃗ ∞)]2 ⟩ + ⟨[ ri (0)] ⟩ − 2⟨( ri (0) ⃗ ⃗ ⟩⟨ ri (⃗ ∞)⟩

= 2[⟨ ri ⃗ 2⟩ − ⟨ ri ⃗⟩⟨ ri ⃗⟩] (4)

since for t → ∞ the position ri⃗ (0), ri⃗ (t) do not exhibit any nontrivial correlation. For each chain the grafting site is used as the origin of a Cartesian coordinate system (x, y, z), with the zaxis in radial direction. We note that ⟨ri⃗ ⟩ = (0,0,zi) with zi the mean distance of the ith monomer from the grafting site of the chain to which it belongs. Thus, the main interest in gi(t) is in intermediate times, before gi(t) has reached its saturation value. The time that needs to elapse to reach saturation is a natural relaxation time to consider. However, since the asymptotic approach of gi(t) to gi(∞) often is somewhat slow, it is computationally advantageous42,62,63 to rather introduce an (arbitrary!) factor 2/3 to define a relaxation time τi for the ith monomer from gi(t = τi) = (2/3)gi(∞)

Figure 5. Same as in Figure 4, but for the case Rc = 7.9, f = 92, σ = 0.118.

(5)

Of course, one expects that a τi defined from eq 5 would be distinct from other choices (where 3/4 or 4/5 instead of 2/3 would be used, for instance) only by a numerical factor of order unity. This expectation has been checked in related work on chain dynamics in bulk melts62,63 and planar brushes.42 Another interesting regime occurs for 1 ≪ t ≤ τi where the grafting constraint is not yet felt. If N and i are sufficiently large to allow for a sufficiently broad regime of times, before saturation of gi(t) starts to set in, one observes a power-law behavior

g i (t ) ∝ t a

(6)

where the exponent a = 1/2 for the Rouse model with ideal chains, no excluded volume, but a ≈ 0.55 in the dilute limit of chains respecting excluded volume.62 However, these power laws are only seen for times where the monomer motion is not yet sensitive to the center of mass motion of the chain. As pointed out by Johner and Joanny,40 for times t that are intermediate between the time of the third Rouse mode τ(3) R and the above relaxation time τi, it is the diffusive motion (due to the relaxation time τ(1) R of the first Rouse mode) which dominates the monomeric motion. For a planar brush they predict hence a linear behavior ⎛ iπ ⎞ 4kBT ⎟ gi(t ) = sin 2⎜ t, ⎝ 2N ⎠ Nζ

τR(3) ≪ t ≪ τi

τ(3) R

(7)

where ζ is a friction coefficient. Since ∝ N while τI ∝ N (if i is comparable to N) for long chains there should be a significant regime where such a linear behavior of gi(t) with t, i.e. exponent a = 1, can be seen for planar brushes. The data of Reith et al.42 are roughly compatible with the linear behavior in time predicted by eq 7. Figures 4−6 show that the effective exponent aeff seen for spherical brushes is indeed closer to a = 1 than to a = 0.55, expected for star polymers in the excluded volume dominated free draining limit. An interesting feature of our data, as well as the corresponding data for planar brushes by Reith et al.,42 is the

Figure 6. Same as Figure 4, but for the case Rc = 8.35, f = 162, σ = 0.18.

The average ⟨(...)⟩ is taken over both the time origin and over all equivalent monomers of all grafted chains. The behavior at very short times t < 1 (where also gi(t) < 1) has not been used, since due to the use of Newtonian dynamics (implicit in MD42,56) at these very short times a ballistic motion of the monomers occurs, gi(t) ∝ t2, which is not of physical interest here. We are only interested in the behavior at large D

2

3


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observation that at times t≪τi the free end (i = N + 1) moves much faster than monomers further inside the chain. Reith et al.42 have extracted from the short time behavior of the mean square displacements an effective local friction coefficient ζi, and showed that ζi is strongly depressed for the first few mobile monomers (i = 2, 3, 4, ...) and strongly enhanced for the last ones (i = N + 1, N, N − 1), while in the chain interior an almost constant plateau value, ζI = ζ independent of i, has been found. Since Reith et al.42 used the same coarse-grained model, described by eqs 1, 2, for their grafted chains, we expect a similar behavior in the present problem. Such a feature is not included in the Rouse description of Johner and Joanny,40 of course. We also note that (due to the strong stretching limit of self-consistent field theory9,17,18 used in ref.40) the motion of the chain end is strictly restricted to distances not exceeding the brush height, while in the simulations of Reith at al.42 and in the present work (Figures 2 and 3) the chain end clearly makes excursions far beyond the brush height, where it does not feel constraints due to the monomers of the other chains. Finally, Figure 7 presents plots of the relaxation time τi as a function of monomer index i for our choices of grafting density

the star polymer limit (where the size of the polymers would be by far larger than the size of the core particle to which the polymers are grafted) and the planar brush limit (where the brush height is small in comparison to the size of the core particle). We have focused on the behavior of individual monomers, considering the variation of properties with the “chemical distance” from the grafting site. The “chemical distance” is measured simply via the index i that labels the monomers along the chain, i = 1 being the grafting site, so i = 2 is the first mobile monomer and i = N + 1 the free chain end. As expected the static fluctuation gi(∞) = ⟨ri⃗ 2⟩ − ⟨ri⃗ ⟩·⟨ri⃗ ⟩ increases monotonously with i. Now the simple scaling argument would predict that the relaxation time τi is reached when the ith monomer has diffused over the distance related to this static fluctuation, τi ∝ gi(∞)/D = gi(∞)(Nζ /kBT )

(8)

One expects that gi(∞) ∝ iN with ν ≈ 3/5 in the star polymer limit, while gi(∞) ∝ iN2 in the planar brush limit, and so always a monotonous increase of τi with the chemical distance i from the grafting site, is predicted. However, this scaling argument has assumed that the friction coefficient ζ does not depend on the chemical distance. Already Reith at al.42 have obtained evidence from their study of flat brushes that this is not true, but they did consider a single (rather large) grafting density only. In the present work, we have varied the grafting density by a factor of almost three, but we found a nonmonotonic variation of τi with the chemical distance i throughout. In particular, the free chain end on average is moving much faster at all times. In part this is due to the fact that the free end has only one neighbor to which it is bound, and in part due to the fact that the free ends are responsible for the “tail” of the total monomer density profile ρ(r) that extends far beyond the brush height. These monomers in the “tail” far outside of the brush do not collide with monomers of other chains, and hence experience a much smaller friction coefficient. Of course, this effect should become less and less relevant, the smaller the grafting density is, but we feel that it will often be relevant under experimental circumstances. The dynamics of chains in brushes is important for applications in “smart materials” with switchable properties due to rearrangement processes of the chain conformations.64−66 Can one obtain direct experimental evidence on such effects? While collective motions of terminally anchored polymers can be probed very elegantly by evanescent-wave dynamic light scattering, in the case of flat brushes,52 the experimental study of individual chain motions in brushes clearly is more difficult. One could graft a small minority fraction of deuterated chains on an assembly of many spherical polymer brushes to measure the dynamic structure factor S(q, t) by neutron spin echo techniques, but clearly such an experiment is very difficult. The present work was motivated in part by the report67 of an experimental study where silica nanoparticles with grafted poly(ethylmetacrylate) chains where investigated by NMR techniques. In these studies, chains enriched with 13C were synthesized and grafted, but the dynamics was not studied in dilute solution, but rather with spherical nanoparticles embedded in a polymer melt.68 Under these conditions of the experiment, it was found that the monomers near the free chain ends relax faster than monomers in the middle of the chains, and this finding was attributed to entanglement effects. Of course, due to the different conditions of the simulation (dilute solution rather than melt near the glass transition) a 2ν

Figure 7. Plot of the relaxation time τi versus monomer indiex i, for the cases Rc = 7.0 (a), 7.9 (b), and 8.35 (c). Various chain lengths are included. Error bars are only shown when they exceed the size of the symbols; dotted curves are guides to the eye only.

and chain length. We see, that the maximum of the relaxation time τi does not occur at the chain end, but further inside along the chain. Typically the maximum relaxation time occurs for i = N/2. This behavior is qualitatively similar to the findings of Reith et al.42 for planar brushes.

4. CONCLUSIONS In the present paper, the dynamics of monomeric motions of polymers grafted in spherical polymer brushes under good solvent conditions has been studied by molecular dynamics simulations. Using well-equilibrated configurations of the brushes for a coarse-grained model in an implicit solvent framework, we have focused on the time-dependence of meansquare displacements, varying both grafting density and chain length of the grafted chains, in a regime intermediate between E


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(24) Lo Verso, F.; Yelash, L.; Egorov, S. A.; Binder, K. Soft Matter 2012, 8, 4185. (25) Wijmans, C. M.; Leermakers, F. A. M.; Fleer, G. J. Langmuir 1994, 10, 4514. (26) Borukhov, I.; Leibler, L. Marcomolecules 2002, 35, 5171. (27) Starr, F. W.; Schroder, T. B.; Glotzer, S. C. Macromolecules 2002, 35, 4481. (28) Cerda, J. J.; Sintes, T.; Toral, R. Marcomolecules 2003, 36, 1407. (29) Kalb, J.; Dukes, D.; Kumar, S. K.; Hoy, R. R.; Grest, G. S. Soft Matter 2011, 7, 1418. (30) Green, P. F. Soft Matter 2011, 7, 7914. (31) Mc Ewan, M. E.; Egorov, S. A.; Ilavsky, S.; Green, D. L.; Yang, Y. Soft Matter 2011, 7, 1418. (32) Lo Verso, F.; Yelash, L.; Egorov, S. A.; Binder, K. J. Chem. Phys. 2011, 135, 214902. (33) Lo Verso, F.; Egorov, S. A.; Binder, K. Macromolecules 2012, 21, 8892. (34) Meng, D.; Kumar, S. K.; Lane, J. M. D.; Grest, G. S. Soft Matter 2012, 8, 5002. (35) Doi, M.; Edward, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, U.K.; 1986. (36) Klushin, L. I.; Skvortsov, A. M. Macromolecules 1991, 21, 1549. (37) Murat, M.; Grest, G. S. Macromolecules 1989, 22, 4054. (38) Binder, K.; Lai, P. Y.; Wittmer, J. Faraday Discuss 1994, 98, 97. (39) Neelov, I. M.; Binder, K. Macromol. Theory Simul. 1995, 4, 1063. (40) Johner, A.; Joanny, J.-F. J. Chem. Phys. 1993, 98, 1647. (41) He, G. L.; Merlitz, H.; Sommer, J.-U.; Wu, C. X. Macromolecules 2007, 40, 6527. (42) Reith, D.; Milchev, A.; Virnau, P.; Binder, K. Macromolecules 2012, 45, 4387. (43) Halperin, A.; Alexander, S. Europhys. Lett. 1988, 6, 329. (44) Halperin, A.; Alexander, S. Macromolecules 1989, 22, 2403. (45) Wittmer, J.; Johner, A.; Joanny, J.-F.; Binder, K. J. Chem. Phys. 1994, 101, 4379. (46) Merlitz, H.; He, G. L.; Sommer, J.-U.; Wu, C. X. Macromol. Theory Simul. 2008, 17, 171. (47) Fredrickson, G. H.; Ajdari, A.; Leibler, L.; Carton, J. P. Macromolecules 1992, 25, 2882. (48) Marko, J. F.; Chabrabarty, A. Phys. Rev. E 1993, 48, 2739. (49) Solis, F. J.; Pickett, G. T. Macromolecules 1995, 28, 4307. (50) Xi, H. W.; Milner, S. T. Macromolecules 1996, 29, 4772. (51) Long, D.; Ajdari, A.; Leibler, L. Langmuir 1996, 12, 1675. (52) Fytas, G.; Anastasiadis, S. H.; Seghrouchni, R.; Vlassopoulos, D.; Li, J.; Factor, B. F.; Theobald, W.; Toprakicoglu, C. Science 1996, 274, 2041. (53) Sikorski, A.; Romiszowski, P. Physica A 2005, 357, 364. (54) Goyal, S.; Escobedo, F. A. J. Chem. Phys. 2011, 135, 184902. (55) Binder, K.; Kreer, T.; Milchev, A. Soft Matter 2011, 7, 7159. (56) Binder, K., Ed. Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Oxford Univ. Press: New York; 1995. (57) Kremer, K.; Grest, G. S. J. Chem. Phys. 1990, 92, 5057. (58) 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. (59) Berendsen, H. J.; van der Spoel, D.; van Drunen, R. Comput. Phys. 1995, 91, 43. (60) van der Spoel, D.; Lindall, E.; Hess, B.; Groenhof, G.; Mark, A. E.; Berendsen, H. J. C. J. Comput. Chem. 2005, 20, 1701. (61) Hess, B.; Kutzner, C.; van der Spoel, D.; Lindall, E. J. Chem. Theory. Comput. 2008, 4, 435. (62) Paul, W.; Binder, K.; Heermann, D. W.; Kremer, K. J. Phys. II (Fr.) 1991, 1, 37. (63) Paul, W.; Binder, K.; Heermann, D. W.; Kremer, K. J. Chem. Phys. 1991, 95, 7726. (64) Sidorenko, A.; Minko, S.; Schenk-Meuser, K.; Duschner, H.; Stamm, M. Langmuir 199, 15, 8349. (65) Merlitz, H.; He, G-L..; Wu, C.-X.; Sommer, J. U. Phys. Rev. Lett. 2009, 102, 115702.

quantitative comparison with the experiment cannot be performed. Nevertheless, we feel that the dynamics of chains in both planar and spherical brushes is an interesting problem, also relevant for various application of brushes. We hope that the present work will be useful for the understanding of future experimental work, as well as for development of more detailed analytical theories.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (F.L.V.) [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to the John von Neumann Institute (NIC) Juelich for a generous grant of computer time on the JUROPA machine of the Juelich Supercomputer Centre (JSC). L.Y. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) under Grant No. PA/473/8 in the early stage of this work.

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Macromolecules

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(66) Merlitz, H.; He, G.-L.; Sommer, J.-U.; Wu, C.-X. Macromolecules 2009, 42, 445. (67) Friedrichs, C.; Graf, R., Spiess, H.-W. The influence of Siliva Nano-Particles on the Local Molecular Dynamics in PEMA Paper presented at the 2nd Conference on “Polymer Solid Contacts: Interfaces and Interphases” Frankfurt, Germany, July 9, 2010, unpublished. (68) Friedrichs, C.; Emmerling, S.; Graf, R.; Spiess, H.-W. J. Chem. Phys. 2013, 138, 12A503.

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Dynamics of Macromolecules Grafted in Spherical Brushes under

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