Molecular Dynamics Simulation of a Silica Nanoparticle in Oligomeric

Article pubs.acs.org/Macromolecules

Molecular Dynamics Simulation of a Silica Nanoparticle in Oligomeric Poly(methyl methacrylate): A Model System for Studying the Interphase Thickness in a Polymer−Nanocomposite via Different Properties Hossein Eslami,*,†,‡ Mohammad Rahimi,† and Florian Müller-Plathe† †

Eduard-Zintl Institut für Anorganische und Physikalische Chemie and Center of Smart Interfaces, Technische Universität Darmstadt, Alarich-Weiss-Straße 4, D-64287, Darmstadt, Germany ‡ Department of Chemistry, College of Sciences, Persian Gulf University, Boushehr 75168, Iran ABSTRACT: Large scale atomistic molecular dynamics simulation for a nanoparticle in oligomeric poly(methyl methacrylate), composed of 20 repeating units, for a long time, up to 100 ns, are performed. Simulations are done for systems up to 87500 atoms, each containing a single bare or surface-grafted nanoparticle of various diameters and grafting densities. The effect of surface area, surface curvature, grafting density, and hydrogen bonding on the alteration of local structural and dynamical properties of the polymer is studied in details. Although atomistic simulations are only feasible for oligomeric chains in contact with surfaces, the results of the present simulation still discriminate the interphase thickness, defined in terms of local and global chain properties. In the case of structural properties, a minimum interphase thickness, ≈ 2 nm, is associated with local properties such as layering of individual polymer monomers. However, when probed in terms of global chain properties like the extension of chains from the interface to the polymer phase, a thicker interphase, three times the radius of gyration of the unperturbed chain (Rg ≈ 1 nm), is observed. Our results on the chain structures are shown to be in good agreement with experiment where available. An examination of the dynamical properties shows that the surface influence on the polymer dynamics depends on the length and time scale of the corresponding bulk property. The change in time scales, in a 0.5 nm thick spherical shell, around the nanoparticle, is shown to cover a broad range from a few tens of a percent (for a short-time dynamical property, like the hydrogen bond formation) to 15− 20 orders of magnitude (for a long-time dynamical property, such as the relaxation of end-to-end vector in grafted chains). Therefore, the influence and the range of surface effects on dynamical properties (interphase thickness) depend on the inherent time scale of those properties. In all cases, a thicker interphase is observed for global structural and long-time dynamical properties for chains in contact with a flatter and more densely grafted surface. Hydrogen-bond formation between polymer and surface decelerates the polymer dynamics. The effect is more pronounced at low temperatures.

1. INTRODUCTION Polymers reinforced with nanoparticles comprise an emerging class of materials due to their extraordinary enhanced properties. Compared to neat polymers, certain polymer nanocomposites exhibit a significant increase in tensile modulus and strength without loss of impact resistance and heat distortion temperature.1 Polymer nanocomposites offer new multifunctional properties, which are not observed with micrometer-size fillers. For example, compared to the pure poly(methyl methacrylate) (PMMA), a higher transparency, an increase in the tensile strength, storage elastic modulus and surface hardness, and improvements in the thermal stability in the PMMA−silica nanocomposites, are reported.2 Also, addition of such fillers as carbon nanotubes to the PMMA increases the Young modulus, and the hardness of the composite.3 © XXXX American Chemical Society

Among polymers, PMMA is the most commercially important acrylic polymer, used in many applications. Because of its high transparency and low density, PMMA is an ideal replacement for glass. Being compatible with human tissue makes PMMA an important material for transplants and prosthetics, especially in the field of ophthalmology. Also because of similarity of its elastic modulus to natural bone, PMMA is used as bone cement in orthopedic surgery.4 Due to the widespread applications of PMMA, nanocomposites have been prepared from it by the addition of nanoparticles such as organically modified clays,5 layered double hydroxides,6 layered hydroxy salts,7 silica,8 silica/titania,9 Fe2O3 and TiO2,10 Received: July 12, 2013 Revised: October 13, 2013

A

dx.doi.org/10.1021/ma401443v | Macromolecules XXXX, XXX, XXX−XXX

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aluminum hydroxide,11 and carbon nanotubes.12 Despite the improvements achieved, the development of polymer nanocomposites is still largely empirical. Therefore, a better understanding of structure−property relationships between polymer and filler is still needed in the improvement of different classes of nanocomposites. Computer modeling and simulation play an increasing role in predicting and designing material properties, and guiding synthesis and characterization. These methods are of particular importance in elucidating the molecular understanding of the structure and dynamics at the interface between nanoparticles and polymer matrix and, hence, the molecular origins of such phenomena as reinforcement, and the impact on the mechanical, thermal, fire, and barrier properties. Although Monte Carlo (MC) and molecular dynamics (MD) simulation methods have been widely applied to study the structure and dynamics of model bead−spring polymer chains in contact with model (spherical) nanoparticles,13 simulation reports on realistic polymer−nanoparticle systems are relatively scarce. Of the limited reports on the MC and/or MD simulations of realistic polymer chains in contact with a nanoparticle surface we may address to the works by Barbier et al.14 on the MD simulation of interface between poly(ethylene oxide) and silica, MD simulations on the interphase structure and dynamics of polystyrene near bare and coated Au nanoparticles, by Milano et al.,15 and that of polyimide near a silica nanoparticle, by Komarov et al.,16 MC simulation of a coarse-grained model of polystyrene-silica nanocomposite by Vogiatzis et al.,17 detailed atomistic18 and coarse-grained19 MD simulations of polystyrene-silica nanocomposite, and atomistic20 MD simulations of short chains and coarse-grained21 MD simulations of long chains of polyamide-6,6 in contact with graphene surfaces, which can also be regarded as the infinite diameter limit of nanoparticles, by this group. The results of all these studies indicate that the filler modifies the polymer structure in its neighborhood. This contribution goes beyond our previous works on nanocomposites,18,19 which involved polystyrene as an essentially apolar polymer. We have performed detailed atomistic simulations of PMMA in contact with silica nanoparticles, as a model for a polar polymer in contact with a polar surface, with the additional ability of hydrogen bond (HB) formation between surface and polymer. To be able to study the effect of surface curvature on the nanocomposites properties, simulations were done on PMMA plus spherical silica nanoparticles of different diameters. As grafting polymer to the surface is one of the methods of compatibilisation, we have also investigated grafted silica particles and compared the results with those for ungrafted surfaces and for bulk polymer. For the sake of computational efficiency, the simulation results in this work are mostly given at higher temperatures (550 K) than the ambient temperatures. However, it is known22,23 that the polymer structure at the interface does not change so much from the high-temperature melt state to the low-temperature glassy state. Therefore, the results of the present work are useful in giving address to fundamental molecular-scale phenomena governing the alteration in structure and dynamics of polymer at the interface, with respect to the bulk polymer.

nanoparticles of diameters 2 and 4 nm were simulated. The spherical silica nanoparticles were made according to the crystal structure of α-quartz24 and all silicon atoms lying outside a spherical surface as well as oxygen atoms not bonded to the retained silicon atoms were deleted. All surface silicon atoms bonded to three surface oxygen atoms were deleted and all the remaining surface oxygen atoms were saturated with hydrogen to satisfy their chemical bonding. The PMMA chains (both grafted and free) were composed of 20 repeating units, for which the chemical structure is shown in Figure 1. We have

Figure 1. Chemical structure of PMMA chains simulated in this work.

studied different nanocomposite systems in which the polymer chains were in contact either with a bare (ungrafted) or with a PMMA−grafted silica nanoparticle. To generate polymergrafted nanoparticles, nearly equal distant hydrogen atoms on the silica surface were chosen and perfectly stretched PMMA chains were grafted to the silica surface oxygen atoms via the end carbon atoms of the PMMA.

3. SIMULATION DETAILS In this work several PMMA-silica nanocomposites, at different grafting densities, ρg, and different nanoparticle diameters, σ, were simulated. The force-field parameters for silica and PMMA were taken from Lopes et al.25 and Kirschner et al.,26 respectively. All simulations were performed using our simulation package, YASP.27 Because of the slow dynamics of polymer at the interface, in this work all simulations were performed at 550 K. The temperature and pressure (101.3 kPa) were kept constant by coupling the system to a Berendsen thermostat and barostat,28 with coupling times of 0.2 and 3.0 ps, respectively. In all simulations, the geometrical center of the nanoparticle coincided with the Cartesian coordinate origin and was located at the center of the simulation box. The atoms of the silica nanoparticle were kept immobile during the simulation. The pressure is estimated using the kinetic energy and the virial of the polymer atoms only, the volume necessary in the transformation is the volume of the simulation cell minus that of the nanoparticle. Our computations show that the calculated pressure, using this method, based on a sufficiently large subvolume of the system is the same as that calculated with fully mobile SiO2 atoms. All nonbonded interactions were truncated at 0.95 nm with a reaction field correction for the Coulombic interactions.29 The effective dielectric constant was taken to be 2.57.26 An atomic Verlet neighbor list was used, which was updated every 15 time steps, and the neighbors were included if they were closer than 1.0 nm. The time step for the leapfrog integration scheme was 1.8 fs. A total of 7 systems containing nanoparticles with σ = 2.0 nm and σ = 4.0 nm, each with grafting densities ρg = 0, 0.5, and 1.0 chain nm−2 as well as a bulk sample (no nanoparticle) were simulated in this work. The characteristics of all systems simulated are summarized in Table 1. Simulations were performed for 30 ns to achieve equilibrium, as verified by the relaxation of the end-to-end

2. MODEL MD simulations were performed on PMMA-silica nanocomposites. To examine the effect of surface curvature on the polymer properties at the interface, systems containing B


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Table 1. Description of Systems Simulated in This Worka 100 × NHB ρg (nm−2)

Nnp

nf

ng

0 0.5 1.0

3258 3233 3208

276 251 226

0 25 50

0 0.5 1

456 450 444

170 164 158

0 6 12

−

0

100

−

ρ (kg m−3)

N σ = 4 (nm) 87438 87388 87338 σ = 2 (nm) 52306 52294 52282 Bulk 30500

L (nm)

free

grafted

1080 1086 1090

9.62 9.60 9.59

11.1 2.1 1.0

0.0 9.1 7.7

1072 1074 1076

8.12 8.11 8.11

12.9 5.6 2.8

0.0 5.3 7.3

1071.4

6.79

0.0

0.0

σ is the nanoparticle diameter, ρg is the grafting density, Nnp is the number of atoms in the nanoparticle, nf and ng are the number of free and grafted chains, respectively, N is the total number of atoms, ρ is the density, L is the dimension of the cubic box, and NHB is the number of hydrogen bonds per donor. a

Figure 2. Snapshot of a simulation box indicating PMMA chains grafted to the silica nanoparticle of σ = 4 nm with ρg = 0.5 nm−2.

radial density profiles in Figure 3 the centers of mass of PMMA monomers (repeat units) are put into spherical shells, of thickness 0.01 nm, centered at the nanoparticle center of mass and the number of monomers is time-averaged in each shell. The results are indicated as a function of radial distance, d, from the nanoparticle surface. In all calculations, the nanoparticle radius is taken into account as the position of nanoparticle surface. It is worth mentioning that defining the nanoparticle surface as an average over the radial positions of H, O, and Si atoms in the outer shell of nanoparticle produces a quite close value to the nanoparticle radius. Organized layers of monomers are formed close to the nanoparticle surface. The layering is more pronounced for bigger (σ = 4.0 nm) than for smaller nanoparticles (σ = 2.0 nm). When part of a chain is in contact with a flat surface, the rest of this chain is at closer distance to a flat surface than to a curved surface, and hence, experiences more attraction. This means that the chain adhesion is

vector of polymer chains whose center of mass were located within 1−2 nm from the nanoparticle surface (see below). After equilibration, long production runs, up to 100 ns, were performed to collect data every 2.0 ps. A snapshot of the simulation box, indicating PMMA chains grafted to a silica nanoparticle (σ = 4.0 nm) is shown in Figure 2.

4. RESULTS AND DISCUSSION 4.1. Polymer Structure in the Interphase. 4.1.1. .Number Density Profiles. It is well-known that fluids near surfaces form organized layered structures. We have observed such layering behavior in the case of polymers in contact with flat20,21 (infinite nanoparticle diameter) and curved surfaces18,19 in our previous atomistic and coarse grained modeling of polymer nanocomposites. Here, the radial number density profiles, normalized by the bulk number density, ρ0, for systems tabulated in Table 1 are shown in Figure 3. To calculate the C


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Figure 3. Normalized density profiles for all (grafted + free) PMMA monomers (top panel) and for grafted and free PMMA monomers (bottom panel), at 550 K and 101.3 kPa, as a function of distance from the silica surfaces. The dashed curves indicate the normalized density profiles for end monomers of free chains. The description of systems is given in the figure’s legend.

energetically more favorable on flatter, rather than on more curved, surfaces. As the grafted and free polymer chains are completely miscible, the overall (grafted + free) monomer density peak heights do not change noticeably with changing grafting density. The only difference is that the first density peak in grafted systems starts at a shorter distance. This is understandable, as the polymer is chemically tied to the surface in grafted systems, while there is a free volume between polymer and surface for free polymer chains. As the nanoparticle surface is not perfectly spherical, especially in the case of particle with σ = 2.0 nm, a small peak at d < 0.2 nm is observed in Figure 3. We have also shown in Figure 3 the distinct monomer density profiles for free and grafted chains. At shorter distances from the nanoparticle surface, the contribution of the grafted chains to the overall density peaks is larger than that of free chains. In other words, only a small fraction of free chains, depending on the grafting density, adsorbs on the nanoparticle surface. Increasing the grafting density stretches the grafted chains away from nanoparticle surface. This is explicable in terms of the higher repulsion between chains grafted at higher than at lower density, which forces chain density to escape in the direction of the surface normal. The density peaks for the grafted chains are more pronounced in systems with bigger nanoparticles. On the contrary, the peaks for free chains are more pronounced in systems with smaller nanoparticles.

Having a fixed density of grafted chains at the surface of a nanoparticle, the space between the grafted chains is larger for higher surface curvature, which leaves more space for the penetration of the grafted corona by free chains. To have a direct comparison of our results with experimental data, obtained at room temperature, we have cooled a nanocomposite sample (containing the ungrafted σ = 4 nm nanoparticle) to 310 K, with a cooling rate of 5 K ns−1. Experimental observations30 indicate that the maximum amount of PMMA adsorbed on a silica surface, from a concentrated toluene solution at 295 K, ranges between 0.8 and 1.1 mg m−2. Integrating over the first density profile peak in PMMA-silica sample at 310 K (d < 0.7 nm, not shown), indicates an adsorption amount of 1.05 mgm−2, which is in a pretty good agreement with experiment. Moreover, X-ray reflectometry experiments31 on density profiles of samples of iPMMA in contact with flat silica surfaces at 296 K shows that the normalized height of the first density peak, ρ/ρ0, is around 2. This is quite comparable to what is observed in our simulations. It is worth mentioning that the samples in this work are cooled down to lower temperatures with a much faster rate than the experimentally relevant cooling rates (a few ten so of kelvin per minute). However, reports in the literature show that despite much larger cooling rates of polymer samples in simulations (compared to experiment), for example, the D


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in systems with the smaller nanoparticle is increasing with the grafting density, while it is decreasing in systems with the bigger nanoparticle. Both observations can be interpreted in terms of decreased repulsions between the chains and, more penetration of the free chains to the nanoparticle surface over a more curved surface. Also the results in Table 1 indicate that the ratio of HBs per donor formed by grafted chains to those formed by the free chains is increasing with increasing the grafting density, which can be attributed to more accessibility of carbonyl O of the grafted chains in close vicinity of the surface. We have also shown in Figure 4 the normalized HB distributions in spherical shells centered on nanoparticle center

calculated glass transition temperatures of long polymer chains are within a few kelvin of experiment.32−34 In the case of the present oligomeric chains, even smaller cooling rate dependency of the calculated properties is expected. To have a better picture of the chain contact to the interface, we have also shown in Figure 3 the radial density profiles for end group monomers. They show a more pronounced structure. Compared to the corresponding distributions of all monomers, the end group distributions are sharper and are shifted to shorter distances from nanoparticle surface. The magnitude of increase in the end-monomer density heights, compared to all monomer distributions, depends on the surface curvature and on the grafting density. The ratios of the peak heights in the end-monomer to those for all monomer distributions for free chains in systems with ρg = 0, 0.5, and 1 nm−2 varies as 1.37, 1.75, and 2.32, respectively, in systems containing σ = 4 nm nanoparticle, while it is 1.34, 1.66, and 1.97, respectively, in systems containing σ = 2 nm nanoparticles. Due to the repulsion between grafted chains over flatter surfaces and the excluded volume interactions between the free and grafted chains, the attachment of free chains to the surface is more probable to occur via chain ends. A similar enrichment of chain ends at a solid interface has also been found in coarse-grained simulations of polyamide at graphene.21 Our results on polymer−surface density profiles indicate that at distances ≈2.0 nm from the surface, the density profiles converge to the bulk density. The same interphase thickness, has been reported in previous atomistic simulations of short polymer chains15,18,20 and in coarse grained17,19,21 and DPD35 simulations of long polymer chains. This means that in the case of a local structural property, like the packing of monomers, the interphase thickness is just a few monomer diameters, regardless of the chain length. Similar length scales for layering beside various surfaces have been observed in surface force apparatus experiments.36 X-ray reflectometry experiments31 on samples of i-PMMA and s-PMMA in contact with flat silica surfaces at 296 K report an interphase thickness (determined in terms of density profile convergence to the bulk value) around 5 nm. 4.1.2. Hydrogen Bonding. A distinct feature of the polymer nanocomposite studied in this work, compared to the previously studied ones18−21 is the possibility of hydrogenbond (HB) formation between the OH hydrogens of the surface and the carbonyl O atoms of the PMMA. There are a few reports in the literature on the simulation of polymer− nanoparticle systems (PEO−TiO2), capable of HB formation.37,38 However, to our knowledge, none of them have studied the hydrogen bonding between polymer and nanoparticle and its effect on the polymer dynamics. In this work, the HBs are counted based on a geometric criterion in which the H···O bond distance is less than a threshold value of 0.3 nm and the donor−hydrogen−acceptor angle, θO−H···O, is bigger than 130°. Based on these criteria, we have counted the number of HBs per donor in close vicinity of the nanoparticle surface. The number of HBs per donor for free and grafted polymer chains at 550 K is given in Table 1. It is seen that in all cases roughly 10% of OH donors at the surface form HBs with the polymer. Unsurprisingly, the number of HBs per donor for free polymer chains decreases with increasing the grafting density, with a faster decrease for systems containing bigger nanoparticles compared to those with smaller nanoparticles. Moreover, the contribution of grafted chains to HB formation

Figure 4. Normalized distribution functions for the number of hydrogen bonds per donor as a function of the distance from the nanoparticle surface.

of mass. The maximum probability occurs at a distance about 0.15 nm from the surface. Because of the fact that the nanoparticle surface is not perfectly spherical, the distributions start even at negative values from the spherically assumed nanoparticle surface. The normalized HB distributions for free polymer chains shift to longer distances from the surface with increasing grafting density. The distributions for grafted chains, on the other hand, shift to shorter distances. For the sake of comparison with experiment, we have also calculated the number of HBs per donor for our PMMA-silica composites at 310 K. In this case the number of HBs per donor (0.65) is quite comparable with the experimental reports of Kulkeratiyut et al.,39 0.62, at 295 K. We have also calculated the enthalpy change for HB breakage from the results of our simulations at 310 and 550 K, using the Gibbs−Helmholtz equation. Because of the lack of experimental data, the calculated enthalpy change, 15.5 kJmol−1, cannot be compared with experiment. However, this enthalpy change is within the range of reported values for HB rupture in water (23 kJ mol−1)40 and in polyamide-6,6 (15.4 kJ mol−1).41 4.1.3. Chain Conformation and Orientation. In order to investigate the orientation of chains in contact with the surface, we have computed the orientational distribution function for vectors connecting Cα of subsequent repeat units along the polymer backbone, Cα−Cα vectors. The orientation profiles are described by a second Legendre polynomial, expressed as: E


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1 ⟨3(u m·u n)2 − 1⟩ (1) 2 where um is the Cα−Cα unit vector, un is a unit vector normal to the surface from the midpoint of the Cα−Cα vector, p2 is the second Legendre polynomial, and d is the distance of the Cα− Cα vector (midpoint) to the surface. We show in Figure 5 the p2 (d) =

Figure 6. Normalized radius of gyration of chains as a function of chains’ center-of-mass distance from the nanoparticle surface. The full and dashed curves represent the results for free and grafted chains, respectively.

Obviously, those grafted chains whose center of mass is at longer distances from the surface must be stretched and show a higher radius of gyration. This effect increases with grafting density and with decreasing surface curvature. As stated earlier, this is due to stronger repulsion between the grafted chains above flatter surfaces which causes the chains to stretch away from the interface. As a measure of distortion of chain structure in the radial and tangential directions, we have calculated the anisotropy in the chain structure, defined as the ratio of tangential to radial components of the radius of gyration, for the sample containing σ = 4 nm particle. In this case, the anisotropy for free chains in system with ρg = 0 nm−2 varies from 4.6 for centers of mass within a distance 0 nm < d < 0.5 nm to 1 for those located at d ≈ 2 nm. For grafted chains in system with ρg = 1 nm−2, the anisotropy varies from 3.1 to 0.19, in the afore-cited regions, respectively. Flattening of the polymer chains beside the surfaces has also been reported in previous simulations.15−21,36 Our results of this section indicate that the interphase thickness, determined in terms of chain’s global structure, is ≈2.5Rg (≈ 2.3 nm). The radius of gyration of the unperturbed chain, R0g = 0.9 nm, is too short to distinguish this interphase thickness from that obtained in terms of local chain structure in section 4.1.1. Calculations on much longer chains using coarse-grained models have, however, established for different polymer−solid combinations unambiguously, that the radius of gyration is the governing length scale for the structural features of entire chains to converge to their bulk behavior.21 4.1.4. Configurational Properties. Particularly useful information regarding the chain structure in the interphase can be obtained by categorizing the chains in terms of the way they contact the surface. We define a surface contact when any monomer is located within a distance 0 nm ≤ d ≤ 0.7 nm from

Figure 5. The second Legendre polynomial for the orientation of the Cα−Cα unit vectors with respect to the unit vectors normal to the surface of nanoparticle from the midpoints of Cα−Cα vectors. The description of systems is given in the figure’s legend. The full and dashed curves represent the results for free and grafted chains, respectively.

profile for free and grafted chains. The Cα−Cα vectors for free chains, with their centers of mass within 0.5 nm from the surface, orient preferentially perpendicular to the surface normal (p2 < 0). The magnitude of flattening depends on the surface curvature; flatter conformations are adopted in contact with flatter surfaces. At distances ≈2.0 nm from the nanoparticle surface, the chains adopt random conformations (p2 = 0). Expectedly, monomers of grafted chains, which are located at longer distances from the surface (d > 1.5 nm), orient preferentially normal to the surface (p2 > 0). More useful information regarding the surface effect on the chain’s global conformations can be obtained from the chains’ mean-squared radius of gyration. We have calculated it for free and grafted PMMA chains, by putting their centers of mass into spherical shells, of thickness 0.1 nm (Figure 6). Both free and grafted polymer chains in close vicinity of the nanoparticle surface are expanded compared to bulk chains. The magnitude of chain stretching depends on the nanoparticle diameter (surface curvature) and on the grafting density. The radius of gyration of the chains near the bigger particle increases by more than 50% above its unperturbed value, 0.9 nm. With increasing surface curvature, the radius of gyration is distorted less. F


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the surface, i.e., within the first monomer density peak in Figure 3. The polymer segments between contacts are categorized in terms of simple configurations: tails bounded by one contact, loops enclosed by two contacts, and trains characterized by adsorbed consecutive monomers. We have shown in Figure 7 the normalized density profiles for tails and loops. The density profile for trains is identical to

distributions extend to longer distances from the surface compared to that containing σ = 2 nm nanoparticle. The trend, however, turns around in the case of higher grafted systems, in which a higher population of tails and trains (in free chains) is observed around the smaller nanoparticles (see Figure 3 for free chains in systems with ρg = 1 chain nm−2, but containing nanoparticles of different diameters). Because the grafted chains crowd the space more over flatter surfaces, a smaller fraction of free chains can penetrate to the surface of big grafted nanoparticles compared to smaller grafted nanoparticle. Similarly, for grafted chains the population of tails and loops depends on the surface curvature; higher populations of tails and loops are observed around flatter surfaces. Because tails and/or trains are attached to the interface, their dynamics is different from the bulk polymer (this is shown in the next section). Therefore, the interphase thickness, defined in terms of the chain extension of attached chains toward the bulk, is about 3.3Rg, which is thicker than the interphase defined in terms of local properties, such as monomer density profiles. A similar difference in the interphase thicknesses was found in our previous coarse grained model of polymers in contact with flat and curved surfaces.18−21 4.2. Polymer Dynamics in the Interphase. 4.2.1. Hydrogen Bond Dynamics. The effect of surface curvature and surface grafting on the dynamics of hydrogen bonds (HBs) can be analyzed in terms of the following correlation function, originally proposed by Rapaport:42

C(t ) =

⟨h(t )h(0)⟩ ⟨h⟩

(2)

Here t is the time and h(t) is a binary function; h(t) = 1 if a donor−acceptor pair is bonded at time t and h(t) = 0 if it is not. The intermittent correlation function C(t) is the probability that a donor−acceptor pair is bonded at time t provided that it was bonded at t = 0 (independent of possible breakings in the interim time). We show in Figure 8 the intermittent correlation function for all systems. The function C(t) always shows a very fast initial decay ( 2 nm from the nanoparticle surface. The curve labeled as no HB represents the results for the same system, but the charges of surface H atoms of silica nanoparticle are turned off, to prevent hydrogen-bond formation between surface and polymer.

Figure 11. Local tangential component of the mean-squareddisplacement for free chains (full curves) compared with its radial component for free chains (dashed curves) and for grafted chains (dotted curves) for the system containing σ = 4 nm with ρg = 1 nm−2.

results indicate that the free chains translate much faster in the tangential direction than in the radial direction. This is expected, as the chains beside the surface adopt flattened pancake conformations, their translation in the radial direction, compared to the tangential direction, is substantially hindered. A comparison of the results for mobility of free and grafted chains in the radial direction shows that the grafted chains are more mobile than the free ones. As the normal Einstein diffusion regime for the chain centers of mass is not reached in these simulations, the MSDT and MSDR values at 30.0 ns in 1.0 nm thick shells are used to rank the chain translation in the interphase. In this regard, the chain translations are compared to that of bulk polymer in Table 2. The surface restricts the entire chain translation, in both tangential and radial directions, up to a distance of 3.0 nm (3.3Rg). The reduction of the mobility, however, depends on the surface curvature (more) and on the grafting density (less). Moreover, the deceleration by the surface is more pronounced for radial translation (across the interphase), than for tangential translation (along the interphase). A comparison of the

containing σ = 4.0 nm nanoparticle. For chains with centers of mass located at d < 0.5 nm, the end-to-end vector only decays 2% over a time span of more than 60 ns. The decay rate of the end-to-end autocorrelation function, however, increases with distance from the surface. Even farthest away from the nanoparticle (d > 2.5 nm), the end-to-end vector decays slower than in a bulk polymer. This means that in terms of the chains’ overall dynamics the whole box simulated in this work belongs to the interphase. However, the relaxation behavior of the shell 2.5 nm < d < 3 nm and the bulk polymer is not very different, so this outermost shell is probably relatively close to bulk. Still, atomistic simulations of bigger systems would be required to confirm this assertion. Surface curvature and grafting density have an effect on the relaxation of the end-to-end vectors for both free and grafted J


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surface-polymer interactions than in their presence (Figure 12). However, the interphase thicknesses reported in this work in the case of a polar polymer-polar filler system, is nearly indistinguishable from those reported in previous literature in the case of apolar polymer-apolar filler systems. The reason is that in atomistic simulations of short polymer chains, the interphase thickness defined in terms of different local and/or global chain properties is about a few nanometers, which is the range up to which the effect of polar−polar interactions spans. However, a comparison of our previous atomistic and coarse grained simulations simulation results show that coarse grained simulations have the ability of discriminating the interphase thickness in terms of local properties (with a smaller interphase thickness) and global properties (with a longer interphase thickness). It is also worth examining the dynamics of tails. Grafted chains, whose monomers extended to long distances (d > 2 nm), have their center of mass in d > 1 nm. For the sake of comparison, we have also included in Figures 12 and 13, the reorientation correlation functions for free-chain tails, which are equally stretched (monomers extend to distances d > 2.0 nm). A comparison of the reorientation correlation function of such tails with unattached chains at the same distance indicates that they have a much slower dynamics. Since they are connected to the interface, they have a dynamics very comparable with that of grafted chains (see Figure 13). Therefore, the interphase thickness defined in terms of the dynamics of chains extended from the interface to the polymer phase is at least ≈3 Rg. We have also tabulated in Table 2 the relaxation times τT for decorrelation of end-to-end vectors of tails whose monomer density extends to d > 2 nm. As indicated already by their correlation functions (see above) the tail relaxation is comparable to that of grafted chains, whose center of mass is located ≈1.5 nm from the surface. Again the τT values, normalized with the relaxation time of the unperturbed chain end-to-end vectors, are bigger for systems containing more densely grafted nanoparticles with a higher surface area.

chains. This is shown for the chain centers-of-mass located within 1 nm < d < 2 nm (Figure 13). Increasing the surface

Figure 13. A comparison of the decay of local reorientation correlation functions for the end-to-end vectors for chains with their center of mass located within 1.0 nm < d < 2.0 nm. The description of the systems is given in the figure’s legend.

curvature (from σ = 4 nm to σ = 2 nm) causes a faster relaxation of the end-to-end vectors. The end-to-end vector of free polymers reorients slower with increasing grafting density (from ρg = 0 nm−2 to ρg = 1 nm−2). The reason is that the grafted chains have a slower dynamics than the free chains, due to their connection to the interface. The slower dynamics of grafted chains hinders the reorientation of free chains, too. The grafted chains relax slower than the free chains for a grafting density of 0.5 nm−2, whereas they are very similar for 1 nm−2. Relaxation times τee were obtained by fitting the relaxation curves by a Kohlrausch−Williams−Watts (KWW) stretched exponential function and integrating analytically (Table 2). They depend on surface area and grafting density. The chain end-to-end vector relaxation for both free and grafted chains is affected to a different extent by the distance from nanoparticle surface. At close distances (d < 0.5 nm), a deceleration by several orders of magnitude compared to the bulk polymer is observed. The dynamics beyond d = 2.5 nm is still slower than bulk, but is in the same order. Furthermore, a higher grafting density causes a stronger dynamic deceleration. A caveat regarding the extremely high correlations times τee in the first two shells (d < 1 nm) in Table 2. They were estimated from KWW fits to correlation functions which have not even started to converge. Therefore, they carry a big statistical uncertainty. For example, fitting these relaxation functions in the range 0 ns < t < 10 ns and 0 ns < t < 50 ns gives relaxation times ranging from 1015 ns to 1020 ns, respectively. We can, however, still conclude that the end-to-end relaxation of a chain at the surface is much larger than in the bulk. It is worth mentioning that the values of β, in the KWW analysis, range between 0.15 and 0.35 and no intercorrelation between β and τ values were detected. To obtain an estimate of the effect of polar interactions on the dynamic deceleration, in the present simulation of polar polymer-polar particle composite, we have turned off the charges on the silica surface H atoms and tuned the charges on the adjacent O atoms to obtain a neutral silica core. The endto-end vectors of chains with their centers of mass located within d < 0.5 nm decorrelate faster in the absence of polar

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CONCLUSIONS Atomistic molecular dynamics simulation for a nanoparticle in oligomeric poly(methyl methacrylate), for a long time, up to 100 ns, are performed. The simulated systems contain nanoparticles of diameters 2 and 4 nm at different grafting densities, ρg = 0, 0.5, and 1 chain nm−2. Polymer monomers form organized layered structures close to the nanoparticle surface up to a distance of ≈2 nm. The degree of organization depends on the surface curvature and on the grafting density. A good comparison with experiment30,31,36 is found regarding the amount of PMMA adsorbed on the silica surface, the height of the first density profile peak, and the interphase thickness (determined in terms of density profile convergence to bulk value). End group monomers are enriched at the surface. This effect is more pronounced in the case of flatter surfaces with a higher grafting density. Both free and grafted chains form hydrogen bonds with the surface, but the contribution of the grafted chains is larger than that of the free chains. Overall, only 1/10 of the surface OH groups are hydrogen bonded at 550 K. However, in complete agreement with experiment,39 at low temperatures (310 K) 65% of surface OH groups are hydrogen bonded with polymer. The chains are shown to adopt flattened conformations near the surface and the degree of flattening depends on the surface curvature. A substantial fraction of the chains, which have at K


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least one contact to the interface, are shown to extend to distances as long as 3Rg from the surface, and hence, extending the interphase to ≈3Rg. Therefore, the results of our simulations still associate different interphase thicknesses with local and global chain properties. In the case of structural properties, a minimum interphase thickness, ≈2 nm, is associated with local properties such as layering of individual monomers. When probed by global chain properties like the radius of gyration or the extension of chains from the interface toward the polymer phase, a thicker interphase, about 3Rg, is observed. Similar to the chain structure, the dynamical properties of the chains are also affected by the surface. The chain translation in the interphase is shown to be anisotropic; the surface slows down the mobility in the radial direction more than in the tangential direction. The degree of slowing of the polymer dynamics by the nanoparticle surface depends on the time scale of the corresponding bulk property as well as on the surface features (surface area, surface curvature, and grafting density). The change in time scales, in a 0.5 nm thick spherical shell, centered at the nanoparticle center, is shown to cover a broad range from a few 10% to 15−20 orders of magnitude. Faster dynamical properties (e.g., hydrogen-bond formation/rupture) are affected less by the surface, and hence, depend less on the surface features. On the other hand, long-time dynamical processes, like the reorientation of chain end-to-end vectors, are affected more by the surface. They are more affected in systems with bigger nanoparticles (higher surface area, lower surface curvature) and more densely grafted surfaces. Therefore, the maximum difference in the time scales of dynamical properties between a surface-adsorbed and a bulk chain is found for the reorientation of end-to-end vectors of chains grafted to the surface of bigger nanoparticle at a high grafting density. Even in ungrafted systems, the dynamics of chains, which have one contact to the surface and extend into the polymer phase, is shown to be influenced considerably by the surface. Because of their contact to the surface, such tails move much more slowly than the bulk polymer and are comparable to grafted chains. We can, therefore, conclude that, in the interphase, dynamics follows structure: Using local structural properties to probe the interphase leads to an interphase thickness of 2 monomer diameters or 2 nm, and so do local relaxation processes involving small units. When either the structure or the dynamics of entire polymer chains is used to study interphase thickness, then it comes out to be about 2 − 3 radii of gyration. This assessment agree with previous studies on other polymer− nanoparticle systems.17−21

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REFERENCES

(1) Ramanathan, T.; Stankovich, S.; Dikin, D. A.; Liu, H.; Shen, H.; Nguyen, S. T.; Brinson, L. C. J. Polym. Sci., Part B: Polym. Phys. 2007, 45, 2097−2112. (2) Sugimoto, H.; Daimatsu, K.; Nakanishi, E.; Ogasawara, Y.; Yasumura, T.; Inomata, K. Polymer 2006, 47, 3754−3759. (3) Luo, J.-T.; Wen, H.-C.; Wu, W.-F.; Chou, C.-P. Polym. Compos. 2008, 29, 1285−1290. (4) Grimsrud, C.; Raven, R.; Fothergill, A. W.; Kim, H. T. Orthopedics 2011, 34, e378−e381. (5) Jeon, I.-Y.; Baek, J.-B. Materials 2010, 3, 3654−3674. (6) Wang, G.-A.; Wang, C.-C.; Chen, C.-Y. Polym. Degrad. Stab. 2006, 91, 2443−2450. (7) Majoni, S.; Su, S.; Hossenlopp, J. Polym. Degrad. Stab. 2010, 95, 1593−1604. (8) Rittigstein, P.; Priestley, R. D.; Broadbelt, L. J.; Torkelson, J. M. Nat. Mater. 2007, 6, 278−282. (9) Wang, H.; Meng, S.; Zhong, W.; Du, W.; Du, Q. Polym. Degrad. Stab. 2006, 91, 1455−1461. (10) Ismail, L. N.; Ahmad, F. A.; Zulkefle, H.; Zaihidi, M. M.; Abdullah, M. H.; Herman, S. H.; Noor, U. M.; Rusop, M. Adv. Mater. Res. 2011, 364, 105−109. (11) Martinez, A. B.; Realinho, V.; Antunes, M.; Maspoch, M. L.; Velasco, J. I. Ind. Eng. Chem. Res. 2011, 50, 5239−5247. (12) Lee, W.-J.; Lee, S.-E.; Kim, C.-G. Compos. Struct. 2006, 76, 406− 410. (13) W. Starr, F.; Schroder, T. B.; Glotzer, Sh. C. Macromolecules 2002, 35, 4481−4492. Ozmusul, M. S.; Picu, C. R.; Sternstein, S. S.; Kumar, S. K. Macromolecules 2005, 38, 4495−4500. Sen, S.; Thomin, J. D.; Kumar, S. K.; Keblinski, P. Macromolecules 2007, 40, 4059−4067. Qiao, R.; Brinson, L. C. Compos. Sci. Technol. 2009, 69, 491−499. Hall, L. M.; Jayaraman, A.; Schweizer, K. S. Curr. Opin. Solid State Mater. Sci. 2010, 14, 38−48. Kalb, J.; Dukes, D.; Kumar, S. K.; Hoy, R. S.; Grest, G. S. Soft Matter 2011, 7, 1418−1425. (14) Barbier, D.; Brown, D.; Grillet, A.-C.; Neyertz, S. Macromolecules 2004, 37, 4695−4710. (15) Milano, G.; Santangelo, G.; Ragone, F.; Cavallo, L.; Di Matteo, A. J. Phys. Chem. C 2011, 115, 15154−15163. (16) Komarov, P. V.; Mikhailov, I. V.; Chiu, Y.-T.; Chen, S.-M.; Khalatur, P. G. Macromol. Theory Simul. 2013, 22, 187−197. (17) Vogiatzis, G.; Voyiatzis, E.; Theodorou, D. N. Eur. Polym. J. 2011, 47, 692−712. (18) Ndoro, T. V. M; Voyiatzis, E.; Ghanbari, A.; Theodorou, D. N.; Böhm, M. C.; Müller-Plathe, F. Macromolecules 2011, 44, 2316−2327. Ndoro, T. V. M; Böhm, M. C.; Müller-Plathe, F. Macromolecules 2012, 45, 171−179. (19) Ghanbari, A.; Böhm, M. C.; Müller-Plathe, F. Macromolecules 2011, 44, 5520−5526. (20) Eslami, H.; Müller-Plathe, F. J. Phys. Chem. B 2009, 113, 5568− 5581; ibid 2010, 114, 387−395; ibid 2011, 115, 9720−9731. Eslami, H.; Mohammadzadeh, L.; Mehdipour, N. J. Chem. Phys. 2011, 135, 064703. (21) Eslami, H.; Karimi-Varzaneh, H. A.; Müller-Plathe, F. Macromolecules 2011, 44, 3117−3128. Eslami, H.; Müller-Plathe, F. J. Phys. Chem. C 2013, 117, 5249−5257. (22) Baschnagel, J.; Binder, K. Macromolecules 1995, 28, 6808−6818. (23) Wei, C. Nano Lett. 2006, 6, 1627−1631. (24) Wells, A. F. Structural Inorganic Chemistry; Oxford Science Publications: New York, 1984. (25) Lopes, P. E. M.; Murashov, V.; Tazi, M.; Demchuk, E.; MacKerell, J.; Alexander, D. J. Phys. Chem. B 2006, 110, 2782−2792. (26) Kirschner, K. N.; Heikamp, K.; Reith, D. Mol. Simul. 2012, 36, 1253−1264. (27) Müller-Plathe, F. Comput. Phys. Commun. 1993, 78, 77−94. (28) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684−3690. (29) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, PA, 1987. (30) Porter, C. E.; Blum, F. D. Macromolecules 2000, 33, 7016−7020.

AUTHOR INFORMATION

Corresponding Author

*E-mail: (H.E.) [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

The support of this work by the Alexander von Humboldt Foundation and the Deutsche Forschungsgemeinschaft (Priority Programme 1369, Polymer-Solid Contacts: Interfaces and Interphases) is gratefully acknowledged. L


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(31) Van der Lee, A.; Hamon, L.; Holl, Y.; Grohens, Y. Langmuir 2001, 17, 7664−7669. (32) Eslami, H.; Müller-Plathe, F. Macromolecules 2007, 40, 6413− 6421. (33) Barrat, J.-L.; Baschnagel, J.; Lyulin, A. Soft Matter 2010, 6, 3430−3446. (34) Eslami, H.; Müller-Plathe, F. Macromolecules 2009, 42, 8241− 8250. (35) Toth, R.; Santese, F.; Pereira, S. P.; Nieto, D. R.; Pricl, S.; Fermeglia, M.; Posocco, P. J. Mater. Chem. 2012, 22, 5398−5409. (36) Israelachvili, J. N.; McGuiggan, P. M.; Homola, A. M. Science 1988, 240, 189−191. Granick, S. Science 1991, 253, 1374−1379. Klein, J.; Kumacheva, E. J. Chem. Phys. 1998, 108, 6996−7009. (37) Barbier, D.; Brown, D.; Grillet, A.-C.; Neyertz, S. Macromolecules 2004, 37, 4695−4710. (38) Hong, B.; Panagiotopoulos, A. Z. J. Phys. Chem. B 2012, 116, 2385−2395. (39) Kulkeratiyut, S.; Kulkeratiyut, S.; Blum, F. D. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 2071ym S.. (40) Smith, G. D.; Bedrov, D.; Borodin, O. Phys. Rev. Lett. 2000, 85, 5583−5586. (41) Karimi-Varzaneh, H. A.; Carbone, P.; Müller-Plathe, F. Macromolecules 2008, 41, 7211−7218. (42) Rapaport, D. C. Mol. Phys. 1983, 50, 1151−1162. (43) Batistakis, C.; Lyulin, A. V.; Michels, M. A. J. Macromolecules 2012, 45, 7282−7292.

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Molecular Dynamics Simulation of a Silica Nanoparticle in Oligomeric

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