Molecular Dynamics Study of an Atactic Poly(methyl methacrylate

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Molecular Dynamics Study of an Atactic Poly(methyl methacrylate) - Carbon Nanotube (PMMA-CNT) Nanocomposite Emmanuel Skountzos, Panagiotis G. Mermigkis, and Vlasis G. Mavrantzas J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b06631 • Publication Date (Web): 31 Aug 2018 Downloaded from http://pubs.acs.org on September 1, 2018

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The Journal of Physical Chemistry

Molecular Dynamics Study of an Atactic Poly(methyl methacrylate) - Carbon Nanotube (PMMA-CNT) Nanocomposite

Emmanuel N. Skountzos,1 Panagiotis G. Mermigkis1 and Vlasis G. Mavrantzas1,2,*

1

Department of Chemical Engineering, University of Patras & FORTH/ICE-HT, Patras, GR 26504, Greece 2

Particle Technology Laboratory, Department of Mechanical and Process Engineering, ETH Zürich, CH-8092 Zürich, Switzerland

*

Author to whom correspondence should be addressed. Tel.: +30-6944-602580, e-

mail: [email protected] (V.G. Mavrantzas)

ABSTRACT Molecular dynamics (MD) is used to simulate a model atactic poly(methyl methacrylate) (PMMA) system in which carbon nanotubes (CNTs) have been randomly dispersed. Our purpose is to elucidate the equilibrium structure and dynamic behavior of PMMA chains at the interface with a CNT. CNTs with different diameters and at different concentrations in the host PMMA matrix are studied and their effect on the equilibrium squared radius-of-gyration and squared end-to-end distance of PMMA chains is examined. We have analyzed PMMA density, structure and conformation both axially and normal to the CNT surface. Our MD simulations indicate that the presence of CNTs causes a small decrease in the size of the polymer chains, which becomes more pronounced as the concentration (volume fraction) and diameter of CNTs in the nanocomposite increases. We also provide a detailed analysis

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of adsorbed PMMA chain conformations in terms of trains, loops and tails, and their statistical properties. An important finding of our work is that PMMA chains tend to penetrate significantly inside the CNTs through their faces; as a result of CNT filling by PMMA chains, the area near the CNT mouths is characterized by significantly higher polymer mass density (almost by 45%) than the bulk of the nanocomposite. Additional simulation results for local and terminal relaxation in the PMMA-CNT nanocomposites reveal that due to strong PMMA-CNT attractive forces, all relaxation times in the interfacial region are significantly prolonged in comparison to the bulk, and the same happens with the diffusive (translational) motion of the chains. The density profile that develops (both axially and radially) in the vicinity of CNTs appears to significantly delay PMMA dynamics at all length scales. How this affects the glass transition temperature of the nanocomposite is also analyzed.

1. INTRODUCTION Carbon nanotubes (CNTs) have attracted considerable attention in the last years as possible fillers for the fabrication of polymer nanocomposites with many promising and potentially unique properties.1-3 Their high degree of flexibility, high surface area, high Young’s modulus in the direction of the nanotube axis, remarkably simple crystal structure, chirality-dependent electronic and vibrational states, and hollow morphology with their molecularly smooth, almost frictionless hydrophobic graphitic walls are some of their most notable attractive features.1-3 CNT-based polymer nanocomposites therefore hold the promise to display extraordinary mechanical,4 thermal,5-7 electrical,5,7-8 rheological,9-12 and permeability properties.13-20 In particular, their atomically smooth graphitic walls which allow for the fast and


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ultra-efficient transport of water render them ideal structures for the design of new membranes for application in desalination, gas purification, and as transport gates.21 Experimentally, it appears that the structure and conformation of CNT-based polymer nanocomposites has not been studied as much as nanocomposites based on other types of fillers. Tung et al.22-23 addressed polystyrene (PS)/CNT nanocomposites at several CNT loadings and different states of dispersion (e.g., isotropic22 versus aligned23). They considered both single-walled (SWCNTs) and multi-walled (MWCNTs) carbon nanotubes with aspect ratios (ratio L/D of CNT length L to CNT diameter D) equal to 44 and 33. By employing small angle neutron scattering (SANS) and small angle X-ray scattering (SAXS) experiments, they found that for the isotropically dispersed CNTs at relatively low loadings (up to 2 wt. %), the size of PS chains remained practically unaffected by the presence of either SWCNTs with R/Rg< 1 or MWCNTs with R/Rg ~ 1 (R denotes the radius of the CNT).22 At larger concentrations, however, Rg increased significantly (up to 36 %) in the PS/SWCNT nanocomposites and remained unaltered or decreased slightly in the PS/MWCNT ones.22 When the experiments were repeated with aligned SWCNTs,23 no significant increase in the size of PS chains was observed for concentrations up to 2 wt. %; this agrees with the results obtained with the randomly dispersed CNTs. However, at higher CNT concentrations, Rg increased by almost 30 %.23 From the point of view of practical applications, an interesting question that arises is how the presence of CNTs changes the glass-transition temperature Tg of the host polymer matrix, because this controls segmental dynamics. In the early work of Pham et al.,24 a slight increase (up to 3 oC) of the Tg of a PS/SWCNT nanocomposite was reported for concentrations up to 3 wt. %. For a higher molecular weight (MW), polydisperse PS nanocomposite with randomly dispersed CNTs, Grady et al.25



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reported an increase of the Tg by 6 to 7 oC. Similar results pointing out to a shift of Tg by not more than 9 oC have been reported by Sterzynskia et al.26 for a poly(vinyl chloride) (PVC) host matrix enhanced with MWCNTs at CNT loadings up to 0.05 % wt.. It is worth mentioning though that the maximum Tg increase measured by these authors for the PVC/MWCNT nanocomposites was with the dynamic mechanical analysis (DMTA) method at very high frequency (1000 Hz). When the DMTA experiments were repeated at lower frequencies (i.e., 1 Hz and 10 Hz) or when differential scanning calorimetry (DSC) was employed, the measured Tg shifts never exceeded 3 oC. Logakis et al.27 reported a slight increase (1 to 2 oC) in the Tg of PMMA/MWCNT nanocomposites for concentrations up to 8 % wt. but, as the authors also explain, the measured differences in the Tg between the neat polymer and its nanocomposites were close to the experimental error. Using DSC, Flory et al.28 studied changes in the Tg of nanocomposites synthesized with low loadings (1 wt. %) of SWCNTs in PMMA. When unmodified nanotubes were used, the Tg of the nanocomposite came out to be the same as that of the neat PMMA (Tg = 99 °C). Aging experiments, on the other hand, indicated a slower approach to equilibrium compared to neat PMMA. When amino-functionalized nanotubes were used, a significant increase (by 17 °C) in the Tg of the nanocomposite was measured relative to that of the neat PMMA implying that modified CNTs in PMMA restrict local segmental dynamics significantly. Pradhan and Iannacchione29 determined the Tg of PMMA/SWCNT nanocomposites using two different experimental techniques: (a) modulation calorimetry (ACC) and (b) modulated differential-scanning calorimetry (MDSC), and contradictory results were provided: the MDSC measurements indicated a shift in Tg by 18 oC whereas the ACC ones provided results that varied from a slight decrease up to a slight increase of Tg. According to the authors,29 it was the prolonged 4 ACS Paragon Plus Environment

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heat treatment of the composite samples in the ACC experiments that caused these inconsistences and deviations. Molecular simulations have also been employed (and rather extensively) to understand the unique properties of polymer-CNT nanocomposites and clarify how they are influenced by details of molecular structure and chain conformation in the vicinity of the interface of the polymer with the nanotube.30-38 Entropic (presence of the solid boundary) and energetic (dissimilarity of polymer-polymer and polymerCNT interatomic interactions or polymer-CNT interfacial bonding) will cause variations in the structure, conformation and dynamics of polymer chains spatially in the neighborhood of CNTs, which will strongly affect the macroscopically exhibited properties of the nanocomposite. Early simulation studies stressed the key role of the specific monomer structure in the polymer chain in determining the strength of its interaction with the nanotube. Yang et al.,30 for example, studied the interaction of several polymer chains and concluded that chains with a backbone containing aromatic rings are promising candidates for the noncovalent bonding of CNTs into composite structures. Wei,32 on the other hand, found that polyethylene (PE) molecules adopt conformations on CNTs that depend strongly on temperature but also on the radius and chirality of the nanotube. Similar results were reported by Zheng et al.33 and Tallury and Pasquinelli35-36 who studied the interaction and degree of wrapping of several polymers with SWCNTs in vacuum. All of the above MD studies focused on the interaction of individual polymer molecules with a single CNT or an array of CNTs. To study the effect of polymer adsorption on the properties of the resulting nanocomposite, one should address bulk, multichain polymer-CNT systems.39-41 For example, Eslami and Behrouz,39 studied systems composed of 10-mer polyamide-6,6 (PA-6,6) chains filled with an infinitely-



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long SWCNT and reported the formation of a highly dense layer around the SWCNT. The conformation and dynamics of polymer chains within that layer were found to be strongly perturbed by the presence of the CNT. Close to the CNT, PA-6,6 chains were found to swell significantly, especially for curved CNTs. Translational and orientational dynamics were decelerated up to three orders of magnitude compared to the bulk, while polymer chains could recover part of their dynamics only for distances greater than twice their radius-of-gyration from the surface of the CNT. In an earlier study, Chakraborty and Roy40 had reported the deceleration of the dynamics of a lowMW polycarbonate (PC) matrix with increasing loading in SWCNTs. Most important was their observation that CNTs showed a clear tendency to buddle as a result of the weak interaction between polymer and filler. More recently, Khare and Khare41 have reported a depression in the value of the Tg by ~ 66 oC for an isotropically dispersed epoxy/CNT nanocomposite, but we should mention that the host matrix consisted of chains made up of only three monomers. The

study

of

the

mechanical

properties

of

CNT-based

polymer

nanocomposites by means of molecular simulations has also been the subject of many reports in the past. Typical examples include the works of Wagner et al.,42 Lordi and Yao,43 Lau and Hui,44 Frankland et al.,45-46 Griebel and Hamaekers,47 Han and Elliott,48 and Arash et al.49 With years, in addition to the mechanical reinforcement of polymer/CNT nanocomposites, people have become interested also in the thermal conductivity of these nanostructures. Babaei et al.,50 for example, have shown that the addition of CNTs into long-chain paraffins leads to a considerable enhancement of their thermal conductivity, which is partly due to the presence of a conductive filler and partly due to the filler-induced alignment of paraffin molecules.


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The atomistic MD simulations of Babaei et al.50 involved paraffins for which equilibration is not an issue. For long chain polymers, however, atomistic-scale modelling of CNT-based polymer nanocomposites is a rather challenging issue owing to the tremendous multiplicity of characteristic time and length scales characterizing these systems. Even in the absence of the nanofiller (i.e., for the pure polymer matrix), the longest relaxation time of a melt with MW above the characteristic entanglement molecular weight (Me) increases according to the reptation theory as MW3.4 which renders the direct MD simulation of truly long, entangled polymer systems a formidable task. One way to overcome this is to resort to coarse-graining, implying the use of a simplified model for the polymer chains.51-53 Resorting to coarse-graining or to simplified models for the representation of the polymer chemical structure is a rather common strategy followed by many researchers in order to cope with the extremely broad spectra of length and time scales governing structure and molecular motion in polymeric materials. However, coarsegraining comes with a number of artifacts:54-55 a) It is not clear at all what features of the atomistic model to preserve in the mapping from atomistic units to beads or particles; b) The range of conditions over which reasonable estimates of the structural, thermodynamic, and conformational properties can be expected is usually unknown; in principle, a coarse-grained potential is valid only at the conditions (e.g., pressure P and temperature T) at which coarse-graining was performed; c) Restoring the atomistic details lost in the averaging procedure by reverse mapping might not be a well-defined problem; d) dynamic methods at the coarse-grained level should be corrected for the entropy loss (extra dissipation) accompanying the neglected degrees of freedom (otherwise they can be totally erroneous). Indeed, as explained by Öttinger,54 any level of coarse-graining necessarily introduces irreversibility and



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additional dissipation which is often forgotten or ignored, and this severely limits the usefulness and range of applicability of coarse-grained models. To handle the irreversible nature of coarse graining, Öttinger proposed resorting to the fundamental principles of nonequilibrium statistical mechanics, as opposed to strictly atequilibrium ideas.54 Despite being computationally very demanding, expensive, and laborious to implement, atomistic simulations offer a detailed and consistent picture of the salient features of the structural and dynamic characteristics of polymers and their nanocomposites. Unlike coarse-grained models, they permit studying the dependence of the macroscopically manifested physical properties on details of the chemical structure of the nanocomposite, which is of primary interest in many technological applications. In addition, atomistic models allow for a more direct comparison with the experiment, which is not typically the case with coarse-grained models. The latter lack the rigor and accuracy of the detailed atomistic representation; thus they can only afford a qualitative (and not a quantitative) comparison with experimental observations and measurements. Atomistic MD simulations have thus also been used recently to study the molecular mechanisms and relative magnitude of driving forces for assembly versus disassembly of SWCNT dispersions in several polymers and solvents56 and bundle formation of SWCNTs in their mixtures with monomers and trimers of polycarbonate at different temperatures.40 They can also be used to parameterize new (continuum) theoretical models for the phase behavior and rheology of polymer nanocomposite melts derived on principles of nonequilibrium thermodynamics.57-58 In the present work, we use detailed atomistic MD simulations to study PMMA-CNT nanocomposites in order to explore the spatial dependence of the


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structural, conformational and dynamic properties of PMMA chains in the interfacial region around CNTs. We are particularly interested in the type and statistics of adsorbed PMMA conformations on CNTs by distinguishing between free and adsorbed chains,59 and separately determining their relaxational and conformational properties. By probing segmental relaxation, we also determine the glass-transition temperature Tg of the simulated materials. This allows us to study how the Tg of the nanocomposite deviates from the Tg of the bulk (pure) polymer. The rest of the material presented in this paper is organized as follows: Section 2 discusses the molecular model employed in our work and describes in detail the systems simulated and the methodology followed. Results from our simulations referring to local mass PMMA density, chain organization and conformation of adsorbed PMMA segments, orientational and local dynamics of the host polymer matrix and the Tg of the simulated systems are provided in Section 3. The paper concludes with Section 4 summarizing the most important findings of this work and discussing future plans.

2. METHODS The selected host polymer is atactic PMMA with degree of polymerization (number of monomers) X = 45, implying a strictly monodisperse sample with MW equal to 4,511.25 g mol-1. Five different model systems were considered denoted as System 15. System 1 corresponds to the pure PMMA matrix while Systems 2–5 correspond to the PMMA-CNT nanocomposites. Their molecular characteristics (number of PMMA chains, number of CNTs, diameter of CNTs, CNT weight fraction and CNT volume fraction) are listed in Table 1. The volume fraction (v/v %) was calculated at the end of each simulation in the NPT ensemble from the volumes V1 of the pure PMMA



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system (System 1) and Vi, i = 2 – 5, of the nanocomposites (Systems 2 - 5) as (Vi – V1)/Vi; it will be referred to in the following as the CNT concentrations C. For all nanocomposites, the length of the CNTs was kept fixed, equal to L = 100 Å. A typical atomistic configuration from the MD simulation with System 2 is shown in Figure 1.

Figure 1. Typical atomistic snapshot of a PMMA-CNT model system (System 2) consisting of 280 PMMA chains with degree of polymerization X = 45 and 12 CNTs. Cell dimensions: (130 × 130 × 130) Å3. Carbon (PMMA), carbon (CNTs) and oxygen atoms are shown in gray, yellow and red colors, respectively. All hydrogen atoms have been omitted for clarity.

Table 1. Molecular characteristics of the model pure PMMA system and its CNTbased nanocomposites studied in this work.

System

PMMA chains

Number of CNTs

1 2 3 4 5

280 280 195 280 211

12 12 3 3

CNT diameter (Å) 10.86 10.86 21.68 21.68

CNT loading (w/w %) 13.06 17.75 6.99 9.07

CNT loading (v/v %), at T = 550 K 5.13 7.06 5.23 7.06

Total number of interacting particles 189,560 205,688 148,143 197,624 150,911

Initial configurations were built using the Amorphous Builder module based on the work of ref 60 integrated in the Scienomics MAPS software,61 followed by 10 ACS Paragon Plus Environment

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static structure optimization using a molecular mechanics algorithm to minimize the potential energy and obtain an equilibrium structure that is completely free of atom overlaps. The resulting minimum-potential energy configuration for each system was then submitted to a long MD simulation in the NPT statistical ensemble at T = 550 K and P = 1 atm, conditions where PMMA and its CNT-based nanocomposites are in their molten state, and thus equilibration of structural and conformational properties at all length scales is easier to achieve. The Nosé-Hoover thermostat62-63 coupled with the Parrinello-Rahman barostat64 were used to maintain temperature T and pressure P fixed at their prescribed values. Additional MD simulations at several other temperatures in the interval [475 K, 400 K] keeping the pressure constant at P = 1 atm were also performed in order to obtain estimates of the glass-transition temperature Tg. Very large, cubic simulation cells were considered (edge lengths of approximately 130 Å) subject to full periodic boundary conditions. For both PMMA and CNTs, interatomic interactions were described by the explicit-atom DREIDING forcefield.65 The same forcefield has been successfully employed in the past for the computation of the mechanical properties of syndiotactic PMMA nanocomposites with functionalized and non-functionalized graphene sheets.66 Refs66-67 provide all details regarding the functional form of the employed forcefield and the values of the energy parameters. Electrostatic interactions were calculated using the particle mesh Ewald (PME) method.68 For the calculation of the van der Waals (vdW) interactions a typical 12-6 Lennard-Jones potential was employed with a cutoff radius equal to 12 Å; tail corrections for the energy and the pressure were included. For the interactions between unlike atoms, the Lorentz–Berthelot rules were used, i.e., arithmetic averages were employed for the calculation of the Lennard-Jones radii σij, σij = (σii+σjj)/2, and geometric averages for the calculation of the Lennard-Jones energies εij, εij = (εii·εjj)0.5,



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with i and j denoting atom types. 1-4 pairwise van der Waals and Coulomb interactions were in all cases considered with a factor of 1. All simulations were executed with the GROMACS software69 and lasted for 800 ns.

3. RESULTS 3.1 MD Simulation Predictions at T = 550 K CNT mobility and state of dispersion. Through several theoretical, simulation and experimental studies, it is well known by now that the state and degree of nanoparticle dispersion can have a crucial effect on the physicochemical properties of the resulting nanocomposite. In atomistic simulations, and given that nanoparticle motion is limited due to the long time scales characterizing polymer relaxation and the slow diffusivity of nanoparticles, it is fair to assume that the initial state of nanoparticle dispersion or aggregation will persist in the course of the simulation, which in turn will affect the simulation predictions for the relevant volumetric, structural, conformational and other properties of the model nanocomposite simulated. To avoid dependence of the predicted properties on the spatial distribution of nanoparticles, for all systems studied in the present MD simulations, the CNTs were initially dispersed randomly in the simulation cell. Also, and despite that they were not kept static but allowed to diffuse in the PMMA matrix according to Newton’s law (as the result of the forces exerted on their carbon atoms by the carbon atoms of other CNTs and by the atoms of PMMA chains), their mobility was quite slow. The corresponding mean-square displacements (msd’s) of their centers-of-mass are shown in Figure S1 of the Supporting Information to this paper and indicate values ranging from 20 to 80 Å2 (after 800 ns of simulation time), which are significantly smaller than the length of the simulated CNTs. As a result, CNTs were not given the time to form bundles, so all results reported in the


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next sections of this paper can be considered to refer to homogeneous dispersions of CNTs.

Density. MD predictions for the density of the pure PMMA and all PMMA-CNT nanocomposites considered here at T = 550 K and P = 1 atm are reported in Table 2. The error bars on the simulation data are very small because of the use of large simulation cells. The results indicate a slightly higher value for the average mass density of all PMMA-CNT nanocomposites in comparison to the pure PMMA matrix at the same temperature and pressure conditions. This can be attributed to two factors: a) First, CNT is a denser material70 than PMMA, thus its introduction into the PMMA matrix increases the total density. For the same reason, for a given CNT diameter, the density of a PMMA-CNT nanocomposite will increase with increasing CNT concentration. b) Second, local packing of polymer chains around CNTs leads to higher densities. Indeed, due to strong energetic interactions between PMMA segments and CNTs, a dense polymer layer forms around the CNTs with density significantly higher than the density of pure PMMA.

Table 2. Density of all simulated model systems (T = 550 K, P = 1 atm). System 1 2 3 4 5

ρ (g cm-3) 1.072 ± 0.002 1.127 ± 0.002 1.152 ± 0.002 1.116 ± 0.002 1.129 ± 0.002

To investigate this effect further, we spatially resolved the mass density of PMMA both radially from the CNT surface and axially with respect to its two faces (entrance and exit regions) by partitioning the corresponding space in 1-Å thick bins. In these 13 ACS Paragon Plus Environment


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calculations of the local mass density of the nanocomposite, only PMMA atoms were considered. Our results are shown in Figures 2-3. Figure 2 reports the local variation of PMMA density as a function of radial distance r from the CNT surface and Figure 3 the corresponding variation as a function of distance z from the faces of the CNT.

Figure 2. Results from the present NPT MD simulations at T = 550 K and P = 1 atm for: (a) the variation of the local PMMA mass density radially from the surface of a CNT for all PMMA-CNT nanocomposites studied here (the dashed line indicates the density of the pure atactic PMMA at the same temperature and pressure conditions), and (b) the corresponding variation of atomic densities for System 4.


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Our calculations show that the local density in the interfacial zone around CNTs is significantly higher than in the bulk. It increases rapidly with increasing distance r from the CNTs reaching a maximum at around 4 to 6 Å from the CNT surface, and then drops off to assume the bulk value. In fact, our graphs indicate two slightly different maxima that are located very close to each other: one at 4 Å and another at 5 Å from the CNT surface. Similar conclusions are drawn by analyzing Figure 2b depicting the corresponding atomic density profiles for carbon (C), oxygen (O) and hydrogen (H) atoms, of System 4.

An interesting point to observe is the two

pronounced peaks in the hydrogen profile. These have to do with the fact that PMMA prefers to lie on the CNTs with its two branches (ester and methyl branches) rather than with its backbone. Thus, the first peak is due to branch hydrogen atoms and the same holds for the first, weak peak in the population of carbon atoms and the peak in the population of oxygen atoms (they all arise from adsorbed ester and methyl branches). On the other hand, the second peak corresponds to hydrogen atoms along the backbones of adsorbed PMMA chains, and the same holds for the second peak in the radial profile of carbon atoms. According to Figure 2a, the local polymer density at a distance approximately 5 Å from the surface of the CNT is higher than the corresponding bulk PMMA density by 40% in the nanocomposites with D = 1 nm (Systems 2 and 3) and by 50 % in the nanocomposites with D = 2 nm (Systems 4 and 5). The slightly oscillatory profile of the PMMA mass radially around the CNT has also been observed in the coarse-grained simulations of Karatrantos et al.51 for a PS/CNT nanocomposite system, and in the atomistic simulations of Eslami and Behrouz39 for a PAA-6,6/CNT nanocomposite. Similar oscillatory profiles have been reported from MD simulations with a graphene-based PMMA nanocomposite66,71,72



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and from MD73 and Monte Carlo74 (MC) simulations of PE melts supported by a graphitic surface. The corresponding axial PMMA density profiles (i.e., normal to the two faces of a CNT) are shown in Figure 3. Again, part (a) of the figure reports the local variation of the total PMMA mass density and part (b) the corresponding variation of the atomic densities. Negative distance values (z < 0) imply that we move away from the CNT face (this is assumed to be located at z = 0) whereas positive distance values (z > 0) imply that we move inside the CNT. Our results are subject to larger error bars (which are not shown for clarity) compared to those depicted in Figure 2, since the volume of the small cylinders considered for the calculation of the axial density is significantly smaller compared to the volume of the cylindrical cells considered for the calculation of the radial density distribution (~ ∆z πD2/4 which is much less than ∆r LπD, because D 0).

(a)

(b) Figure 4. Typical atomistic snapshot of a PMMA chain that has fully penetrated a CNT (result obtained from the NPT MD simulation with System 4 at T = 550 K and P = 1 atm). Both radial (a) and axial (b) views with respect to the CNT orientation are 18 ACS Paragon Plus Environment

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provided. Backbone carbons of PMMA chains are shown in yellow while hydrogens have been omitted for clarity.

Adsorption of PMMA on CNTs. Motivated by the findings of our analysis on the radial and axial PMMA density profiles, which provide also an estimate of the thickness of the adsorbed layer, we proceed to examine the conformation of PMMA chains in the immediate neighborhood of a CNT. To this, we first need to distinguish between free and adsorbed PMMA chains. Following previous work,74 we define a polymer chain as adsorbed if at least one of its (non-hydrogen) atoms lies inside the adsorbed layer (region I in Figure 2a). In contrast, a PMMA chain is defined as free when all of its (non-hydrogen) atoms lie outside it (region II in Figure 2a). The time evolution of the fraction of adsorbed and free PMMA chains in the course of our MD simulations with Systems 2-5 is given in the Supporting Information to this manuscript (Figures S2 – S4). Then, from the equilibrated part of the trajectory (practically from the last 100 ns), we calculated the mean values of adsorbed and free chains and the results for the systems are summarized in Table 3. In the table we have also included the mean values of the fraction of adsorbed backbone torsional angles and adsorbed ester branches. In our work, a torsional backbone angle is characterized as adsorbed if all non-hydrogen atoms associated with it are within the adsorption layer (region I in Figure 2a); and the same definition applies for adsorbed PMMA branches.

Table 3. Mean values of the fraction of adsorbed and free chains, adsorbed and free backbone torsional angles, and adsorbed and free ester side groups (results from the MD simulations with Systems 2-5 at T = 550 K and P = 1atm).



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Fraction of Adsorbed Free System Adsorbed Free Adsorbed Free torsional torsional chains chains branches branches angles angles 0.85 0.15 0.11 0.89 0.11 0.89 2 0.95 0.05 0.15 0.85 0.16 0.84 3 0.50 0.50 0.06 0.94 0.07 0.93 4 0.62 0.38 0.08 0.92 0.09 0.91 5

According to Table 3, the majority of PMMA chains in all four nanocomposites studied in our work are adsorbed. Furthermore, for a given concentration of the nanocomposite in CNTs, the smaller the CNT diameter, the larger the percentage of adsorbed PMMA chains (i.e., the larger the polymer amount adsorbed). This can be explained by the fact that in Systems 2 and 3, the total surface area A of CNTs available for PMMA chain adsorption is twice the corresponding surface area in Systems 4 and 5 (A = 409 nm2 for Systems 2 and 3 as compares to A = 204.5 nm2 for Systems 4 and 5). This leads to the extra accumulation of PMMA chains on the surface of the CNTs with D = 1 nm. The same trend is observed for the adsorbed backbone dihedral angles and adsorbed ester branches.

Conformation of individual PMMA chains. Significant insight into the conformation of PMMA chains in their CNT nanocomposites can be obtained by computing their mean-squared end-to-end distance Ree2 , and mean-squared radius-of-gyration Rg2 , with the brackets denoting an average over all PMMA chains in the system. The time evolution of

Ree2

and

Rg2

is depicted in Figure 5; the corresponding average

values (from the equilibrated part of the MD trajectory) are summarized in Table 4. Furthermore, for Systems 2-5, separate Ree2 and Rg2 values for free and adsorbed PMMA chains are reported. The corresponding graphs displaying the time evolution


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of the Ree2 and Rg2 for free and adsorbed PMMA chains are shown in Figures S5S6, respectively, of the Supporting Information to this manuscript. According to our atomistic MD simulations, the presence of CNTs causes a small decrease in the size of PMMA chains which is more pronounced in the nanocomposite with the largest CNTs in the highest concentration (7.06 v/v %) in the nanocomposite (System 5). In the literature, Karatrantos et al.51 have reported that when the CNT radius is smaller than the polymer radius-of-gyration, CNTs cause no significant change in the size of PS chains, and the same holds for the simulations of Rissanou and Harmandaris72 for PMMA melts filled with single-sheet graphene.

Figure 5. Time evolution of the instantaneous values of the mean-squared (a) end-toend distance Ree2

and (b) radius-of-gyration

Rg2

of PMMA chains, as obtained 21

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from the present NPT MD simulations at T = 550 K and P = 1 atm for all model systems studied.

Table 4. MD simulation results for the mean-squared end-to-end distance, Ree2 , and mean-squared radius-of-gyration,

Rg2 , of PMMA chains as obtained from the

present NPT MD simulations at T = 550 K and P = 1 atm, for all systems studied. For the CNT-based nanocomposites we also report the corresponding separate values for adsorbed and free PMMA chains.

Ree2 (Å2)

System 1 2 3 4 5

Adsorbed PMMA chains 1425 ± 41 1419 ± 44 1594 ± 59 1197 ± 61

Free PMMA chains 1259 ± 120 997 ± 152 1366 ± 66 1248 ± 102

Rg2 (Å2)

All PMMA chains 1481 ± 43 1400 ± 39 1401 ± 64 1479 ± 44 1216 ± 84

Adsorbed PMMA chains 240 ± 4 241 ± 3 254 ± 4 222 ± 4

Free PMMA chains 220 ± 12 208 ± 20 230 ± 8 226 ± 8

All PMMA chains 247 ± 5 237 ± 5 240 ± 5 242 ± 4 223 ± 6

We have also examined how CNTs affect the size of PMMA branches by computing the mean-squared of the branch end-to-end vector,

Rbr2 . The latter is

defined as the vector connecting the carbon atom at the branch point on the PMMA backbone with the methyl carbon atom of the ester side group. The results can be seen in Table 5. Again, for all nanocomposites, separate Rbr2

values were computed for

adsorbed and free branches. It is found that the size of ester branches both for free and adsorbed segments remains practically unaltered by the presence of CNTs. However,

if one looks at the corresponding distribution of the backbone dihedrals, then one notices an enhancement of trans conformations for adsorbed PMMA segments. A typical example is shown in Figure 6, taken from the simulations with System 3 (for 22 ACS Paragon Plus Environment

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the rest of systems, the graphs are reported in the Supporting Information to the present paper, see Figures S7 - S9, and exhibit similar trends); the individual contributions from adsorbed and free angles to the total distribution are also included in the figure. The graph shows a non-negligible enhancement of the trans states for adsorbed backbone dihedrals; free segments, on the other hand, behave exactly as in

the corresponding bulk unconstrained systems. A similar weak enhancement of trans states in the distribution of the φ1 dihedrals for adsorbed segments has been reported by Daoulas et al.74 for a thin PE film adsorbed on a graphite surface through MC simulations.

Table 5. Same as in Table 4 but for the mean-squared magnitude, Rbr2 (Å2), of the PMMA branch vector. System

1 2 3 4 5

Adsorbed PMMA branches 12.16 ± 0.3 12.17 ± 0.3 12.14 ± 0.2 12.13 ± 0.3

Free PMMA branches 12.20 ± 0.2 12.20 ± 0.2 12.19 ± 0.2 12.20 ± 0.2

All PMMA branches 12.20 ± 0.2 12.20 ± 0.2 12.20 ± 0.2 12.19 ± 0.2 12.19 ± 0.2



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Figure 6. Distribution of backbone dihedrals as obtained from the present NPT MD simulations at T = 550 K and P = 1 atm with Systems 3 and 1. The black curve shows the result averaged over all PMMA backbone dihedrals in System 3, the red curve the result when only adsorbed PMMA backbone dihedrals in System 3 are considered, and the green curve the result when only free PMMA backbone dihedrals in System 3 are considered. The blue curve, on the other hand, corresponds to the result obtained for the pure PMMA matrix (System 1).

Adsorbed Melt Conformations. As already mentioned, motivated by previous

simulation works which examined the conformation of polymer chains adsorbed on several types of organic,74,76 inorganic,76-77 flat74,76-77 or curved surfaces,76 we have carried out an analysis of adsorbed PMMA chains in terms of trains, loops and tails.59 A methyl methacrylate (MMA) monomer is considered as adsorbed only when at least one of its (non-hydrogen) atoms resides inside the adsorbed layer (region I in Figure 2a). According to Scheutjens and Fleer,59 such an adsorbed segment can belong to a train, loop or tail. Trains are consecutive monomer sequences that are adsorbed on the surface of the CNTs along their entire path (Figure 7a). Loops are the intermediate segments along the polymer chain which connect two trains but lie outside the interfacial area (Figure 7b). Tails (Figure 7c) are those PMMA segments at the end of the chain that lie outside the adsorbed layer with their non-terminal monomer connected to a train. The length of a train, loop or tail is measured in terms of the number of MMA monomers that belong to each one of these three characteristic structures.


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(a)

(b)

(c)

Figure 7. Definition of train (a), loop (b), and tail (c) conformations. Backbone carbon atoms in the three characteristic conformations are highlighted in yellow.

The time evolution of the instantaneous number and length of trains, loops and tails per adsorbed PMMA chain in the simulated PMMA-CNT nanocomposites are

reported in the Supporting Information to the present paper (Figures S10 – S13). The corresponding mean values are listed in Table 6 and provide a rather detailed picture of PMMA chain adsorption on CNTs. Our simulations show that the larger the diameter of the CNT the less the number but the longer the trains formed. This



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implies a stronger PMMA adsorption for Systems 4 and 5. We can see this in Figure 8 showing the atomistic configuration of two adsorbed PMMA chains. The first has been taken from the simulation with System 2 and provides an example of a slightly adsorbed chain (it consists of only three short trains and two loops). The second has been taken from the simulation with System 5 and provides an example of a strongly adsorbed chain (its entire backbone is adsorbed on the CNT surface).

Table 6. MD results for the average population and average length (number of MMA monomers) of trains, loops and tails per adsorbed PMMA chain (T = 550 K, P = 1 atm). System 2 3 4 5

Population trains loops tails 3.0 2.0 1.3 3.5 2.5 1.2 2.6 1.6 1.5 2.7 1.7 1.4

trains 4.6 4.8 5.3 5.4

Length loops 5.7 5.0 4.3 4.5

tails 14.4 12.5 17.0 17.1

(a)

(b)


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Figure 8. Typical atomistic snapshots of an adsorbed PMMA chain consisting of three short trains and two loops (a) and one long train (b) along its backbone (results obtained from the NPT MD simulation with Systems 2 and 5, respectively). Nonadsorbed backbone carbon atoms are shown in black while backbone carbon atoms participating in trains are shown in yellow.

Pandey et al.76 have shown how curvature at length-scales comparable to the Kuhn length of the polymer affects the formation of train segments in polymer nanocomposites with both atomistic and coarse-grained models.78 It would be therefore of interest to examine whether train segments are preferentially formed along the main axis of the nanotube rather than along the angular direction when the curvature is very high. To this, we calculated the relative orientation of all adsorbed trains in the four nanocomposite systems with respect to the axis of the CNT onto

which they were adsorbed by computing the dot product utrain ⋅ uCNT

( = cos (θ ) ) of

the two unit vectors u train (along the train end-to-end vector) and uCNT (along the average CNT axis) and making the corresponding histogram (θ denotes the angle between the two unit vectors). The results for the probability distribution of cos (θ ) are shown in Figure 9 as a function of CNT diameter (curvature) and CNT concentration in the nanocomposite (volume fraction), and indeed they provide evidence for preferential orientation of the formed trains along the main axis of the CNTs for all PMMA-CNT nanocomposites simulated. The fact that the maxima in Figure 9 are not exactly at the values cos (θ ) = −1 and cos (θ ) = +1 but slightly shifted is due to the flexibility (or curvature) of the short CNTs considered in our work at the conditions of T = 550 K and P = 1 atm which causes continuous fluctuations in their 27 ACS Paragon Plus Environment


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orientation with time. The tendency for a preferential axial adsorption seems to be independent of the CNT curvature (or, equivalently, CNT diameter) for the range of CNT diameters covered in our study, but simulations with CNTs of smaller and larger diameter than those addressed here would help clarify this issue even further.

Figure 9. MD simulation predictions for the relative orientation of trains with respect to the main axis of the CNT onto which they have been adsorbed. The figure shows the computed histogram of the values of cos (θ ) where θ denotes the angle between the unit vector u train along the train end-to-end vector and the unit vector uCNT along the average CNT axis, for all simulated PMMA-CNT nanocomposites (T = 550 K, P = 1 atm). The symmetry of the graph with respect to the value cos (θ ) = 0 reflects the excellent equilibration of the conformational characteristics of adsorbed train conformations in our MD simulations.

More quantitative information on the structure of adsorbed PMMA segments at the surface of CNTs is obtained by computing the statistics of the population and size of the three adsorbed structures. Figure 10 shows the probability distribution of the number of trains, loops and tails per adsorbed chain. From Figures 10a and 10b it 28 ACS Paragon Plus Environment

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becomes clear that for all nanocomposites, a significant amount of trains (thus also of loops, since the number of trains equals always the number of loops plus one) per adsorbed chain develops. The maximum number of trains and loops per adsorbed

PMMA chain is 10 and 9, respectively. From the conformational analysis of tails (Figure 10c) it turns out that only ~ 10 % of adsorbed PMMA chains have both of their terminal monomers adsorbed on CNT surfaces (i.e., they have no tails). The corresponding probability of either one or both terminal MMA monomers to be nonadsorbed is ~ 45 %.



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Figure 10. Probability distribution of the number of: (a) trains, (b) loops and (c) tails per adsorbed chain for all PMMA-CNT nanocomposites studied in this work at T = 550 K and P = 1 atm.

Figure 11 presents the distribution of the length of trains, loops and tails. Figure 11a shows that the length of trains exhibits a non-monotonic behavior characterized by the appearance of two local maxima at 1 and 3 MMA monomers. The same non-monotonic behavior had been previously reported and thoroughly explained by Daoulas et al.74 A similar trend had been observed in the MC simulations of PE on flat and spherical silica surfaces by Pandey and Doxastakis;76 these authors had also reported that such a trend should vanish for highly curved nanoparticles (such as fullerenes C60).76 Long trains consisting of more than 15 adsorbed monomers (which corresponds to 30% of the total length of the PMMA chains studied here) are also observed but their population is relatively low. In a previous study,74 we had found that for a PE oligomer (containing 80 carbon atoms along its backbone) the probability of full adsorption on a flat graphite surface was significantly higher (although it decreased rapidly with increasing chain length). It appears that the bulky ester branches of PMMA and the cylindrical geometry of CNTs


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prohibit polymeric chains from full adsorption on CNTs. The computed distribution of the length of tails revealed a non-negligible probability for a tail to consist of 44 monomers (i.e., only one terminal MMA to be adsorbed) for all four nanocomposite systems studied. The dominant length for loops (Figure 11b) is less than two monomers, and long-length loops are rarely observed.



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Figure 11. Same as with Figure 10 but for the distribution of the length of trains, loops and tails.

Despite the overall similarities in the probability distributions of the number (Figure 10) and length (Figure 11) of trains, loops and tails among the four PMMACNT systems examined, essential differences exist. For example, at fixed CNT loading, as the CNT diameter decreases: (a) the number of trains increases but their length decreases, (b) the number of tails remains practically constant but their length decreases, and (c) the average number and length of loops increases. On the other hand, upon increasing the CNT volume fraction in the nanocomposite while keeping their diameter constant: (a) more trains are formed but their length remains practically unaltered, (b) the number of tails remains almost unaffected but their size decreases, and (c) the average number of loops increases while their size decreases.

Orientational and Local Dynamics. We turn our attention now to the study of the

dynamic properties of the simulated PMMA-CNT nanocomposites. We quantify chain terminal relaxation by calculating the time decay of the autocorrelation function


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(ACF) of the chain end-to-end unit vector u ee ( t ) ⋅ u ee ( 0 ) , and local relaxation by computing the decay of the corresponding ACF of the branch end-to-end unit vector u br ( t ) ⋅ u br ( 0 ) .

How fast

the

two

ACF

functions,

u ee ( t ) ⋅ u ee ( 0 )

and

u br ( t ) ⋅ u br ( 0 ) drop to zero provides a good measure of the rate of chain terminal and local relaxation, respectively. Part (a) of Figure 12 shows the function u ee ( t ) ⋅ u ee ( 0 ) -vs.-t at the temperature of simulation (T = 550 K) for all systems studied here. Parts (b) and (c) of the same figure show the same curves separately for adsorbed and free chains, respectively. For comparison, in all cases, we also show the corresponding u ee ( t ) ⋅ u ee ( 0 ) -vs.-t curve for bulk PMMA at the same temperature (pink curve). We explain here that for a given time interval, the ACF of adsorbed and free chains is calculated over only those PMMA chains that remained continuously adsorbed or free, respectively. That the number of chains that contribute to the calculation of the ACF for adsorbed and free chains is a dynamic quantity (it fluctuates during the simulation) explains why the ACF for free chains for System 3 in Figure 12c extends only up to 400 ns despite that the total simulation time was 800 ns. Indirectly, this indicates that during the simulation with this System and after approximately 400 ns, all PMMA chains came in contact with a CNT at least once.



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Figure 12. Decay of the time autocorrelation functions of the chain end-to-end unit vector at T = 550 K and P = 1 atm for all model systems studied here, averaged over: (a) all chains, (b) only adsorbed chains, and (c) only free chains, present in the simulation cell.

The presence of CNTs in the host matrix causes significant deceleration of the orientational dynamics of PMMA chains (Figure 12a), due to their strong adsorption on the surface of CNTs. As expected, this is more pronounced in the ACFs of adsorbed chains (Figure 12b). Interestingly enough, our data suggest that free chains in the nanocomposites exhibit a faster relaxation than bulk PMMA chains (see Figure 12c). For Systems 2 and 3 this can be possibly attributed to numerical uncertainties 34 ACS Paragon Plus Environment

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arising from the small number of free chains present in these systems (see also Table 3). However, for Systems 4 and 5 for which free chains constitute a good fraction (approximately 50 %) of the total PMMA chains, numerical errors due to poor statistics can be safely excluded; in this case, definitely, free chains exhibit a faster orientational relaxation than bulk chains. This apparently unexpected result should be attributed to the smaller density values characterizing locally areas where several CNTs come together. As we can see from Figure 2a, at relatively large distances away from CNTs (e.g., beyond 10 Å), the density attains values smaller than the average system density, which accelerates the diffusive motion of non-adsorbed chains and can explain the faster dynamics of free chains. Comparable results have been reported by Vogiatzis and Theodorou regarding polymer diffusion in PS/C60 nanocomposites.79 That chain dynamics in polymer nanocomposites is highly heterogeneous has also been reported by Karatrantos et al.52 for a highly attractive CNT-based polymer system through coarse-grained MD simulations. The corresponding

u br ( t ) ⋅ ubr ( 0 ) -vs.-t curves quantifying local (side group)

relaxation for all systems studied here are shown in Figure 13. Separate calculations for adsorbed and free branches have been carried out following the definition discussed before. Similar to Figure 12, branch relaxation is significantly decelerated compared to that in the bulk polymer, for all nanocomposites. As expected, the higher the fraction of adsorbed ester branches the slower their local relaxation. The simulation curves of Figure 13 were fitted using a Kohlrausch-Williams-Watts (KWW) stretch exponential function of the form:

  t β  P ( t ) = exp -      τ KWW  

(1)



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where τKWW denotes the characteristic time and β the characteristic stretching exponent. The characteristic relaxation time τc is estimated from the integral below the P(t)-vs.-t curve and is given analytically by the following equation:

∞

τ c = ∫ P ( t ) dt = τ KWW 0

1 Γ  β 

(2)

β

Figure 13. Decay of the time autocorrelation functions of the chain branch unit vector at T = 550 K and P = 1 atm for all simulated systems.

The values obtained from such a procedure for the constants τKWW and β for all simulated systems are provided in Table S1 of the Supporting Information to this manuscript. The corresponding values of τc as extracted from eq 2 are given in Table 7. For the four nanocomposites, we have also included in Table S1 the values of τKWW and β computed by separately analyzing the local relaxation of free PMMA branches, and the same for the values of τc in Table 7. The latter confirm that local relaxation in the PMMA-CNT nanocomposites is systematically slower than in pure PMMA, because τc attains significantly larger values in all nanocomposites. In System 3, in particular, the value of τc is ~ 5 times larger than in the bulk PMMA. An increase in the relaxation times of free branches (side ester groups) in all nanocomposites is 36 ACS Paragon Plus Environment

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observed as well. This can be explained by the fact that although these are free branches, most of them belong to chains that are adsorbed, characterized therefore by a slower relaxation than truly free branches. Please note that no relaxation times are reported in Tables 7 and S1 for adsorbed branches in the four nanocomposites because these were observed to exceed by orders of magnitude the simulation time for each system, thus rendering their exact calculation through KWW fittings highly uncertain. It is also worth mentioning that similar dynamic heterogeneities for adsorbed segments have been studied by Pandey et al.80 who reported that classification as “adsorbed” for such segments based on distance for atomistic models provides an incomplete description. Specifically, adsorbed monomers more than a Kuhn length apart from the “end” of a train segment can decorrelate with an order of magnitude slower dynamics than the “ends” of train segments. Such effects would explain our low beta values in Table S1 given the potential changes of the length of the train segments discussed previously and the ability of nanotubes to provide “flat” surfaces along the main axis.

Table 7. MD predictions for the values of the characteristic relaxation time τc quantifying PMMA branch orientational relaxation in the simulated systems. For Systems 2-5, we also report the τc values corresponding only to free branches (T = 550 K, P = 1 atm).

τc (ns) System 1 2 3 4 5

Free PMMA branches 10.6 ± 0.4 15.5 ± 0.5 6.7 ± 0.3 7.3 ± 0.2

All PMMA Branches 4.3 ± 0.2 15.4 ± 0.5 24.9 ± 1.0 11.9 ± 0.7 13.5±0.5



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3.2 Estimation of the Glass-Transition Temperature From the point of view of molecular simulations, the glass-transition temperature Tg of a polymer can be estimated by performing MD simulations at progressively lower temperatures to obtain the segmental relaxation times τc as a function of temperature, and then fitting the plots obtained with a Williams-Landel-Ferry (WLF) equation:

 −C1 (T − Tg )    C2 + T − Tg   

τ c (T ) = τ c,∞ exp 

(3)

where C1 and C2 are numerical parameters with values for polymers approximately equal to 17.44 and 51.6 K, respectively (global WLF parameters). We have followed such a procedure here and the results obtained for the segmental relaxation times τc as a function of temperature T for the pure PMMA matrix and its CNT-based nanocomposites with D = 1 nm (Systems 2 and 3) are shown in Figure 14. The figure also shows the corresponding WLF fits (dashed lines) to the three sets of simulation data. The predicted Tg value for the pure PMMA system is 362 ± 11 K. For high-MW PMMA, experimentally reported values of Tg vary from 372 K to 381 K.27,28,81 For polymers, the Tg shows a strong dependence on MW, especially in the regime of low MWs, due to the free volume around chain ends. This dependence is usually expressed through the Fox-Flory82 equation according to which the Tg increases with MW as

Tg = Tg,∞ −

K Mn

(4)

In eq 4, Tg,∞ corresponds to the glass-transition temperature of an infinite-MW polymer, M n denotes the number-average molecular weight and K is a polymerspecific constant whose value is related to the free volume in the polymer. For 38 ACS Paragon Plus Environment

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syndiotactic and isotactic PMMA, such a dependence has been previously reported by Ute et al.83 For 100% atactic PMMA, to the best of our knowledge, no such information is available. Recently, Genk and Tsui84 estimated the Tg of PMMA samples with MW ranging from 2,500 g mol-1 up to a very high MW values. Through interpolation, the estimated Tg value for PMMA with MW = 4,500 g mol-1 is 365 K, which is very close to the one estimated from our MD simulations (equal to 362 ± 11 K).

Figure 14. Temperature dependence of the characteristic relaxation time τc for branch side group motion, from the simulations with Systems 1, 2 and 3. The dashed lines denote the best fits to the simulation data with a WLF function (eq 3).

As far as the various PMMA-CNT nanocomposites are concerned, the predicted values from the WLF fits to the simulation data in Figure 14 are Tg = 365 ± 16 K for System 2, and Tg = 368 ± 16 K for System 3. The errors in the reported two Tg values were calculated by: a) running new MD simulations (typically three) for Systems 2 and 3 at each temperature, i.e., starting from a new configuration (typically this was the final one of the previous MD simulation at the same temperature for the



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given system) for about the same time (700 to 800 ns); b) fitting the resulting curve with a KWW function to obtain the characteristic correlation time, τc (T) , at each temperature through eq (2) above; and c) using the WLF equation, eq (3) above, to fit the new data and obtain the new Tg value. The standard deviation (about 16 K) of the three different predicted Tg values for the two systems is the reported error in the computed Tg values. Overall, our Tg predictions for Systems 2 and 3 are consistent with the general knowledge in the field that CNT inclusions at small concentrations in polymer matrices cause a slight increase of the Tg.27-29

4. CONCLUSIONS The MD simulation results presented here provide detailed information about the equilibrium structural and dynamic properties of PMMA-CNT nanocomposites, especially in the interfacial domains. Because of the strong attractive forces between PMMA segments and CNT atoms, the local mass polymer density radially from the CNT surface is significantly higher than in the corresponding pure PMMA at the same temperature and pressure conditions, exhibiting a weak oscillatory behavior typical of a melt adsorbed on a solid surface with two characteristic maxima. The highest value is observed at approximately 5 Å from the CNT surface and is about 40 to 50% larger than the density of the corresponding pure polymer system. We found that PMMA chains exhibit a strong tendency to enter inside the CNTs, a phenomenon which becomes more pronounced as the CNT diameter increases. For example, increasing the CNT diameter from D = 1 nm to D = 2 nm was seen to allow a few PMMA chains to fully penetrate some CNTs. To the best of our knowledge, this is the first time that a polymer with bulky side groups along its backbone (such as PMMA) is found to exhibit such a tendency. Obviously, this is a very important result because the 40 ACS Paragon Plus Environment

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polymer mass accumulated at the entrance and exit regions (the mouths) of a CNT will significantly delay the transport of small molecules (such as water) through the nanocomposite. This issue will be addressed in detail in a future work. For the PMMA-CNT nanocomposites addressed in this work, our MD simulations indicated small changes in the size (average mean-squared end-to-end distance and radius-of-gyration) of PMMA chains compared to the pure polymer, which were more pronounced for the nanocomposites with the higher volume fraction in CNTs and the larger diameter. As far as adsorbed PMMA chains are concerned, their conformational properties were analyzed in detail in terms of trains, loops and tails; our analysis revealed appreciable differences in the statistical properties (population and length) of these characteristic structures from nanocomposite to nanocomposite. Analysis of the time decay of appropriately defined ACFs revealed a significant slow-down of the relaxation rates describing terminal and local dynamics of PMMA chains near CNTs. At fixed CNT loading, this slow-down was more pronounced in the nanocomposites with smaller CNT diameters because of the considerably larger amount of adsorbed PMMA mass (owing to their larger surface area per unit volume). Surprisingly, free chains exhibited faster dynamics compared to PMMA chains in the pure bulk, a result that was attributed to confinement phenomena among CNTs leading locally to regions with less density than the average system density. As far as local relaxation is concerned, the corresponding relaxation times were measured to be considerably longer than in the corresponding bulk. By extending the simulations to progressively lower temperatures, computing the corresponding relaxation times describing side chain relaxation and fitting the resulting τc-vs.-T curves with a WLF function, we were able to estimate the glass-



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transition temperature Tg of the simulated systems. Our MD prediction for the Tg of the pure bulk PMMA was in excellent agreement with reported experimental values for the same MW. For the corresponding nanocomposites, the predicted Tg values were slightly higher (by ~ 6 oC) compared to the Tg of the bulk PMMA material. In the future, we plan to discuss issues related with the formation of polymer bridges between the CNTs and the diffusive behavior of small penetrants in the PMMA-CNT nanocomposites, as a function of the concentration and geometric characteristics of CNTs.

Supporting Information Additional MD simulation results from all PMMA-CNT model nanocomposite systems studied in this work for the msd of the CNTs centers-of-mass, the time evolution of several quantities (e.g., percentage of adsorbed and free PMMA chains, percentage of adsorbed and free PMMA ester branches, percentage of adsorbed and free backbone dihedral angles, mean-squared end-to-end distance and mean-square radius of gyration of adsorbed and free PMMA chains, average number and length of trains, loops and tails, etc.) in the course of the MD simulation, and the distribution of backbone dihedrals in the various systems are provided in the Supporting Information to this paper.

ACKNOWLEDGMENTS Financial support for this study has been provided by the project “Multiscale Simulations of Complex Polymer Systems” (MuSiComPS) supported by the Limmat Foundation, Zürich, Switzerland. The work was supported by computational time granted from the Greek Research & Technology Network (GRNET) in the National


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HPC facility-ARIS-under project pr004011. To this, we feel indebted to Dr. Dimitrios Dellis from GR-NET, Greece, for his invaluable technical support regarding MD run execution on ARIS.

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