Carbon Nanotube

Apr 9, 2014 - The carbon atoms of CNT were kept fixed during the simulation. ...... Sanket A. Deshmukh , Benjamin J. Hanson , Qian Jiang , Melissa A...
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Molecular Dynamics Simulation of a Polyamide-66/Carbon Nanotube Nanocomposite Hossein Eslami* and Marzieh Behrouz Department of Chemistry, College of Sciences, Persian Gulf University, Boushehr 75168, Iran ABSTRACT: Atomistic molecular dynamics simulations are performed on oligomeric polyamide-6,6 chains, composed of 10 chemical repeat units, at a carbon nanotube (CNT) interface. The effect of surface curvature on the structure and dynamics of polymer is studied by simulating systems containing CNTs of various diameters. It is shown that polymer at the CNT interface organizes into layered structures. The hydrogen bonding in the polymer is influenced by the CNT surface. In proximity to the CNT surface, the hydrogen bonds (HBs) are weaker than the corresponding unperturbed bonds and their density is lower than that of the bulk sample. On the other hand, over the region where organized layered structures are formed, stronger HBs with a higher density than that of the bulk sample are found. An analysis of chain orientation at the interface shows that the monomers (repeat units) very close to the CNT surface wrap around the tube. However, at distances corresponding to higher densities than the bulk density, the segments orient parallel to the CNT axis (along the CNT). The wrapping costs higher energies in the case of smaller diameter CNTs (more curved surfaces). It is shown that while the CNT surface perturbs the local chain properties up to a distance of ∼2 nm from the surface, perturbation in the global chain properties, such as the radius of gyration, extends to farther distances (a few times the radius of gyration of the unperturbed chain, R0g). The chain translation at the interface is found to be anisotropic, depending on the surface proximity and surface curvature. This is due to the formation of extended conformations (along the CNT), facilitating smoother chain translation parallel to the CNT surface, compared to that in the radial direction. The magnitude of dynamics deceleration caused by the CNT surface depends on the surface proximity, surface curvature, and the time scale of the unperturbed dynamical property of interest. The dynamics decelerates more in the case of long-time dynamical properties (in the bulk) for chains at closer distances to flatter CNT surfaces. While the ratio of relaxation time to the corresponding bulk quantity for HB formation/rupture is increased by a factor of 3 in a cylindrical shell of thickness 0.6 nm around the flatter surface studied in this work, the above ratio for the decorrelation of chain’s end-to-end vectors is increased by 3 orders of magnitude in a cylindrical shell of thickness 1.0 nm on the same surface. In this respect, the interphase thickness depends on the time scale of the dynamical property of interest. Our observations show that the surface effect on a short-time dynamical property, like the HB dynamics, extends to distances as long as 1.5 nm, while it extends to a few R0g (from the surface) for a long-time dynamical property, like the relaxation of the entire chain expressed in terms of chain’s end-to-end vector relaxation. polymers with the inclusion of low volume fractions of CNTs.4 Nanofillers with high aspect ratio also exhibit dramatic changes in permeability of small gas molecules in the polymer.5 Although many experimental studies, as addressed above, have revealed much information on the properties and applications of polymer nanocomposites, a full grasp of the potential of such composites is still developing, and there exist open unresolved questions on the molecular nature of interactions between polymer and filler. For example, while it is shown that the viscosity of poly(propylene) is reduced dramatically through the addition of a minute amount of silica nanoparticles,6 no change in the viscosity and the chain mobility is observed for a polystyrene−polyisoprene copolymer

1. INTRODUCTION Polymer−nanotube composites have been widely investigated using various techniques to develop high performance materials, which combine the features of polymers with nanotubes. Nanoscale fillers, unlike micrometerscale fillers, offer excellent properties to a polymer matrix without decreasing its mechanical properties. Besides the significant improvements in mechanical properties of the polymer nanocomposites over those of the parent polymers, the inclusion of nanofillers also offers new multifunctional properties, such as electrical and thermal properties, which are not observed with micrometer-size fillers.1,2 For example, it is shown that inclusion of electrically conductive nanofillers, such as carbon nanotubes (CNTs) and graphene, beyond a threshold level of loading significantly increases the electrical conductivity of the polymer.3 There are also reports in the literature indicating improvements in the thermal properties of © 2014 American Chemical Society

Received: February 17, 2014 Revised: April 6, 2014 Published: April 9, 2014 9841

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in contact with layered nanosilicates.7 On the other hand, a viscosity enhancement is reported for nanoclay−poly(propylene) composites.8 Obviously, very sensitive instrumentation is needed to characterize thin polymer films, constituting only a small volume or weight fraction of the system but having a significant impact on the behavior of the system, in order to answer such not fully resolved questions. Molecular simulation methods, on the other hand, can complement the experimental studies, furthering our understanding of the molecular origin of such phenomena as reinforcement and impact on the mechanical, thermal, fire, and barrier properties. So far extensive atomistic simulations have been done to investigate particle−polymer nanocomposites and grafted particles in free chains. Early investigations consist of Monte Carlo (MC) and molecular dynamics (MD) simulation studies of model bead−spring polymer chains in contact with model (spherical) nanoparticles.9,10 However, a limited number of more recent simulations on realistic polymer−nanoparticle systems are also reported. For example, the interfaces between poly(ethylene oxide) and silica,11 polystyrene and gold nanoparticles,12 polystyrene and silica,13−15 and poly(methyl methacrylate) and silica16 have been studied in recent simulations. Recently, we have done detailed atomistic MD simulations of structural,17,18 rheological,19 and thermal properties20,21 of polyamide-6,6 (PA-6,6) in contact with graphene surfaces as well as the coarse-grained MD simulations22,23 of the structure and dynamics of PA-6,6 at the graphene and vacuum interfaces. As a continuation of our previous studies, in this work we do detailed atomistic MD simulation of PA-6,6 in contact with CNTs. The motivations of this study stem from the following facts: (1) The interface between polymers and CNTs is present in many CNT−polymer nanomaterials such as polymer nanocomposites and CNT sensors and in applications such as functionalization with large organic or biological molecules24,25 and transport in CNT channels.26 Performance of the fiber-reinforced nanocomposite in these applications critically depends on the structure of polymer chains at the interface. (2) The aspect ratio of the CNTs in CNT-based nanocomposites plays an important role on their properties. However, it is difficult to control and measure the effect of aspect ratio on the properties of nanocomposites. The results of the present computer simulation can provide insights into the effect of CNT diameter (surface curvature) on the structural and dynamical properties of polymer at the interface. (3) Among polymers, polyamides have excellent properties including high thermal stability, excellent mechanical strength, and high resistance to organic solvents, making them suitable for tailoring nanocomposites of varieties of engineering applications.27,28

Figure 1. Chemical structure of PA-6,6 chains simulated in this work.

Table 1. Description of Systems Simulated in This Worka composite PA-6,6 PA-6,6 PA-6,6 PA-6,6

+ CNT (6, 0) + CNT (10, 0) + CNT (17, 0) (bulk)

NC

N

ρ (kg m−3)

Lx = Ly (nm)

336 560 952 0

31 856 32 080 32 472 31 520

985.2 981.5 978.2 995.3

7.244 7.279 7.353 6.766

a

In all systems 80 PA-6,6 chains, consisting of 10 chemical repeat units (394 atoms), are simulated. The CNT length is identical with the dimension of the simulation box along the z axis (Lz = 5.983 nm). Nc and N are the number of C atoms involved in the CNT and the total number of atoms, respectively, and Lx and Ly are the dimensions of the simulation box along the x and y axes, respectively.

tube axis and the transverse directions. In all systems the center-of-mass of CNT coincides with the Cartesian coordinates origin, located at the center of the simulation box, and the largest axis of CNT is along the z axis of the Cartesian coordinates. The CNT length is identical with the dimension of the simulation box along the z axis (Lz = 5.983 nm). A snapshot of the simulation box for PA-6,6 containing CNT (17, 0) is given in Figure 2.

2. MODEL AND SIMULATION DETAILS In this work atomistic MD simulations for a CNT in oligomeric PA-6,6, composed of 10 chemical repeat units, were performed. To be able to study the effect of surface curvature on the PA6,6 properties at the interface, systems containing CNT (6, 0), CNT (10, 0), and CNT (17, 0) of diameters 0.475, 0.786, and 1.333 nm, respectively, were simulated. The chemical structures of PA-6,6 oligomers are shown in Figure 1. Simulations were performed for four systems: three polymer nanocomposite systems and a bulk polymer sample. The details of systems simulated are given in Table 1. All systems were composed of an infinitely long CNT, simulated in a simulation box, replicated using periodic boundary conditions along both the

Figure 2. Snapshot of a simulation box indicating PA-6,6 chains in contact with a CNT (17, 0).

The force field parameters for CNT and PA-6,6 chains were taken from the literature.17 All simulations were done at 500 K and 101.3 kPa, using the simulation package YASP.29 The temperature and pressure (101.3 kPa) were kept constant by coupling the system to a Berendsen thermostat and barostat.30 The time constants for coupling the system to the thermostat and barostat were 0.2 and 3.0 ps, respectively. The carbon atoms of CNT were kept fixed during the simulation. The 9842

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specified cylindrical shells around the CNT surface. A comparison of density profile peaks for all atoms and those of monomers shows that some fine structures, observed in the first peak of the former case, are not resolved in the latter case. Moreover, the effect of surface curvature on the polymer layering is more pronounced in the latter case; the system containing bigger diameter CNT (flatter surface) corresponds to the shaper density profile peak. As it is observed in our previous study,16 the segments involved in a chain have, on average, a closer distance to a flat surface than to a curved surface. Therefore, the chains in contact with the bigger diameter (flatter) CNT are subject to more attractions than those in contact with the smaller diameter (more curved) CNT. Also, compared to the density of all atoms, the surface influence on the density of monomers extends to longer distances from the CNT surface. 3.1.2. Hydrogen Bonding. Hydrogen bonding in PA-6,6 is an essential factor in determining the structure and dynamics of the polymer. Experimental observations indicate that nearly all the amide groups that separate a sequence of methylene groups in nylons are hydrogen-bonded.33 Because of the formation of a relatively strong hydrogen bond (HB) network, properties of nylons can be highly anisotropic. To study hydrogen bonding in present nanocomposites, we have shown in Figure 4 the

nonbonded interactions were truncated at 0.90 nm, and the neighbors were included if they were closer than 1.0 nm. A reaction field correction for the Coulombic interactions, with an effective dielectric constant of 5.5, was applied.31 All bond lengths were constrained using a SHAKE algorithm.32 The time step for the leapfrog integration scheme was 2.0 fs. All polymer−CNT nanocomposites were relaxed, verified by the relaxation of polymer end-to-end vectors for chains with their centers of mass within a distance of up to 2 nm from the tube surface, over a time span of 20 ns. After this initial equilibration cycle, simulations were further done for another 20 ns for the sake of data collections. During the simulation the trajectories of all atoms were recorded every 1 ps.

3. RESULTS AND DISCUSSION 3.1. Polymer Structure in the Interphase. 3.1.1. Polymer Layering beside the CNT Surface. The results of previous simulations on fluids in contact with flat and curved surfaces show that organized fluid layers beside the surfaces are formed.15−23 In this work the profiles for number density, ρ, calculated in cylindrical shells (of thickness 0.01 nm) around the nanotube surface are normalized with the bulk number density, ρ0, and are depicted in Figure 3. Here the density

Figure 3. Normalized density profiles for all atoms (top panel) and for centers of mass of monomers (bottom panel) at 500 K and 101.3 kPa as a function of distance from the CNT surface. The description of systems is given in the figure. Figure 4. Normalized density profiles for amide H, O, and N at the interface of CNT (6, 0) (top panel), the second Legendre polynomial (red curves) for the orientation of methyleneamide plane normal, shown in the figure, and the first Legendre polynomial (dark curves) for the orientation of NH bond vector with respect to the CNT surface normal, shown in the figure (middle panel), and normalized density profiles for HBs per amide H atom (bottom panel) as a function of distance from CNT surface. For the sake of consistency, in middle and bottom panels the horizontal axis indicates the distance of amide H atom from the surface. The description of systems is given in the figure.

profiles are plotted as a function of radial distance, d = (x2 + y2)1/2, from the tube surface. The results in Figure 3 for the density profile of all atoms show that organized layers of polymer are formed up to distances as long as 2.0 nm from the nanotube surface. A slight effect of the CNT curvature (surface area) on the density profile of all atoms is seen in this case (see the inset of Figure 3). We have also plotted the density profiles for the individual monomers (repeat units) by counting the number of monomers whose centers of mass are located in 9843

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p1 passes through zero at d = 0.46 nm followed by a maximum at d = 0.52 nm. This maximum is indicative of preferentially parallel orientations of the NH bond, normal to the plane of methyleneamide, and the CNT surface normal vectors and corresponds to the position of maximum in the amide H density profile peak. To study the effect of NH and methyleneamide plane orientations on the hydrogen bonding, we have also counted the number of HBs as a function of radial distance of amide H atoms from the surface. Here, the HBs are counted based on a geometric criterion in which the H···O bond length, lH···O, is less than a threshold value of 0.297 nm and the donor− hydrogen−acceptor angle, θN−H···O, is bigger than 130°. The threshold length and angle are chosen based on previous examination of HB lengths and angles.34,35 We have counted the number of HBs per amide H in cylindrical shells around the CNT axis and reported the results in Figure 4. A comparison of the results for HB and amide H density profiles indicates that a small fraction of H atoms very close to the CNT surface (d ≤ 0.25 nm) are capable of HB formation. At d = 0.25 nm the ratios of density to the corresponding bulk value are 0.6, 0.1, and 0.1 for amide H, O, and HBs, respectively. This means that a majority of amide H atoms in this region are not hydrogenbonded because of the lack of O atoms, and hence, the density of HBs closely matches the density of O atoms. Such a restriction in HB formation is due to the preferentially parallel orientations of methyleneamide groups along the CNT surface, forcing the H atoms very close to the surface (away from the O atoms). The number of HBs, however, increases with increasing distance from the surface. At d = 0.43 nm, the number of HBs per amide H is ∼10% higher than the corresponding bulk valve. At this distance the density of amide H atoms is ∼5% less than the corresponding bulk value. However, such H atoms are surrounded by a region (within the HB threshold length) enriched from O atoms, facilitating HB formation. The preferentially antiparallel orientation of NH bond vectors along the surface normal at this distance means that these H atoms mainly form HBs with O atoms locating at shorter distances to the CNT surface (see the density profile peak for O atoms in Figure 4). However, at d = 0.53 nm, corresponding to the position of maximum in the amide H density profile peak, the amide H atoms are surrounded by a depleted region from O atoms. Here, the preferentially parallel orientations of NH bond vectors along the surface normal suggest that in this region the H atoms form HBs with O atoms locating at longer distances from the CNT surface. Here, the lower density of O atoms, compared to amide H atoms, lowers the density of HBs per amide H. A slight effect of surface curvature on the density of HBs is seen in Figure 4. Increasing the surface area imposes more restrictions on the HB formation very close to the surface. However, a slight increase in the number of HBs is seen at distances corresponding to the position of maximum in the HB density profile peak (0.43 nm). This is due to the formation of sharper O layers at the interface of flatter CNT, compared to the interface of more curved CNT. To examine the surface effect on the lH···O and θN−H···O, we have plotted in Figure 5 the density profiles, normalized with the corresponding bulk values, for HBs choosing a shorter threshold HB length, lH···O < 0.25 nm, and a larger threshold angle, θN−H···O > 150°, i.e., counting stronger HBs, compared to the case where lH···O < 0.297 nm and θN−H···O > 130°. It is seen that very close to the CNT (6, 0) surface, because of

density profiles, normalized with the corresponding bulk values, for N, H, and O atoms of amide groups, which are involved in hydrogen bonding, for the sample of PA-6,6 containing CNT (6, 0) as a typical example. Here, the maxima in the O and N density profile peaks are located at d = 0.33 nm and d = 0.43 nm, respectively. However, compared to N and O density profiles, the peak for amide H atoms is much wider; it starts from a shorter distance to CNT surface followed by a wide shoulder at 0.32 nm and a maximum at 0.52 nm. Obviously, the layering behavior of N, O, and H atoms, seen in Figure 4, perturbs the HB structure at the interface, from that of the bulk sample. To understand the reason for closer proximity of amide H, compared to N and O atoms, to the CNT surface, we have studied the orientational behavior of methyleneamide groups, −CH2CONH−, expressed in terms of the second Legendre polynomial, p2, defined as p2 (d) =

1 (3⟨u1·u 2⟩2 − 1) 2

(1)

where u1 is an outward pointing unit vector normal to the CNT surface, pointing to the geometrical center of methyleneamide group (surface normal), and u2 is a unit vector normal to the methyleneamide plane at its geometrical center, shown in Figure 4. For the sake of consistency, here the radial distance d is defined as the distance of the amide H atom from the CNT surface. Shown in Figure 4 is the second Legendre polynomial for the orientation of methyleneamide plane normal with respect to the CNT surface normal. The results show that amide H atoms close to the CNT surface are connected to methyleneamide groups whose surface normal is oriented parallel (p2 ≈ 0.6) to the CNT surface normal; i.e., the plane of methyleneamide groups is parallel to the CNT surface. By increase of the distance of methyleneamide groups from the CNT surface, they adopt more and more random orientations. At d = 0.52 nm (the position of maximum in the amide H density profile peak), p2 shows its second maximum. This indicates that the amide H atoms at d = 0.52 nm are connected to methyleneamide groups which are preferentially parallel to the CNT surface. To show how the orientation of methyleneamide groups affects the NH bond orientation and hence on the HB structure, we have shown in the same figure the first Legendre polynomial p1 (d) = ⟨u1·u3⟩

(2)

for the orientation of NH bond vectors with respect to the surface normal. Here p1 is the first Legendre polynomial, u1 is the CNT surface normal, pointing to the midpoint of NH bond, u3 is the NH bond unit vector (pointing from N to H), and d is the distance of amide H atom from the surface. The results in Figure 4 indicate that NH bonds close to the surface have antiparallel orientations (p1 = −1) with respect to the CNT surface normal. Such an orientation is due to the adoption of parallel orientations of methyleneamide groups along the CNT surface. Because of the connection of bulky polymer backbone to N, energetically favorable parallel orientation of the methyleneamide group along the CNT surface forces the small amide H atoms toward the CNT surface. Therefore, the NH vector close to the CNT surface is an inward-pointing vector. This is in agreement with the density profile peaks for amide H, N, and O atoms, i.e., the H atoms are closer to the CNT surface than N and/or O atoms. By increase of the distance of NH bonds from the CNT surface, 9844

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Figure 6. Second Legendre polynomial for the orientation of monomer end-to-end vector along the CNT axis and along the CNT surface normal as a function of the monomer’s center-of-mass distance from the surface.

Figure 5. Effect of variations in the threshold values of hydrogen bond lengths, lH···O, and the donor−hydrogen−acceptor angle, θN−H···O, on the normalized hydrogen bond density profiles at the CNT (6, 0) interface. The hydrogen bond lengths and angles are depicted in the figure.

angle are shown in Figure 7. Here, the factor sin(θ) is the Jacobian of transformation of Cartesian to spherical coor-

geometrical restrictions, the number of shorter length and flatter HBs is reduced. At d = 0.43 nm, corresponding to the position of maximum HB density profile, however, the number of shorter length and flatter HBs is increased. This indicates that because of the formation of compact organized polymer layers beside the CNT surface, HBs formed in the higher density regions beside the CNT are stronger than the corresponding unperturbed bonds. Expectedly, geometrical restrictions allow formation of weaker HBs (compared to those in the bulk sample) in proximity to the CNT surface. 3.1.3. Chain Conformation. The degree of alteration in chain conformations is examined in terms of the orientations of monomers’ end-to-end unit vectors with respect to CNT surface normal and/or CNT axis. The second Legendre polynomial, eq 1, quantitates this orientation. Shown in Figure 6 is the orientational behavior of the monomer’s end-to-end unit vectors, with respect to both surface normal and tube axis, as a function of monomer’s center-of-mass distance from the CNT surface. The results indicate that polymer segments close to the CNT surface have perpendicular orientations with respect to both vectors. While the monomers very close to the CNT surface (with perpendicular orientations with respect to both surface normal and CNT axis) must wrap around the tube, the monomers located in higher density regions (see Figure 3) prefer extended conformations along the tube axis. The region (0.35 nm < d < 0.72 nm) over which the monomer end-to-end vectors orient parallel to the tube axis (p2 > 0) correlates with the higher density region in Figure 3. The effect of surface curvature on the orientation of monomer end-to-end vector is more pronounced for chains in the vicinity of more curved CNT surfaces. The coiling behavior of monomers whose centers of mass are within the first density profile peak (d ≤ 0.63 nm) can be influenced by the surface structure. The coiling of PA-6,6 monomers around the CNTs is investigated in terms of the probability, P(θ), of observing the wrapping angle, θ, between the monomer end-to-end vector and the CNT axis. Plots of probability density, P(θ)/[sin(θ) dθ], as a function of wrapping

Figure 7. Normalized probability distribution function for the wrapping angle, θ, between the CNT axis and monomer (repeat unit) end-to-end vectors. The description of systems is given in the figure.

dinates. As the results show, a distinct peak at θ = 0° is seen for all systems. For the smaller diameter CNT (6, 0) the monomer coiling becomes less important. It is shown that the energy required for coiling36 scales as sin4(θ)/r2, where r is the coiling radius of the chain. To investigate the effect of CNT diameter (curvature) on the coiling angle distribution, we have shown in Figure 8 plots of ln(P(θ)) versus sin4(θ) for our three nanocomposite systems. As analytical theory predicts, the slopes of lines decrease for chains in contact with flatter CNT surfaces. This means that the chains at the more curved CNT interfaces adopt energetically favorable extended conformations along the tube axis. Increasing the tube diameter increases the probability of chain wrapping around the CNT. 9845

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increasing surface area. The same effect has been observed in previous simulations in the literature.14−16 A comparison of the interfacial length scales, obtained in terms of density profiles (see section 3.1.1), and those obtained in this section, in terms of global chain structure, indicates that the interphase is thicker in the latter case than in the former case. For a global chain property (such as the chain’s radius of gyration) the interphase thickness exceeds the dimensions of the present simulation box. Simulations of much bigger systems, which are not feasible to do at atomistic resolution, are required to discriminate the interphase thickness defined in terms of global chain properties.23 3.1.4. Chain Configuration at the Interface. The chain structure at the interface can be addressed by analyzing the way they contact the CNT surface. On the basis of our analysis of polymer layering at the interface, here, a surface contact is assumed when an atom is located within the first density profile peak in Figure 3, i.e., d ≤ 0.63 nm. The consecutive atoms belong to one of the three categories: those in contact with CNT surface (trains) and those located in d > 0.63 nm but delimited between one (tails) and two (loops) surface contacts. The normalized tails and loops density profile peaks for three nanocomposites are shown in Figure 10. The density profile for

Figure 8. Natural logarithm of the probability of observing a wrapping angle, θ, versus sin4(θ) for systems shown in the figure.

We have further studied the effect of CNT surface on the global chain conformations by investigating the perturbation in the chains’ root-mean-squared radius of gyration, Rg, from the corresponding bulk value, R0g. We have calculated the local root-mean-squared radius of gyration values by putting the chains’ centers of mass into cylindrical shells of thickness 0.1 nm and showed the results in Figure 9. The results show that

Figure 10. Normalized density profiles for atoms involved in loops and tails for systems shown in figure as a function of distance from the CNT surface.

trains is identical to the first density peak in Figure 3. The results in Figure 10 indicate that the tails and loops also show layered structures in the polymer−CNT interphase. It is seen that a fraction of tails and loops even extend over the whole simulation box. Expectedly, the extension of chains, from the interface to the polymer phase, is more pronounced in the case of tails. Moreover, the chains in contact with the bigger diameter (flatter) CNT extend to farther distances from the interface than those in contact with the smaller diameter (more curved) CNT. This can be interpreted in terms of decreased repulsions between the chains over flatter surfaces and hence more penetrations of chains to the flatter CNT surface. Because of the fact that such tails and/or loops are connected to the interface, they have a different relaxation behavior than the unperturbed (bulk) chains. Therefore, the interphase thickness defined in terms of the configurations introduced

Figure 9. Normalized z-component of the radius of gyration as a function of the chain’s center-of-mass distance from the CNT surface. The description of systems is given in the figure.

for centers of mass in proximity of the CNT surface, the chain conformations, compared to the conformations of unperturbed chains, have been substantially altered. The z-component of radius of gyration of chains at the interface of the smaller diameter CNT increases by a factor of 2 above its unperturbed value, 1.6 nm. The magnitude of chain stretching depends on the nanotube diameter (surface curvature). While the chains at the interface of more curved CNTs extend along the CNT axis, those at the interface of more flat CNT surfaces wrap around the tube. This is due to the increased surface interactions with 9846

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nm). Moreover, the decay rate is slower on flatter surfaces. This is due to more hindered translational diffusion of polymer chains in contact with flatter surfaces (see section 3.2.2). However, the decay rate of C(t) within 0.63 nm < d < 1.26 nm is roughly independent of surface curvature and resembles the bulk sample relaxation. To have a quantitative measure of the effect of surface proximity on the relaxation behavior of C(t), we have fitted the relaxation curves in Figure 12, in the range

here is thicker than that defined in terms of more localized structural properties (for instance, the density profile peaks for HBs and for monomers depicted in Figures 4 and 3, respectively). This is in accordance with the results of previous atomistic and coarse grained models of polymers in contact with flat and curved surfaces.16,23 The interphase thickness defined in terms of chain extension from the interface to the polymer phase extends to ∼2−3 R0g. Therefore, when expressed in terms of long scale (global) structural properties, even the whole simulation box in atomistic simulations is the interphase. 3.2. Polymer Dynamics in the Interphase. 3.2.1. Hydrogen Bond Dynamics. The dynamics of HBs, formed between amide groups, is analyzed through investigation of the following time correlation function, introduced by Rapaport.37 C(t ) =

⟨h(0) h(t )⟩ ⟨h(0)⟩

(3)

where t is the time, C(t) is the intermittent correlation function, h(t) is the population variable, being unity when a particular hydrogen−oxygen pair is hydrogen-bonded at time t and being zero otherwise, and the brackets indicate the average over all hydrogen−oxygen pairs and over all time origins. The intermittent correlation function is the probability that a randomly chosen donor−acceptor pair is hydrogen-bonded at time t provided that it was bonded at t = 0 (independent of possible breakings in the interim time). In Figure 11 the variation of function C(t) with time for different cylindrical shells (of thickness 0.63 nm) around the

Figure 12. Comparison of relaxation behavior of different dynamical properties, with the corresponding bulk value, as a function of reciprocal distance from the CNT surface. The open, half filled, and filled markers indicate the results for PA-6,6 chains in contact with CNT (6, 0), CNT (10, 0), and CNT (17, 0), respectively.

1−600 ps, with a Kohlrausch−Williams−Watts (KWW)38 stretched exponential function and calculated the structural relaxation time of HBs (τHB) by analytic integration. The results are compared with the relaxation time of HBs in the bulk PA6,6 (τ0HB = 55.5 ps) in Figure 12. A maximum perturbation in τHB from τ0HB is observed for HBs located within d < 0.63 nm over the flatter CNT surface. In this case, the dynamics of HB formation/rupture, compared to the corresponding bulk process, decelerates by a factor of 3. The interphase thickness, defined in terms of a fast dynamical process, such as the HB dynamics, is ∼1.5 nm. 3.2.2. Chain Translation. A good measure of overall chain mobility at the interface is the chain’s center-of-mass meansquared displacement (MSD), defined as MSD = ⟨(r(t ) − r(t0))2 ⟩

Figure 11. Intermittent correlation function for hydrogen bond formation and rupture. The solid and dashed curves show the results for hydrogen bonds located within 0 nm < d < 0.63 and 0.63 nm < d < 1.26 nm, respectively. The description of systems is given in the figure.

(4)

where r(t) is the chain’s center-of-mass position vector at time t. We have shown in Figure 13 the MSD curves for different 1.0 nm thick cylindrical shells around the CNT (10, 0) as a typical example. In the calculation of MSD(t) curves, the average is taken over all centers of mass and over all time origins for centers of mass which remain continuously in a specified shell. When a center of mass travels out of a preexisting shell to a new shell, the time origin for that center of mass in the new shell is set to zero. This means that the MSD(t) in each shell is calculated for the time the center of mass spends in that shell. The results in Figure 13 show that chain mobility at the interface follows a short-time (t < 1 ps) ballistic regime, over which the slope of MSD versus t is close to 2, followed by a

CNT is given. The intermittent correlation function has an initial fast decay (