Effect of particle size and grafting density on the mechanical properties of polymer nanocomposites

Polymer 54 (2013) 5222e5229

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Effect of particle size and grafting density on the mechanical properties of polymer nanocomposites Huikuan Chao, Robert A. Riggleman* Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 April 2013 Received in revised form 26 June 2013 Accepted 3 July 2013 Available online 13 July 2013

End-grafting polymer chains to nanoparticles in polymer nanocomposite is a widely used method to disperse inorganic particles in a polymeric matrix in order to improve the material properties. While many fundamental studies have investigated how various factors influence the dispersion or aggregation of the nanoparticles, the effect of grafting on the resulting material properties has received considerably less attention. In particular, the effect of nanoparticle curvature and grafting density on the mechanical properties in polymer nanocomposites remains elusive. In this study, we develop a coarse-grained model of a polymer glass containing grafted nanoparticles and examine the resulting effects on the mechanical properties. By carefully designing the parameters of our polymer nanocomposites model, we can maintain dispersion of the nanoparticles whether they are grafted with polymer chains or not, which allows us to isolate the effect of end-grafting on the resulting mechanical properties. We examine how the nanoparticle size and grafting density affect the elastic constants, strain hardening modulus, as well as the mobility of the polymer segments during deformation. We find that the elastic constants and yield properties are enhanced nearly uniformly for all of our nanocomposite systems, while the strain hardening modulus depends weakly on the grafting density and the nanoparticle size. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Polymer nanocomposites Mechanical properties Molecular simulation

1. Introduction Polymer nanocomposites (PNCs) are materials obtained by dispersing nanoparticles in a polymer matrix, and due to the large surface-to-volume ratio between the nanoparticles and the polymer, substantial property changes can be observed for relatively low concentrations of particles [1,2]. Often the desired property changes depend critically on maintaining good dispersion of the nanoparticles in the polymer matrix, and aggregation of the nanoparticles can lead to the loss of the enhanced properties of the material [2,3]. This is particularly important for inorganic nanoparticles, which typically have very strong interactions between the particle cores that can readily lead to aggregation if the surfaces of the nanoparticles are not functionalized. One popular method to promote dispersion is to end-graft polymer chains onto the particle surface that are either chemically identical to or have an affinity for the matrix chains. However, it remains unclear how material properties are affected by end-grafting polymer chains onto the surface of the nanoparticles.

* Corresponding author. E-mail address: [email protected] (R.A. Riggleman). 0032-3861/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.polymer.2013.07.018

At low grafting densities when the grafted chains are wellseparated on the nanoparticle surface, the coreecore interactions between the nanoparticles will only be prevented very close to the grafting points. In this case, the particles will interact with each other as “patchy” attractive particles, and can lead to a variety of equilibrium self-assembled structures such as strings and sheets [4,5]. Significant mechanical reinforcement in the melt state has been found when these structures percolate the polymer matrix [6]. At the limit of high grafting density, the interactions between the particle cores are effectively shielded, and the dispersion state is dictated by the entropy of the grafted and the matrix chains. Depending on the particle curvature and relative length of the grafted chains to the matrix polymers, the grafted chains can be wetted by matrix chains which penetrate the brush layer, causing the grafted chains to adopt an extended conformation and thus stabilize the particles’ dispersion in the matrix [7e9]. When the melt chains become smaller or comparable to that of grafted chains, the entropic penalty for the matrix chains to penetrate the grafted chains can be compensated by the translational entropy of matrix chains, which leads to a “wet” the brush layer. Increasing particle curvature (smaller radius) has an similar to decreasing grafting density, which reduces crowding between the grafted chains and facilitates melt chain penetration [9].

H. Chao, R.A. Riggleman / Polymer 54 (2013) 5222e5229

Previous molecular simulation studies of PNCs with grafted nanoparticles have focused on the effect of grafting density, particle radius, and the length of matrix chain on configuration of the grafted chains [10e13]. Other studies also investigated the interaction between two particles as a function of the relative chain length between grafted and matrix chains [14]. However, these studies had either one or two grafted nanoparticles immersed in the polymer matrix, which neglects any possible many-body effects. Studies of PNCs with multiple particles typically either use bare particles [15,16] or grafted nanoparticles in an implicit matrix to study the self-assembly [17]. Therefore, to the best of our knowledge, simulation studies of PNCs containing multiple grafted nanoparticles in an explicit polymer matrix remain uncommon. In most applications, understanding the role that the nanoparticles have on the resulting mechanical properties of the polymer will be of central importance in both the final applications as well as the processing of the original sample. In the melt state, it remains unclear how nanoparticles can alter the entanglement network of the polymer matrix, and some experimental reports have shown that the plateau modulus can increase [18,19] or decrease [20]. Although molecular models have shown that nanoparticles can alter the primitive path network in the melt state [15,21], the direct connection to the corresponding plateau moduli has not yet been established. Recent experiments have demonstrated that a percolating network of matrix polymer chains and nanoparticles leads to optimal reinforcement in the melt [6]. In the glass state, it is commonly thought that the interactions and conformational changes in the polymer chains located in the interfacial area between the polymer and the nanoparticles dominate the influence of the nanoparticles on the mechanical properties [22], and this qualitative picture has been confirmed in simulations [23]. However, the focus in these studies has been on the elastic properties, and it is unclear how other mechanical behaviors are affected (e.g., the yield stress and post-yield behaviors). One of the mechanical behaviors in glassy polymer systems that has received significant attention recently is strain hardening, which happens when the glassy polymers experience large (typically compressive) deformations far beyond the yield point. Early studies [24] in strain hardening used rubbery elasticity theory that treated the glassy polymers as a cross-linked network spaced by entanglement length, Ne. However, as pointed out by Kramer [25], the temperature dependence of the hardening modulus is inconsistent with the view that entanglements are the origin of strain

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hardening. Furthermore, simulation studies have shown that strain hardening could happen in a glassy polymer system whose chain length is below the entanglement length [26e28]; these studies provide evidence that the origin of strain hardening in polymer glasses is due to the deformation of the polymer chains. Beyond the yield point when the material exhibits plastic flow, an increasing stress is required to drive plastic flow while maintaining the chain connectivity [29]. The molecular theory of Chen and Schweizer was able to capture many of the effects of strain hardening on the mechanical and dynamical response of polymer glasses by incorporating the effects of deformation on the packing and orientation of the polymer chains [30]. If anisotropic polymer packing is the origin of strain hardening, then one may expect that grafting chains to the surface of a nanoparticle may further restrict their ability to align and allow for plastic flow. Therefore, in a PNC containing grafted nanoparticles, one may expect that strain hardening is more pronounced due to the increased restriction on some of the chains’ freedom to align with the deformation. To that end, the present work aims to investigate the effect of grafted chains on the mechanical reinforcement of a glassy model polymer nanocomposite containing grafted nanoparticles using non-equilibrium molecular simulations. By designing a model system that remains dispersed whether the nanoparticles are grafted or not, we can directly probe the role that chain grafting plays on the mechanical response of glassy polymer nanocomposites containing grafted particles. We examine the effects of grafting density and particle curvature on the elastic constants, yield stress, and the strain hardening behaviors. We find that grafting the polymer chains to the nanoparticles does lead to an enhancement in strain hardening, while grafting has a negligible effect on the elastic constants and yielding behaviors. In addition, during the deformation we find that the dynamics of the grafted chains near the grafting sites are reduced compared to both the matrix chains and the ends of the grafted chains, offering a possible explanation for the increased strain hardening modulus. Given that more pronounced strain hardening is often associated with more ductile materials [31], our results imply that grafted nanoparticles could potentially be used to design tougher materials. 2. Methods The coarse-grained polymer model is similar to the canonical Kremer-Grest model [32], and each chain consists of 40 Lennarde

Fig. 1. (a) Visualization of our PNC system with a particle radius R ¼ 3.0s and grafting density 0.2. The size of the monomers of the grafted chains (blue and transparent) and matrix chains (yellow and transparent) are reduced to show the distribution of particles within the simulation box. Note that the surface beads of the particles are not shown. Images of isolated nanoparticles and their grafted chains with R ¼ 3 and G ¼ 0.3 (b) and R ¼ 3 with G ¼ 0.1 (c) are also shown, where the particle surface beads are in red, and the grafted chains are in blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Table 1 Parameters of each system examined: effective particle radius R, grafting density G, number of nanoparticles in the system Np, total number of grafted chains M, number of melt chains N, and the glass transition temperature Tg. System

R

G

Np

M/N

Tg

1 2 3 4 5 6 7 Pure polymer

1.5 3.0 3.0 3.0 3.0 4.5 4.5 e

0.20 0.40 0.20 0.10 0.05 0.00 0.20 e

270 34 34 34 34 10 10 0

540/657 1088/109 544/653 272/925 136/1061 0/1197 400/797 0/1400

0.422 0.428 0.430 0.426 0.427 0.420 0.424 0.422

" UðrÞ ¼ 4ε

Jones (LJ) interaction sites connected by stiff harmonic bonds. The nanoparticle cores are modeled as spherical shells with Lennarde Jones interaction sites distributed evenly on the surface (see Fig. 1(c)). The volume averaged mass density of the particle is set to r0 ¼ m/s3, and the ratio between the total mass of the surface beads (ms) and the particle center (mc) is fixed at 1.5. This ensures that our particle model, which is a hollow shell of LJ interaction sites with a single site in the center, has the same moment of inertia as a particle with the same radius and a homogeneous mass density of r0. Polymer chains are end-grafted to surface sites that are selected randomly from all sites on the surface of a particle, and each particle has a distinct distribution of grafting sites on its surface. The positions of the individual LJ particles that comprise our nanoparticles’ surfaces are maintained by a network of stiff harmonic bonds which maintains the distance between particle center and

1

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12

s

 

r  Dp

s r  Dp

6 # ;

r < rc þ Dp

(1)

Here, Dp is the diameter of the particle, r is the centerecenter distance between two particles, and the cut-off radius rc ¼ 2.5s for all pair-wise interactions in the system. All numerical values presented below are in dimensionless units reduced by the Lennarde Jones parameterspffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the polymer monomers (e.g., T ¼ kB T * =ε; t ¼ t * ε=ms2 , etc. where the * indicates values in laboratory units). All of our systems are at a fixed particle core volume fraction of 5.5%. Detailed information about the model PNCs with different nanoparticle radii and grafting densities are listed in Table 1. An image of a typical simulation box as well as configurations of two nanoparticles are shown in Fig. 1(a) and (b). For reference, the radius of gyration of polymer chains is Rg z 3.30 in the pure polymer melt. All systems are first equilibrated using molecular dynamics (MD) simulation at a high temperature T ¼ 1.5, P ¼ 0.0 in the NPT ensemble with time step dt ¼ 0.002 to disperse particles in the simulation box. The double-rebridging algorithm is introduced into the simulation to facilitate equilibration of polymers [33e36]. Equilibration of our particle distributions was assessed by ensuring that the motion of the particles had entered the diffusive regime

gPM(r)

φ(r)/φsurf

interaction sites on the surface. The interaction between a polymer monomer and individual site on the particle surface has the same energy as that between polymer monomers, and the expanded LennardeJones potential is adopted to describe non-bonded pairwise interaction between particles.

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Fig. 2. Density profiles and radial distribution functions of chains in systems with constant grafting density G ¼ 0.2 with different particle radii: R ¼ 1.5s (black circles), 3.0s (red squares), and 4.5s (green diamonds). (a) Monomer density profile as a function of position from the particle center of grafted chains that are grafted to the reference particle, normalized by f near the surface. (b) Radial distribution function of the matrix chains around the particles. (c) Radial distribution function between particle centers and the grafted chains belonging to other nanoparticles. (d) Radial distribution function between the particle centers. In (a), (b) and (c) all functions are shifted to the surface of particle, while in (d) the functions are shifted to the left by a distance of the diameter of particle in each system. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

H. Chao, R.A. Riggleman / Polymer 54 (2013) 5222e5229 2 ðtÞft 1 , the nanowhere their mean-squared displacement rnp particles had diffused a distance larger than their diameter, and that our equilibration time is at least one order of magnitude longer than the rotational relaxation time of the nanoparticle with highest grafting density. After equilibration, two uncorrelated configurations of each system are selected and then cooled down below glass transition temperature Tg (see Table 1) to T ¼ 0.37 at cooling rate of 104 followed by aging for t ¼ 4  103. The value of Tg in Table 1 is obtained by calculating the density as a function of temperature and identifying the temperature at which the thermal expansion coefficient changes from a liquid to a bulk-like value. Finally, all systems are subjected to a uniaxial compressive deformation in the x-direction at a constant true strain rate ε_ ¼ 1  104 while the normal stresses in uncompressed directions are held at zero. All the simulations described above are executed in LAMMPS simulation package [37].

3. Results and discussion 3.1. Brush configuration We begin by investigating the structure of our PNCs as we vary the nanoparticle core size at a constant grafting density in the melt state far above Tg at T ¼ 1.5. Fig. 2(a) plots the density profile of the grafted chains as a function of position away from the particle surface for nanoparticles of varying radius at a constant grafting density; the density profiles are normalized by their surface values. We find that as the radius of the particle increases from 1.5s to 4.5s, the grafted chains become increasingly stretched away from the particles’ surface and adopt a more extended configuration; the

chains grafted to particles with a lower curvature (larger radius) are more crowded by the neighboring grafted chains and therefore stretch farther from the nanoparticles’ surfaces. Fig. 2(b) plots the radial distribution function calculated between the particle centers and the matrix polymer monomers, and we find that as the particle radius increases, the matrix chains do not penetrate the brush layer as much, consistent with the view that the grafted chains are more crowded on the larger particles. Similarly, Fig. 2(c) plots the radial distribution function between the particle centers and the monomers from chains grafted to other particles, which are also increasingly excluded from the brush layer at the particle curvature decreases and the grafted chains become more crowded. Finally, Fig. 2(d) plots the pair distribution function between the particle centers. For the smallest particles (Rp ¼ 1.5), the particle positions are essentially uncorrelated and g(r) quickly settles to g(r) ¼ 1 at short distances. However, for the larger particles, we find the onset of weak longerange correlations in the nanoparticle positions, as indicated by the features that arise in g(r), although we do not observe peaks larger than g(r) z 1.6. We next compare the brush profiles for particles as a function of the grafting density G but at fixed particle radius, Rp ¼ 3.0. Fig. 3(a) shows the average density of the grafted chains as a function of distance from the surface of the particle to which they are grafted. As one would expect, as the grafting density is increased, the chains become increasingly crowded and stretch farther from the particle surfaces. A previous simulation study [12] examining systems with longer grafted and matrix chains shows that the density profile of long grafted chains exhibits a transition from concave to a convex parabolic shape as the grafting density increases, which coincided with the transition from wet to dry brush structure. Fig. 3(b), (c),

1.5

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Fig. 3. Density profiles and radial distribution functions in systems with fixed radius R ¼ 3.0s and varying grafting density with G ¼ 0.05 (black circles), 0.1 (red squares), 0.2 (green diamonds), and 0.4 (blue triangles). (a) Monomer density profile as a function of position from the particle center of grafted chains that are grafted to the reference particle. (b) Radial distribution function of matrix chain around the particles. (c) Radial distribution function between particle centers and the grafted chains belonging to other nanoparticles. (d) Radial distribution function between the particle centers. All the curves in (a), (b) and (c) are shifted by the particle radius, while in (d) gPP(r) is shifted by the particle diameter. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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and (d) together show that increasing the grafting density also decreases the density of the matrix chains and the grafted chains from other particles that are near the surfaces of the particles considered here. Furthermore, we see that the matrix chains fill in some of the volume between the particles, as the peak in the distribution function between the particle centers and the matrix chains (Fig. 3(b)) is on a slightly shorter length scale than the peak between particle centers (Fig. 3(d)). When we examine the correlations between particle centers (Fig. 3(d)), for the lower grafting densities there is essentially no correlation in the particle positions other than excluded volume, while for higher grafting densities the nanoparticle positions develop some weak correlation (g(r) > 1), even though they remain well dispersed. We emphases that the maximum values in g(r) are relatively small in all cases, and the particles remain well-dispersed.

Table 2 Tabulated mechanical properties of our various PNC systems. Strain hardening modulus, Gs

R ¼ 1.5,G ¼ 0.2 R ¼ 3, G ¼ 0.05 R ¼ 3, G ¼ 0.1 R ¼ 3, G ¼ 0.2 R ¼ 3, G ¼ 0.4 R ¼ 4.5, G ¼ 0 R ¼ 4.5, G ¼ 0.2 Pure polymer

0.21 0.24 0.25 0.27 0.33 0.23 0.42 0.22

       

0.01 0.02 0.01 0.03 0.01 0.03 0.05 0.01

Plastic flow stress, sflow 0.78 0.73 0.73 0.74 0.71 0.74 0.61 0.66

       

0.01 0.04 0.02 0.04 0.01 0.02 0.03 0.04

Young’s modulus, E 34.78 34.39 34.12 35.04 34.51 37.02 37.60 31.06

       

Yield stress, sY

2.63 0.68 2.01 3.54 2.01 1.70 2.38 0.12

1.03 1.03 1.01 1.03 1.04 1.04 1.02 0.90

       

0.03 0.01 0.01 0.02 0.04 0.01 0.02 0.01

pure polymer, but there is no clear trend that is beyond our uncertainty in the changes of the yield stress with particle size and grafting density. Interesting trends emerge in the analysis of the strain hardening modulus. For systems with the same grafting density but varied nanoparticle size (Fig. 4(a)), the strain hardening modulus tends to increase with nanoparticle size, while systems with smaller particles have a slightly higher plastic flow stress. For systems with same particle radius but different grafting density, Fig. 4(b) and the data in Table 2 shows that the strain hardening modulus gradually increases with G. Finally, we compare our system with the largest hardening modulus, the PNC where Rp ¼ 4.5 and G ¼ 0.2, with a PNC containing bare nanoparticles (G ¼ 0) of the same radius. The stress response in Fig. 4(c) and the tabulated data in Table 2 demonstrates that simply adding the nanoparticles increases the elastic modulus and yield stress, while it is the grafting of the polymer chains that leads to enhanced strain hardening. The results indicating that bare nanoparticles do not significantly affect the strain hardening modulus are consistent with previous simulation results [41]. In order to further investigate the contribution of the interaction between particles and grafted chains to strain hardening, the stress calculated in our simulations is decomposed into its various components. The contribution to the stress tensor s that arises from P intermolecular forces can be written as s ¼ ð1=VÞ ij rij 5f ij , where the sum goes over all particle pairs i and j, and rij and fij are the separation and the force between the particles, respectively. By restricting the sum only to pairs of particles that make up the different components of our system (e.g., the nanoparticle surface beads, the matrix polymers, or the grafted chains), we can isolate the contribution of each interaction to different regions of the mechanical response. The normalized contribution from the nonbonded interactions between the nanoparticles and the grafted chains is presented in Fig. 5(a). This figure shows the interaction

3.2. Mechanical properties Fig. 4 shows the mechanical response for our polymer nanocomposite systems under compressive deformation at a constant rate, where we plot the measured stress against the ideal rubber elasticity factor, g(l) ¼ 1/ll2, where l is the macroscopic stretch. Early in the deformation, the PNC exhibits an elastic response (g(l) < 0.15) followed yielding and strain softening. Finally, at larger strains (g(l) > 0.3), our polymer glasses enter the strain hardening regime, and the stress resumes an increasing trend as strain continues to grow. The stress during strain hardening is often described by Ref. [38]

s ¼ Gs gðlÞ þ sflow ;

System

(2)

where s is the normal stress in the test, Gs is hardening modulus, and sflow is the plastic flow stress. As discussed above, several recent studies have demonstrated that it is not appropriate to think of strain hardening in terms of an entropic network [25,26,39,40]; nevertheless, it is common to plot the mechanical response in this manner to extract the hardening modulus. The various mechanical properties extracted from the curves in Fig. 4 are summarized in Table 2. First, we note that adding any of our nanoparticles to the polymer matrix increases the Young’s modulus; however, the grafting density does not appear to significantly affect the reinforcement, and the size only plays a limited role in the increase of the elastic constants in the glassy state. Apparently simply adding the nanoparticles stiffens the matrix, regardless of the grafting conditions. Similarly, we find that all of the nanoparticle systems exhibit a higher yield stress relative to the

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Fig. 4. (a) g(l)s results from systems with same grafting density (G ¼ 0.2) but different particle radius, R ¼ 1.5s (red), 3.0s (green), 4.5s (blue), and the pure polymer (black). (b) systems with same particle radius (R ¼ 3.0s) but different grafting density from G ¼ 0.05 (red), 0.1 (green), 0.2 (blue), 0.4 (yellow), and the pure polymer (black). (c) Systems with R ¼ 4.5s and G ¼ 0.3 (green), R ¼ 4.5s with G ¼ 0 (bare particles) (red) particles and the pure polymer (black). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

H. Chao, R.A. Riggleman / Polymer 54 (2013) 5222e5229

Bond Autocorelation

between the nanoparticle and the grafted chains increases with the particle size, and this effect is only observed in the strain hardening region. This result is correlates with the depletion of matrix from particle surfaces with increasing particle size as shown in Fig. 3(b). The depletion of the matrix chains may lead to stronger interactions between the grafted chains and the nanoparticle surface. Similarly, the stress contribution from the non-bonded interactions between the grafted chains is shown in Fig. 5(b). For low grafting densities (G ¼ 0.05,0.1), we find that the non-bonded interaction between beads in the grafted chains does not contribute significantly to the stress increase during strain hardening. However, for higher grafting densities (G ¼ 0.2,0.4), the non-bonded interactions between the grafted chains contribute significantly to strain hardening. This is correlated with the crowding of grafted chain and depletion of matrix near particle surface as shown in Fig. 3(a) and (b), which indicates that crowding between the grafted chains can contribute to strain hardening. We have investigated the other components of the stress tensor, such as the contributions of the bond stress (not shown), which did not exhibit significant changes in behaviors across the various PNC systems investigated here.

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10000

(3)

where C0 is a pre-exponential factor, s and b are fitting parameters. From these parameters, we can calculate effective relaxation time, seff as seff ¼ sG(1/b)/b, where G(x) is the Gamma function. In this analysis, a fixed time window tw ¼ 500 is used to sample Cb(t) and measure seff during the deformation process; at the strain rates we employ, this corresponds to an applied strain of Dε ¼ 0.05 during the course of a single time window. It is noted here that previous studies [41] have shown that the effective relaxation time obtained from short time window during deformation process can be treated as a relative change in the segmental dynamics during the simulation. First, for the biggest nanoparticle size of R ¼ 4s, we compare the evolution of the dynamics in the PNC systems with bare particles to those with grafted nanoparticles; the mechanical response for these systems is shown above in Fig. 4(c). Fig. 7(a) shows that the relaxation of the grafted chains near the grafting points is slower than the relaxation of the matrix chains. Additionally, the relaxation of the segments of the grafted chains near the grafting points is

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We quantify the changes in the mobility by fitting Cb(t) in various time windows over the course of our simulation to the Kohlrausch-Williams-Watts (KWW) equation,

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Fig. 6. Bond autocorrelation function Cb(t) for systems with constant grafting density 0.2 for systems with particle radius R ¼ 1.5 (red), 3.0 (green), 4.5 (blue), and the pure polymer system (black). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Finally, we examine the dynamics of our polymer nanocomposites during compressive deformation. Several recent studies have shown that the inclusion of nanoparticles can impact the dynamics of polymer glasses during active deformation [15,41e 43]; however, for spherical particles, when a comparison was made at constant strain rate, the dynamics in a pure polymer glass and the nanocomposite were comparable [41]. Here, we test whether this picture continues to hold for the grafted nanoparticles. We measure the dynamics during deformation by examining the bond autocorrelation function, Cb(t), of first four bonds from the grafted end of chain. The bond autocorrelation function is defined as < P2 ½ b bðtÞ$ b bð0Þ >, where P2 is the second order Legendre function, b and bðtÞ is a unit vector aligned along the bonds of the grafted chains. Previous studies have shown that the bond autocorrelation could be used to measure the dynamics of glassy polymer chains during deformation, and the trends agree very well with experiments [44e46]. In Fig. 6, we show Cb(t) for systems with different particles sizes. The first time window, t < 1000, corresponds to the elastic deformation region; in this window Cb(t) decays more slowly for systems containing larger particles, indicating lower mobility (slower dynamics) in the systems with larger particles. Moreover, the dynamics in the pure polymer system has the fastest bond dynamics, as observed by a faster decay of Cb(t).

0

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3.3. Dynamics during deformation

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Fig. 5. Different contributions to the stress tensor plotted against g(l). (a) Stress contributions from the non-bonded interaction between particles and the grafted chains for the systems with G ¼ 0.2 and R ¼ 1.5 (black), 3.0 (red), and 4.5 (green). (b) Stress contributions from the non-bonded interactions between the monomers of the grafted chains for the systems with R ¼ 3 and grafting densities G ¼ 0.05 (black), 0.1 (red), 0.2 (green), and 0.4 (blue). All the stress values are normalized by the total number of grafted chains, M. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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0.8

(b)

Fig. 7. (a) seff as a function of strain obtained from the first four bonds of the grafted end (blue, left-facing triangles) in systems R ¼ 4.5, bonds on the matrix polymers that are within 4s of the nanoparticle surface where (G ¼ 2) (orange, up-facing triangles), all the bonds in both grafted and matrix chains in system with particle radius R ¼ 4.5s and grafting density G ¼ 0.2 (red squares), all the bonds in matrix chains in system with R ¼ 4.5s and G ¼ 0 (green diamonds), and all bonds from chains in pure polymer system (black circles). (b) Results obtained form bonds of the first four bonds from the grafted end of chain in systems with particle radius R ¼ 1.5 (red circles), 3.0 (green squares), 4.5 (blue triangles), and pure polymer (black squares). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

slower than that of the grafted chains that are farther from the nanoparticle surface. However in the system with bare nanoparticles, the dynamics near the nanoparticles’ surfaces are identical to the chains far from the particles’ surfaces. Therefore, we conclude that the grafting is the origin of the slower dynamics, not the proximity to the nanoparticle surface. Finally, we can examine how the mobility imparted by the deformation changes for the chains grafted to nanoparticles of different sizes, which is examined in Fig. 7(b). At a given deformation rate, we see that the first four bonds near the particle surfaces move more slowly when they are grafted to larger nanoparticles. Combined with our results above, we find that the larger nanoparticles have grafted chains that are increasingly crowded by their neighbors, exhibit a more pronounced tendency to strain harden, and have slower segmental dynamics in the grafted chains near their grafting points. Our finding of slower dynamics in the systems that exhibit more pronounced strain hardening is consistent with the activated barrier hopping theory of Chen and Schweizer, which ascribes strain hardening to reduced mobility in polymer glasses [30].

4. Conclusions In this work, we have developed a simple model system to study the mechanical properties of polymer nanocomposites containing grafted nanoparticles. At higher temperatures, our results are largely consistent with the previous work of Meng et al. [47]; our work is distinct in that we have many nanoparticles in our simulations, and any potential many-body effects could be taken into account. At fixed grafted density, smaller particles decrease the crowding between the grafted chains, similar to a reduction in the effective grafting density proposed by Trombly and Ganesan [7]. Our results in the glass state indicate that the addition of nanoparticles to the polymer matrix is sufficient to increase the Young’s modulus and yield stress, and the grafting density plays only a limited role on these material properties. However, grafting the chains does appreciably affect the ability of the polymer to strain harden, which could potentially lead to materials less prone to brittle failure. We have also shown that regions of slower mobility during deformation arise in our polymer nanocomposites with more pronounced strain hardening, which is consistent with recent theories associated with strain hardening. In summary, we find that increasing the crowding between the grafted chains leads to an increase in the effective height of the brush, more pronounced strain hardening, and slower dynamics in the grafted chains near

their grafting sites. Furthermore, grafting the nanoparticles does not remove the reinforcement of the Young’s modulus EY and the yield stress provided by bare nanoparticles. References [1] Caseri WR. Nanocomposites of polymers and inorganic particles: preparation, structure and properties. Materials Science and Technology 2006;22(7):807e17. [2] Krishnamoorti R, Vaia RA. Polymer nanocomposites. Journal of Polymer Science Part B e Polymer Physics 2007;45(24):3252e6. [3] Mackay ME, Tuteja A, Duxbury PM, Hawker CJ, Van Horn B, Guan ZB, et al. General strategies for nanoparticle dispersion. Science 2006;311(5768): 1740e3. [4] Kumar SK, Krishnamoorti R. Nanocomposites: structure, phase behavior, and properties. Annual Review of Chemical and Biomolecular Engineering 2010;1: 37e58. [5] Akcora P, Kumar SK, Moll J, Lewis S, Schadler LS, Li Y, et al. Gel-like mechanical reinforcement in polymer nanocomposite melts. Macromolecules 2009;43(2): 1003e10. [6] Moll JF, Akcora P, Rungta A, Gong S, Colby RH, Benicewicz BC, et al. Mechanical reinforcement in polymer melts filled with polymer grafted nanoparticles. Macromolecules 2011;44(18):7473e7. [7] Trombly DM, Ganesan V. Curvature effects upon interactions of polymergrafted nanoparticles in chemically identical polymer matrices. The Journal of Chemical Physics 2010;133:154904. [8] Srivastava S, Agarwal P, Archer LA. Tethered nanoparticleepolymer composites: phase stability and curvature. Langmuir 2012;28(15):6276e81. [9] Green PF. The structure of chain end-grafted nanoparticle/homopolymer nanocomposites. Soft Matter 2011;7(18):7914e26. [10] Klos J, Pakula T. Interaction of a spherical particle with linear chains. ii. chains end-grafted at the particle surface. Journal of Chemical Physics 2003;118(16): 7682e9. [11] Klos J, Pakula T. Computer simulations of chains end-grafted onto a spherical surface. Effect of matrix polymer. Macromolecules 2004;37(21):8145e51. [12] Kalb J, Dukes D, Kumar SK, Hoy RS, Grest GS. End grafted polymer nanoparticles in a polymeric matrix: effect of coverage and curvature. Soft Matter 2011;7(4):1418e25. [13] Ndoro TVM, Voyiatzis E, Ghanbari A, Theodorou DN, Böhm MC, MüllerPlathe F. Interface of grafted and ungrafted silica nanoparticles with a polystyrene matrix: atomistic molecular dynamics simulations. Macromolecules 2011;44:2316e27. [14] Smith GD, Bedrov D. Dispersing nanoparticles in a polymer matrix: are long, dense polymer tethers really necessary? Langmuir 2009;25(19):11239e43. [15] Riggleman RA, Toepperwein GN, Papakonstantopoulos GJ, Barrat JL, de Pablo JJ. Entanglement network in nanoparticle reinforced polymers. The Journal of Chemical Physics 2009;130:244903. [16] Papakonstantopoulos GJ, Doxastakis M, Nealey PF, Barrat JL, de Pablo JJ. Calculation of local mechanical properties of filled polymers. Physical Review E 2007;75(3):031803. [17] Akcora P, Liu H, Kumar SK, Moll J, Li Y, Benicewicz BC, et al. Anisotropic selfassembly of spherical polymer-grafted nanoparticles. Nature Materials 2009;8(4):354. U121. [18] Zhang Q, Archer LA. Langmuir 2002;18:10435. [19] Kopesky ET, Haddad TS, McKinley GH, Cohen RE. Miscibility and viscoelastic properties of acrylic polyhedral oligomeric silsesquioxane-poly(methyl methacrylate) blends. Polymer 2005;46:4743.

H. Chao, R.A. Riggleman / Polymer 54 (2013) 5222e5229 [20] Sternstein SS, Zhu A-J. Macromolecules 2002;35:7262. [21] Toepperwein GN, Karayiannis NC, Riggleman RA, Kröger M, de Pablo JJ. Influence of nanorod inclusions on structure and primitive path network of polymer nanocomposites at equilibrium and under deformation. Macromolecules 2011;44:1034e45. [22] Hergeth W-D, Steinau U-J, Bittrich H-J, Simon G, Schmutzler K. Polymerization in the presence of seeds. Part iv: emulsion polymers containing inorganic filler particles. Polymer 1989;30:254. [23] Papakonstantopoulos GJ, Yoshimoto K, Doxastakis M, Nealey PF, de Pablo JJ. Local mechanical properties of polymeric nanocomposites. Physical Review E 2005;72(3):031801. [24] Argon AS. Theory for low-temperature plastic-deformation of glassy polymers. Philosophical Magazine 1973;28(4):839e65. [25] Kramer EJ. Open questions in the physics of deformation of polymer glasses. Journal of Polymer Science B: Polymer Physics 2005;43:3369e71. [26] Hoy RS, Robbins MO. Strain hardening of polymer glasses: effect of entanglement density, temperature, and rate. Journal of Polymer Science Part B e Polymer Physics 2006;44(24):3487e500. [27] Hoy RS, Robbins MO. Strain hardening in polymer glasses: limitations of network models. Physical Review Letters 2007;99(11):117801e4. [28] Lyulin AV, Balabaev NK, Mazo MA, Michels MAJ. Molecular dynamics simulation of uniaxial deformation of glassy amorphous atactic polystyrene. Macromolecules 2004;37(23):8785e93. [29] Hoy RS, Robbins M. Strain hardening in bidisperse polymer glasses: separating the roles of chain orientation and interchain entanglement. The Journal of Chemical Physics 2009;131:244901. [30] Chen K, Schweizer KS. Suppressed segmental relaxation as the origin of strain hardening in polymer glasses. Physical Review Letters 2009;102(3):38301. [31] Bazhenov S, Li J, Hiltner A, Baer E. Ductility of filled polymers. Journal of Applied Polymer Science 2003;52(2):243e54. [32] Kremer K, Grest GS. Dynamics of entangled linear polymer melts e a molecular dynamics simulation. Journal of Chemical Physics 1990;92:5057e86. [33] Auhl R, Everaers R, Grest GS, Kremer K, Plimpton SJ. Equilibration of long chain polymer melts in computer simulations. Journal of Chemical Physics 2003;119(24):12718e28. [34] Sides SW, Grest GS, Stevens MJ, Plimpton SJ. Effect of end-tethered polymers on surface adhesion of glassy polymers. Journal of Polymer Science Part B e Polymer Physics 2004;42(2):199e208.

5229

[35] de Pablo JJ, Laso M, Suter UW. Simulation of polyethylene above and below the melting point. The Journal of Chemical Physics 1992;96:2395. [36] Karayiannis NC, Mavrantzas VG, Theodorou DN. A novel Monte Carlo scheme for the rapid equilibration of atomistic model polymer systems of precisely defined molecular architecture. Physical Review Letters 2002;88: 105503. [37] Plimpton SJ. Fast parallel algorithms for short-range molecular-dynamics. Journal of Computational Physics 1995;117(1):1e19. [38] Haward RN, Thackray G. Use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics. Proceedings of the Royal Society of London Series A e Mathematical and Physical Sciences 1968;302(1471): 453e72. [39] Govaert LE, Engels TAP, Wendlandt M, Tervoort TA, Suter UW. Does the strain hardening modulus of glassy polymers scale with the flow stress? Journal of Polymer Science Part B: Polymer Physics 2008;46(22): 2475e81. [40] Hoy RS, Robbins MO. Strain hardening of polymer glasses: entanglements, energetics, and plasticity. Physical Review E 2008;77(3):031801. [41] Riggleman R, Toepperwein G, Papakonstantopoulos G, de Pablo J. Dynamics of a glassy polymer nanocomposite during active deformation. Macromolecules 2009;42(10):3632e40. [42] Toepperwein GN, Schweizer KS, Riggleman RA, de Pablo JJ. Heterogeneous segmental dynamics during creep and constant strain rate deformations of rod-containing polymer nanocomposites. Macromolecules 2012;45(20): 8467e81. [43] Toepperwein GN, Riggleman RA, de Pablo JJ. Dynamics and deformation response of rod-containing nanocomposites. Macromolecules 2011;45(1): 543e54. [44] Riggleman RA, Lee HN, Ediger MD, de Pablo JJ. Free volume and finite-size effects in a polymer glass under stress. Physical Review Letters 2007;99(21):215501. [45] Riggleman RA, Lee HN, Ediger MD, de Pablo JJ. Heterogeneous dynamics during deformation of a polymer glass. Soft Matter 2009;6(2):287e91. [46] Lee HN, Riggleman RA, de Pablo JJ, Ediger MD. Deformation-induced mobility in polymer glasses during multistep creep experiments and simulations. Macromolecules 2009;42(12):4328e36. [47] Meng D, Kumar SK, Lane JMD, Grest GS. Effective interactions between grafted nanoparticles in a polymer matrix. Soft Matter 2012;8(18):5002e10.