Molecular Dynamics Simulation Study of Polymer Nanocomposites

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Molecular Dynamics Simulation Study of Polymer Nanocomposites with Controllable Dispersion of Spherical Nanoparticles Zijian Zheng, Guanyi Hou, Xiuyang Xia, Jun Liu, Mesfin Tsige, Youping Wu, and Liqun Zhang J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b06482 • Publication Date (Web): 09 Oct 2017 Downloaded from http://pubs.acs.org on October 12, 2017

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The Journal of Physical Chemistry

Molecular Dynamics Simulation Study of Polymer Nanocomposites with Controllable Dispersion of Spherical Nanoparticles Zijian Zheng1, 2, Guanyi Hou1, Xiuyang Xia1, Jun Liu 1*, Mesfin Tsige3*, Youping Wu1, Liqun Zhang1 1

Key Laboratory of Beijing City on Preparation and Processing of Novel Polymer

Materials, State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, Beijing, 100029, China 2

Hubei Collaborative Innovation Center for Advanced Organic Chemical Materials, Key

Laboratory for the Green Preparation and Application of Functional Materials, Ministry of Education, Hubei Key Laboratory of Polymer Materials, School of Materials Science and Engineering, Hubei University, Wuhan, 430062, China 3

Department of Polymer Science, The University of Akron, Akron, Ohio 44325, United States

ABSTRACT Through coarse-grained molecular dynamics simulation, we construct a novel kind of end-linked polymer network by employing dual end-functionalized polymer chains that chemically attach to the surface of nanoparticles (NPs), so that the NPs act as large cross-linkers. We examine the effects of the length and flexibility of polymer chains on the dispersion of NPs, and the effect of the chain length on the stress-strain behavior and the segment orientation during the deformation process. We find that the stress

*Authors to whom correspondence should [email protected] or [email protected]

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addressed.

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addresses: 1

ACS Paragon Plus Environment


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upturn becomes more prominent with the decrease of the chain length, attributed to the limited extensibility of the chain strand connecting two neighboring NPs. In addition, this end-linked polymer nanocomposite (PNC) is shown to have a temperaturedependent stress-strain behavior that is contrary to traditional physically mixed PNCs, whose mechanical properties deteriorate with increasing temperature. This is due to the stability of the dispersion of NPs and higher entropic elasticity at higher temperature for the former, while the latter has poorer interfacial interaction at higher temperature, leading to less reinforcing efficiency. By imposing a dynamic oscillatory shear deformation, we obtain a dynamic hysteresis loop for end-linked and physically mixed dispersions. Interestingly, the end-linked system possesses a much smaller hysteresis loss than does the physically mixed system, with the latter exhibiting a more prominent decrease with increasing temperature, due to less interfacial contact. Our results demonstrate that end-linked PNCs combine attractive static and dynamic mechanical properties, and exhibit an unusual response to temperature, which could find potential applications. 1. Introduction In contrast to traditional randomly cross-linked polymer networks1-4, end-linked polymer networks have attracted much recent interest, because they permit improved structural design and tuning of properties.5-9 For instance, Urayama et al.10 experimentally studied the mechanical damping characteristics of end-linked poly(dimethylsiloxane) (PDMS) in irregular networks, formed using bi- and monofunctional end-reactive precursor linear PDMS chains linked with a tri-functional cross2


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linker. It was found that this irregular network structure possesses good temperatureand frequency-insensitive damping and elasticity properties, contrary to the conventional damping elastomers. Kenkare et al.5 adopted discontinuous molecular dynamics simulation to study the structure and relaxation of tri- and tetra-functional polymer networks, finding that the structure and properties of the polymer networks are shown to depend heavily on the manner in which the network is initially constructed. Genesco et al.11 combined simulation and experiment to study the mechanical property of end-linked elastomers with a bimodal distribution of molecular weight, and they found that enhanced mechanical properties such as a stress upturn with increased strain and increased toughness result from the limited extensibility of the short chains. Bhawe et al.12 used Monte Carlo simulation to study the effects of chain stiffness and entanglements on the mechanical properties of end-linked networks, finding that the stiffer chain networks are more entangled and have a higher elastic modulus. Meanwhile, Escobedo et al.13 used Monte Carlo simulations to probe the effect of the entanglements on the deformation behavior of end-linked polymeric networks, revealing that the effect of entanglements on the network elastic response decreases with the strain, supporting the entanglement-slippage concept.14 Gilra et al.15 observed through Monte Carlo simulation that the segments of polymer chains near the ends of elastic chains in the network deform much more than that of the segments in the center of chains. It is noted that all the above studies focused on constructing an end-linked polymer network by utilizing small molecule cross-linkers. Recently, however some research 3



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has focused on polymer networks end-linked through added nanoparticles. For instance, experimentally, Bokern et al.16 used silver NPs as cross-linkers to prepare thermoplastic elastomers. Through a computational and multi-scale model, Balazs et al.17 used rigid NPs grafted with polymer chains, which contain reactive functional end groups to form either labile or permanent chemical bonds with each other, leading to NPs interconnected through dual cross-links with these two labile and permanent bonds. The simulation results indicate that NP network with a few unbreakable or permanent bonds can survive strains far greater than can networks without these connections. This suggests a strategy for preparing polymer grafted NPs networks with remarkable strength and ductility. In this work, we aim to construct computationally an end-linked polymer network using dual end-functionalized (each chain is functionalized on both ends) polymer chains to be chemically attached to the surface of NPs, which also results in a NP chemical network. Moreover, we change the length and flexibility of the polymer chains to regulate the inter-particle distance between NPs. Actually, some experimental work has been carried out on such systems. For instance, Fischer et al.18

found using

this approach (nanoparticles acting as netpoints) that they can achieve a stable and uniform distribution of NPs at high volume fraction, and furthermore, the separation distance between distributed NPs can be tuned within the range of 5 to 20 nm, by changing the molecular weight of the attached polymer chains. This could be a key step for the preparation of PNCs for magnetic storage and photovoltaic devices. Besides tuning the distribution of NPs, here we also investigate the static and dynamic 4


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mechanical properties of this end-linked polymer network, compared to those obtained by the traditional physically mixed PNCs. Results from our simulations show that this end-linked polymer network has excellent static and dynamic mechanical performance, which could provide some guidelines for the fabrication of “green” tires, i.e., tires that produce low energy dissipation during rolling. 2. Simulation model and technique To simulate spherical NPs, we use a coarse-grained model similar to fullerene or “bucky ball”, which is composed of sixty beads. The bead size used for the nanoparticles is the same as that used for the polymers. Note that the diameter of the NP is 7.2σ (where σ defines the length scale). In our simulation, we treat each NP as a rigid body by using the fix rigid command of LAMMPS, which makes each NP translate and rotate as one entity. Namely, we do not consider the intra-NP interactions between the beads of each nanoparticle. Note that there exist 20 NPs in our simulation system. Each polymer chain is modeled by the standard bead-spring model developed by Kremer and Grest.19 In most cases, there exist 300 polymer chains and each polymer chain contains 30 beads. To examine the effect of chain length, polymer chains with 60, 90 and 120 beads are also studied. We note that although most of these polymeric chains are not that long compared to the entanglement length of Ne=85,20 they show the qualitative static and dynamic behavior of long chains. By mapping the coarse-grained model to a real one, each bond in the model Kremer-Grest chains corresponds to 3-6 covalent bonds along the backbone of a realistic chemical chain. Since it is not our aim 5



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to study any specific polymer, the mass m and diameter σ of each bead is set to be unity, which implicitly defines the units of mass and length in the simulations. The nonbonded interactions between the different kinds of beads are modeled through a truncated and shifted Lennard-Jones (LJ) potential as follow

 12  6   4 [( )  ( ) ]  C U (r )   r r  0 

r  rcutoff r  rcutoff

(1)

where the LJ interaction is cut off at the distance r = rc and C is a constant, which guarantees that the potential energy is continuous at the cut-off distance. r is the separation distance between two polymer beads and ε is the energy scale of the model. The interaction between adjacent bonded polymer beads is modelled by a stiff finite extensible nonlinear elastic (FENE) potential:

U FENE  0.5kR02ln[1  (

r 2 ) ] (2) R0

where k=30ε/σ2 and R0=1.5σ, guaranteeing a certain stiffness of the bonds while avoiding high-frequency modes and chain crossing. Here r is the separation distance between two connected polymer beads. The angle between three consecutive polymer beads is modeled by a harmonic potential:

U angle    k   0 

2

(3)

where kθ is the bending energy, θ is the instantaneous angle between three consecutive beads and θ0 is the equilibrium angle. In this work, the kθ is set to 0 for most situations, except when considering the effect of the chain flexibility, in which case kθ is set to 500. To model the flexibility of polymer chains, we varied the equilibrum angle θ0. In our 6


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work, we construct and compare two typical systems, namely I (an end-linked system) and II (a physically mixed system). For the end-linked system, the spherical NPs act as the cross-linking sites by becoming bonded to the dual end-groups of each polymer chain, while for the physically mixed system they only act as nano-reinforcing units. The parameters of the interaction potential energy for polymer-polymer, polymer-NP and NP-NP beads are listed in Table 1. Table 1. Interaction potential parameters for polymer-polymer, polymer-nanoparticle (NP) and nanoparticle (NP) –nanoparticle (NP) beads. Polymer-polymer Nonbonded LJ potential ε=1.0 σ=1.0 rcutoff=1.12σ

Boned FENE potential k=30ε/σ2 R0=1.5σ

Polymer-NP

NP-NP

LJ potential ε=1.0 σ=1.0 rcutoff=1.12σ

System I Non-bonded LJ potential ε=1.0 σ=1.0 rcutoff=2.5σ (end polymer beads)

ε=1.0 σ=1.0 rcutoff=1.12σ (other polymer beads)

System II Non-bonded LJ potential ε=5.0 σ=1.0 rcutoff=2.5σ

The NPT ensemble is adopted in our MD simulations, where the temperature is fixed at T*=1.0, and the pressure is set to be P*=0.01 for dilute state (the density of the system is equal to 0.2) and P*=2.0 for melt state (the density of the system is equal to 0.8) by using the Nose-Hoover thermostat and barostat.21 The velocity-Verlet algorithm was used to integrate the equations of motion with a time step δt=0.001τ, where τ denotes the Lennard-Jones time unit τ = (mσ2/ε)1/2. Periodic boundary conditions are used in all three directions. Before cross-linking, the structures are equilibrated under an NPT ensemble over a time long enough that each chain has moved at least 2Rg. After this,

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for system I, the crosslinking was done dynamically where at each given time during the simulation when the end bead of a polymer and NPs are within 1.25σ a chemical bond is generated and the run is continued until all the crosslinking between polymers and NPs are completed. This cross-linking process is shown in Figure 1. For system II, the cross-links are added between polymer beads from different polymer chains all at once, thus embedding the NPs in a cross-linked polymer matrix. It is noted that both Systems I and II have the same cross-linking density with the total number of the crosslinked bonds in each system equal to 600. After the cross-linking process, the structures obtained are further equilibrated with the NPT ensemble, guaranteeing that our simulated systems have been fully and properly equilibrated.

Figure 1. Snapshots show the formation of system I (the end-linked system) as a function of the simulation time, showing how polymer end groups (shown in red) are attached to NPs (the large blue spheres) during the course of a simulation. 0 τ denotes the initial state to impose the end-linked process. We display the polymer chains in orange dot.

We use the following approach to perform the tensile deformation. The pure (particle-free) and filled systems are deformed by changing the box length to L0a in the z direction and to L0a-1/2 in the x and y directions, with the volume of the box held constant. The interactions between atoms in the basic cell and the image atoms across the cell wall transmit the deformation to the atoms in the basic cell. The strain rate is 8


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specified as  =0.001/τ. The average stress σ in the z direction is obtained from the deviatoric part of the stress tensor σ=(1+μ)(-PZZ+P)≈3(-PZZ+P)/2, where P=ΣiPii/3 is the hydrostatic pressure22 and μ is Poisson's ratio, taken to be 0.5. For the oscillatory shear deformation, the SLLOD equations of motion23 with Lees-Edwards “sliding brick” boundary conditions24 are typically used. Alternatively, a continuously deforming, nonorthogonal simulation box can be used25, which is implemented in the LAMMPS package. The upper xy plane parallel to x and y axis of the simulation box is shifted along the x direction so that each point in the simulation box can be considered as having a “streaming” velocity. This position-dependent streaming velocity is subtracted from each atom's actual velocity to yield a thermal velocity, which is used for the temperature computation and thermostatting. The shear strain is defined as  =δx/LZ(0), where the offset δx is the transverse displacement distance in the shear direction (x direction for xy deformation) from the unstrained position, and Lz(0) is the box length perpendicular to the shear or x direction. In our simulations, the value of the shear strain amplitude was set to  0 =2.0. The period of the oscillatory shear is 100τ, and thus, the corresponding frequency  equals to 0.01 in units of τ-1. The average shear stress is obtained from the deviatoric part of the stress tensor σs=Pxy=Pyx.26 All MD runs are carried out through the large scale atomic/molecular massively parallel simulator (LAMMPS), which is developed by Sandia National Laboratories.27 More simulation details can be found in our previous work.28-30 3. Results and discussion 9



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3.1 Structural properties The persistent challenge in the study of polymer nanocomposites is in tuning the interaction between polymer chains and NPs31-34 in order to control the distribution of NPs in the polymer matrix. It is well known that achieving a good dispersion of NPs in polymer matrices is a prerequisite to obtaining good mechanical properties of polymer nanocomposites.35-37 We therefore firstly study the effect of the chain length on the dispersion of NPs for system I, namely the end-linked system. We consider chain lengths of 30, 60, 90 and 120 beads. The snapshots for a single chain with its two end-groups chemically attached to NPs for various chain lengths are shown in Figure 2. Evidently, the inter-particle distance between NPs should increase with increasing chain length. To further quantify the dispersion of NPs, we compare the radial distribution functions (RDF) between the centers of mass of the NPs for different chain lengths, as shown in Figure 3a. From Figure 3a, we observe for each chain length a single peak, at a distance much larger than that for direct contact r=7.2σ, indicating no direct contacts between NPs occur. Since the RDF is derived from the statistical average of the distance between different NPs, the location of the peak of the RDF demonstrates the most probable distance between NPs in the system. Therefore, the peak of the RDF can approximately represent the average distance between NPs. The average distance between NPs is 15.76σ, 20.04σ, 23.69σ and 25.58σ, for chain lengths of 30, 60, 90 and 120, respectively. Figure 3b shows that the average distance between NPs depends linearly on the root-mean10


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squared end-to-end distance (Rend) of the polymer, which is statistically averaged in the NPT ensemble simulation. Therefore, the chain length controls the distance between NPs, which is consistent with the experimental results from Fischer et al.18, who used polystyrene of different molecular weights to control the distance between CdSe nanoparticles.

Figure 2. Snapshots display a single polymer chain with its two functionalized terminal groups attached to the two neighboring NPs.

Figure 3. (a) Radial distribution function for the centers of the mass of NPs for different polymer chain lengths. (b) The average distance between NPs (D) as a function of the root-mean squared end-to-end distance of the polymer (Rend).

Besides the effect of the chain length, we also study how the chain flexibility could influence the dispersion state of NPs. In fact, some experimental work has been performed to study the effect of the flexibility of the grafted polymer chains on the dispersion of NPs. For example, Zhu at al.38 prepared poly(methyl methacrylate)-grafted-nano-silica (PMMA-g-silica) and a copolymer of styrene

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(St), n-butyl acrylate (BA) and acrylic acid (AA)-grafted-nano-silica (PSBA-gsilica) hybrid nanoparticles by using a hetero-phase polymerization technique in an aqueous system. Their work demonstrated that the chain flexibility of the grafted polymer in the hybrid nanoparticles could significantly influence the dispersion behavior of hybrid nanoparticles in the PVC matrix, as well as its thermal and dynamic mechanical properties. Here we change the chain flexibility by varying the equilibrium bending angle θ0 from 0o to 140o, while fixing the chain length at 30 beads. With increasing equilibrium angle, the polymer chains become more extended, which can be verified by the root-mean squared end-toend distance (Rend) at different equilibrium angles. For instance, the Rend is equal to 9.38σ, 10.4σ, 11.35σ and 12.30σ at the equilibrium angles of 0o, 120o, 130o and 140o, respectively. Figure 4a shows the effect of the chain flexibility on the dispersion of NPs. As we expect, with the increase of the equilibrium angle, the average distance between NPs also increases, which approximately scales linearly with Rend, as illustrated in Figure 4b.

12


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Figure 4. (a) Radial distribution function of the center of the mass of NPs for various equilibrium bending angles for chains of 30 beads. (b) The average distance between NPs (D) as a function of the root-mean squared end-to-end distance (Rend).

3.2 Static mechanical properties After examining the effects of the chain length and chain flexibility on the dispersion of NPs, we next study the mechanical properties of both system I (endlinked) and system II (physically mixed). Figure 5a shows that the stress-strain curves in a uniaxial extension in system I have decreasing stress growth, or strain hardening, as chain length increases. For instance, the stress at the strain ε=300% for chain length 30 is almost seventy times higher than that for chain length 120. The up-turn of the stress-strain curve results from the limited chain extensibility, which sets in at smaller strain, for shorter chains. To further analyze the limited chain extensibility, we characterize in Figure 5b the bond orientation along the deformation direction as a function of strain. Here we use the second-order Legendre polynomials P2 to characterize the bond orientation as follows:





p2 (cos )  3 cos2   1 / 2

(4)

where θ denotes the angle between a given element (two connected monomers in the chain) and the reference direction, which is the stretching direction. P2 can range from -0.5 to 1, with P2=-0.5 indicating a perfect orientation perpendicular to the reference direction, whereas P2=1.0 means a perfect alignment parallel to the reference direction. When the segments are randomly oriented, we get P2= 0. From Figure 5b, a much higher bond orientation is observed for 30-beads chains

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at any strain. With increasing chain length, the bond orientation decreases, supporting the result shown in Figure 5a. Snapshots of the deformation process of a single polymer chain attached to the neighboring NPs are shown in Figure 5d. Clearly, with the increase of the strain, the polymer chain is gradually straightened. This tendency can be further verified by the change of Rend during uniaxial tension as shown in Figure 5c. When the strain reaches 300%, the 30beads chains become fully straightened, while the chain with 120 beads is only slightly straightened. These snapshots support the stress-strain behavior, displayed in Figure 5a, and show that shorter elastic strands lead to increases in strain hardening.

14


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Figure 5. (a) The stress-strain curves of the end-linked system (system I) with different chain lengths. (b) The bond orientation of polymer chains in uniaxial tension for various polymer chain lengths. (c) The root-mean squared end-to-end distance, all as a function of tensile strain in uniaxial tension for various polymer chain lengths. (d) Snapshots of the deformation state of single polymer chain connected between two NPs. The big spheres composed of small blue spheres denote the NPs, and the red beads represent the polymer chain.

It is known that temperature has a strong effect on the mechanical properties of polymer nanocomposites, attributed to structural evolution in response to the temperature change.39,40 Here we change the simulated temperature from T*=1.0 to T*=5.0 for both systems I and II. Interestingly, these two systems exhibit opposite temperature-dependent behavior, as displayed in Figure 6a and Figure 6b. For system I, with increasing temperature, the strain hardening is enhanced, which is counter-intuitive because traditional PNCs have much reduced mechanical property at increased temperature. The temperature-dependent behavior of System II, however, is that of a traditional PNC.41 We explain this observation as follows: for system I, the NPs act as the large chemical crosslinkers, and the elasticity results from the entropy of polymer chains. At a given strain, the polymer chain entropy increases with temperature, as illustrated by the well-known formula42:

  n1kT ( 

1



2

)

 RT Mc

( 

1

2

)

(5)

where n1 is the number of network chains per unit volume, k is the Boltzmann constant, λ is the draw ratio, ρ is the density, and MC is the average molecular weight between cross-links. This equation implies that the stress increases 15



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linearly with temperature. In all, the discovery of the end-linked system is consistent with the above equation.

Figure 6. Stress-strain curves at different temperatures for (a) end-linked system (I) and (b) physically mixed system (II). Note that the polymer chain contains 30 beads.

Furthermore, the NPs are still uniformly dispersed in the polymer matrix, as displayed in Figure 7a. Obviously, the first peak of the RDF between the mass center of the NPs is shifted to a bit larger distance at higher temperature, validating better dispersion for system I. For system II, the NPs act as reinforcing units, and the interaction between NPs and polymer chains is entirely physical. With increasing temperature, poorer interfacial contact exists due to the increasing distance between NPs and polymer, leading to less reinforcing efficiency. To characterize the change of the interfacial interaction between NPs and polymer chains as the temperature increases, we calculate the total NPpolymer interaction energy for system II, as shown in Figure 7b. The results indicate that the total interaction energy between NPs and polymer decreases gradually for system II, indicating less interfacial contact at higher temperature. 16


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Figure 7. (a) Radial distribution function of NP cores at different temperatures for system I. (b) The total Polymer-NPs interaction energy (E) as a function of temperature for physically mixed system (II).

3.3 Dynamic mechanical properties Besides the static mechanical properties, dynamic mechanical properties are also very important for PNCs. Hysteresis loss (HL), which is the loss of mechanical energy during one cycle of strain, is an important parameter that characterizes the visco-elasticity of elastomeric polymer materials. Moreover, HL is the main origin of rolling resistance of rubber tires, which undoubtedly is disadvantageous for reducing fuel consumption and improving fuel efficiency in the tire industry. In general, the HL of PNCs comes from the internal filler-filler, filler-polymer and polymer-polymer friction. Here, we compare the HL in different cases.

3.3.1 End-linked system (system I) with different chain lengths For the end-linked system, also denoted by system I, we study the effect of the chain length on the HL, which is shown in Figure 8. Here we only show the stress-strain hysteresis loop for the chain lengths of 30 and 120, in Figure 8a and Figure 8b, respectively. To further quantitatively compare the effect of the chain 17



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length on the visco-elasticity, we can directly calculate the HL, namely the area inside the hysteresis loop in one tension-recovery cycle. Figure 8c shows that as the length of polymer chain increases, the HL decreases. The reason is attributed to the fact that polymer chains undergo a relative large deformation for short chain lengths, and therefore a greater friction between polymer chains occurs. Moreover ， all the four simulation systems contain 300 polymer chains with various chain length and 20 NPs. The shorter polymer chain corresponding to the higher filler loading. These two reasons lead to larger HL for short chain length of end-linked system.43

Figure 8. Hysteresis loop of end-linked system (I) with chain lengths (a) 30 beads and (b) 120 beads under shear at T*=1.0. (c) Comparison of the hysteresis loss (HL) for different chain lengths.

3.3.2 End-linked system (system I) vs physically mixed system (system II) Next we compare the HL of systems I and II under dynamic oscillatory shear at the temperature T*=1.0, as shown in Figure 9a and Figure 9b. We observe that compared to the end-linked system (system I), the physically mixed system (system II) exhibits a larger HL, attributed to the physical polymer-filler network. For system I the low HL results from good dispersion of NPs and chemical interfacial interaction between NPs and polymer chains. In addition, the 18


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simulation data reported by Eslami44 verify that the end-phenylene groups have much higher mobility than the phenylene groups in the backbone, which can result in a higher HL if the end groups of the polymer chain are not immobilized. In our simulation work, the end-groups are chemically tethered to the surface of NPs, significantly decreasing its mobility, which also further contributes to the decrease of the HL. This is also verified by the experimental work in preparing environmentally “green” tires. Through anionic polymerization, Zhao et al.45,46 successfully synthesized end modified styrene-butadiene rubber (SSBR), and their results show that this SSBR nanocomposite displays much lower heat builtup, which is a crucial requirement in making green tires.

Figure 9. Hysteresis loop of these two systems at different temperatures (T*=1.0～5.0) under shear. (a) the end-linked system, (b) the physically mixed system. Note that the polymer chain contains 30 beads.

3.3.3 The effect of the temperature on the HL of systems I and II In addition, the effect of the temperature on the hysteresis loop of these two systems is also considered, and the results are shown in Figure 9a and Figure 9b. Note that the chain length contains 30 beads, and the temperature is changed from 19



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T*=1.0 to 5.0. We calculated the HL from the hysteresis loop and is shown in Figure 10. The result indicates that the HL of both systems I and II decreases with increasing temperature, with the change being more significant for system II. The lack of temperature sensitivity of the HL for the end-linked system (system I), can be explained by the uniformity and stability of the NP dispersion. In the physically mixed system (system II), however, with increasing temperature, the system exhibits poorer interfacial contact between the polymer chains and the NPs, as indicated by Figure 7b, leading to less filler-polymer friction, corresponding to less HL.

Figure 10. Comparison of the hysteresis loss (HL) for (a) end-linked system (I), (b) physically mixed system (II) at temperatures ranging from T*=1.0 to T*=5.0.

3.3.4 The effect of the interfacial interaction on HL Lastly, we study the effect of the interfacial interaction between NPs and polymer on the HL at the same dispersion state of NPs, as shown in Figure 11a. Evidently, the stress-strain curve is enhanced with increasing strength of the physical interaction between the terminal groups and the NPs. We observe that when the interaction strength reaches ε=100.0, the stress-strain behavior becomes 20


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equivalent to that of the end-linked system. This is attributed to the fact that stronger physical interaction between polymer chains and NPs promotes the orientation of polymer chains, as shown in Figure 11b. Considering the approximately equivalent stress-strain behavior and bond orientation vs strain for the end-linked system and the system with the interaction strength ε=100, we compare the HL. From Figure 11c and Figure 11d, we find that the end-linked system has much smaller HL than does the system with ε=100, despite having nearly the same static stress-strain behavior in Figure 11a. The reason is attributed to the fact that although there exist strong terminal adsorptions between the terminal groups and the NPs, there still exist adsorption and desorption processes between terminal groups and NPs, resulting in the high HL. However, for the end-linked system, the terminal groups are chemically linked on the surface of NPs, there exist only polymer chain friction, leading to the smaller HL. We should point out that although the system with ε=10 has a much smaller HL than that with ε=100, the static stress-strain behaviour of the former is weak, see Figure 11a. From this analysis, we can infer that the end-linked system has the potential to possess both good static and dynamic properties.

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Figure 11. (a) Stress-strain curves, (b) bond orientation of polymer chains, (c) hysteresis loop and (d) comparison of hysteresis loss (HL) for: ε=10.0, indicating weak terminal adsorption, ε=100 indicating strong terminal adsorption, and “end-linked,” meaning that the end groups of each polymer chain are chemically bonded (i.e., permanently) to the NPs. Note that T*=1 and the polymer chain contains 30 beads. The black line is for System I and the blue and red lines are for System II.

3.4 Extending to the melt state 3.4.1 Effect of the chain length The system considered so far was equilibrated at P*=0.01, corresponding to a

dilute system (the density of the system is equal to 0.2). Here we turn our attention to study the melt state. On the basis of the above equilibrated state, we further equilibrate the system at a higher pressure of P*=2.0, making the density equal to 0.8. Firstly, we study the effect of the chain length on the RDF of NPs, as shown in Figure 12. Similar to the dilute system, the nearest neighbor distance between the NPs increases with the 22


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increase of the polymer molecular weight. It is worth to mention that when the polymer chain length between the NPs is equal to 30, the NPs exhibit a strong aggregation state, indicating that we should employ much longer chain length to make the NPs disperse well in this melt state for this end-linked method.

Figure 12. Radial distribution function g(r) between NPs for different polymer chain length.

Following this we study the stress-strain behavior at different chain lengths, as shown in Figure 13a. Clearly, the stress-strain curve decreases in the melt state compared to that of the dilute state at the same chain length, attributed to the reduced bond orientation in the melt state, as displayed in Figure 13b. Moreover, the stressstrain curve decreases gradually with the increase of the polymer molecular weight, which is consistent to the dilute system. We also examine the hysteresis loop and the hysteresis loss (HL) under the dynamic periodic shear deformation for different chain length, as displayed in Figure 13c and Figure 13d. In line with the dilute system, the HL decreases gradually with increasing the polymer chain length. Furthermore, the HL is several times higher than that of the dilute system for the same chain length, which

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is attributed to much greater internal molecular friction between polymer chains and polymer chains, polymer and NPs in the melt state.

Figure 13. (a) The stress-strain curves of the end-linked system (system I) with different chain lengths. (b) The bond orientation of polymer chains in the uniaxial tension for various polymer chain lengths. (c) Hysteresis loop of end-linked system (I) with different chain lengths under the dynamic periodic shear deformation at T*=1.0. (d) Comparison of the hysteresis loss (HL) for different chain lengths.

We further study the effect of the temperature on the stress-strain behavior for these two systems such as end-linked and physically mixed systems. Similar to the dilute case, the stress-strain behavior increases gradually with increasing the temperature for system I, as displayed in Figure 14a, attributed to the higher entropy at much higher temperature. It is worth to mention that this phenomenon is valid for both the dilute and melt state. We further study the stress-strain curves at different temperatures for system II, as illustrated in Figure 14b. Consistent with 24


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the dilute state, the stress-strain curves decrease gradually with increasing the temperature.

Figure 14. Stress-strain curves at different temperatures for (a) end-linked system (I) and (b) physically mixed system (II) under melt state.

Lastly, we study the hysteresis loop (Figure 15a and Figure 15c) and HL (Figure 15b and Figure 15d) in the melt state for these two systems at different temperatures. For both of the two systems, the HL decreases gradually with increasing temperature, attributed to the lower polymer-polymer and polymer-filler molecular friction at higher temperature. In general, we conclude that the static and dynamic mechanical properties in the melt state exhibit the same trend with those of the dilute case.

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Figure 15. Hysteresis loop of these two systems at different temperatures (T*=1.0～5.0) under the dynamic periodic shear deformation . (a) the end-linked system, (c) the physically mixed system. Comparison of the hysteresis loss (HL) for (b) end-linked system (I), (d) physically mixed system (II) at temperatures ranging from T*=1.0 to T*=5.0.

4. Conclusions We adopted coarse-grained molecular dynamics simulations to investigate a novel kind of end-linked polymer nanocomposite obtained by employing dual end-functionalized polymer chains that are cross-linked to the surfaces of NPs. Each NPs are chemically attached to multiple end-groups of polymer chains. Firstly, we verified that changing the length and flexibility of polymer chains is an effective way to regulate the average distance between NPs, and we found a linear relationship between this distance and the root mean squared end-to-end 26


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distance Rend of the polymer. In addition, we found that the Hysteresis loss (HL) of this end-linked system decreases with increasing chain length. This is attributed to the fact that shorter polymer chain undergoes a relatively large deformation during the deformation process, and this tendency is further verified by the bond orientation. Secondly, we compared the temperature dependent stress-strain behavior and HL of this end-linked system (system I) with the traditional PNCs, such as a physically mixed system that is then cross-linked (system II). For the end-linked system, the stress as a function of strain is enhanced with increasing temperature, which is counter-intuitive because traditional PNCs have worse mechanical properties (lower stresses) as the temperature increases. The HL decreases as the temperature increases for both systems I and II. We designed two physically mixed systems, both of which have the same dispersion state of NPs with that of the end-linked system. These two physically mixed systems have different interfacial interactions. We found that even for the same static stress-strain behavior, the end-linked system exhibits much lower hysteresis loss. We also considered a melt state that has a much higher density which showed a similar static and dynamic mechanical property as a function of the polymer chain length and temperature are obtained as the dilute system. In general, the end-linked system could combine good performance of both static and dynamic mechanical properties, and exhibit a counter-intuitive increase in modulus with temperature.

Acknowledgment 27



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The authors acknowledge financial supports from the operation expenses for universities' basic scientific research of central authorities(buctrc201715 and JD1611), the National 973 Basic Research Program of China 2015CB654700(2015CB654704), The Foundation for Innovative Research Groups of the NSF of China (51221002 and 51521062), the National Natural Science Foundation of China (51333004, 51403015 and 21674010), the Major International Cooperation(51320105012) of the National Nature Science Foundation of China. M.T. acknowledges financial support from the National Science Foundation under Grant DMR-1410290. The super-calculation center of “Tianhe number two” and the cloud calculation platform of Beijing University of Chemical Technology (BUCT) are both greatly appreciated. References (1) Sommer, J. U.; Schulz, M.; Trautenberg, H. L. Dynamical Properties of Randomly Cross‐ Linked Polymer Melts: A Monte Carlo Study. I. Diffusion Dynamics. The Journal of chemical physics 1993, 98, 7515-7520. (2) Barsky, S. J.; Plischke, M. Order and Localization in Randomly Cross-Linked Polymer Networks. Physical Review E 1996, 53, 871. (3) Horkay, F.; McKenna, G. B.; Deschamps, P.; Geissler, E. Neutron Scattering Properties of Randomly Cross-Linked Polyisoprene Gels. Macromolecules 2000, 33, 5215-5220. (4) Eikerling, M.; Kornyshev, A. A.; Stimming, U. Electrophysical Properties of Polymer Electrolyte Membranes: A Random Network Model. The Journal of Physical Chemistry B 1997, 101, 10807-10820. (5) Kenkare, N.; Smith, S.; Hall, C.; Khan, S. Discontinuous Molecular Dynamics Studies of End-Linked Polymer Networks. Macromolecules 1998, 31, 5861-5879. (6) Meissner, B.; Matějka, L. Kinetic Limitation in the Formation of End-Linked Elastomer Networks. Polymer 2005, 46, 10618-10625. (7) Lang, M.; Göritz, D.; Kreitmeier, S. Length of Subchains and Chain Ends in Cross-Linked Polymer Networks. Macromolecules 2003, 36, 4646-4658. (8) Bhawe, D. M.; Cohen, C.; Escobedo, F. A. Effect of Chain Stiffness and Entanglements on the Elastic Behavior of End-Linked Elastomers. The Journal of chemical physics 2005, 123, 014909.

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Table of Content Graphics

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Molecular Dynamics Simulation Study of Polymer Nanocomposites

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