Abstract
Based on the geometries from molecular dynamics simulations and a package compiled by us, the interactions between graphene nanosheet (GNS) and nine types of flexible polymers have been investigated with force field. Both the van der Waals (vdW) interaction and the electrostatic interaction (EI) for two same polymer chains and between a polymer and a GNS were calculated and compared. The effect of cut-off distance was explored. It was found that the cut-off distance plays a significant role in EI energy, but a less important role in vdW energy when the cut-off distance is over 9.5 Å. The reasonable cut-off distances for EI and vdW interactions for simulation are suggested.
1 Introduction
Since its discovery by Novoselov et al. (1), graphene has attracted an enormous amount of interest owing to its unique structure and mechanical (2), electrical (3) and thermal (4) properties. Therefore, one of the most potential applications of graphene is the polymer nanocomposites. Wang et al. (5) reported that, by adding 1 wt% of graphene, the tensile strength for graphene oxide/poly(methyl methacrylate) (GO/PMMA) composites increased by up to 60.7% than that for pure PMMA. This may be ascribed to the interfacial interaction between graphene and PMMA. Kim et al. (6) reported that the electrical conductivity of un-functionalized linear low-density polyethylene (LLDPE) is higher than that of modified ones, when the thermally reduced graphene oxide (TRG) content is at <3 wt%. Experiments have been conducted to study the interfacial characteristics of carbon-based nanofillers, using theoretical and computational methods (7–11). Lau et al. (7) discussed the molecular adhesion between water and graphene using molecular dynamics (MD) simulation and found that it matched reasonably with experiment results. Our group has simulated the interactions between polyethylene/polypropylene/polystyrene/poly(phenylacetylene)/poly(p-phenylenevinylene) (PE/PP/PS/PPA/PPV) and single-walled carbon nanotubes (SWCNT) (8). It was found that the interaction energies between PPA and SWCNTS are greatly influenced by their repeat unit arrangements and conformations, while the interaction strength between the PP/PS/PPV molecules and SWCNTs is nearly independent of these factors. Tallury and Pasquinelli (9) used force field-based molecular mechanics calculations to determine the interfacial energies and whether the polymers wrap along the diameter of the SWCNT or not. They discovered that PS and PMMA, which have a remarkably smaller value of interaction energy than other polymers, have a poor wrapping behavior. The interfacial mechanical properties of the polymer/graphene composites were also investigated by MD simulations. Rahman et al. (10) found that the Young’s modulus and shear modulus of the graphene-epoxy system were comparatively 25–40% higher than those of neat epoxy resin, and Ebrahimi et al. (11) studied the effect of the roughness of graphene and the pull-out velocity. Lv and Xue (12–14) have investigated the effect of functionalization on the interfacial energy of graphene composites. Their results show that the chemical modifications of graphene not only can increase the shear stress of the composites at the appropriate density of chemical attachment (12, 13), but also can improve the glass transition temperature (Tg) of the graphene/PMMA composites (14).
Nonbond interactions, including electrostatic interactions (EIs) and van der Waals (vdW) interactions, are some of the important factors for the dynamic behavior of a composite system. Rahman et al. (15) have found that the physisorption process is dominated by the vdW interactions in epoxy-functionalized functionalized graphene and chitosan composites. The nonbond interactions are the main calculation in MD simulation. The cut-off distance is a key factor that impacts accuracy and efficiency (16–18). Huang et al. (18) studied the effect of cut-off distance used in MD simulations on fluid properties in both NVT and NPT ensembles and discovered that, in the NPT ensemble, the cut-off distance plays a key role in determining fluid equilibrium structure, density and self-diffusion coefficient. Therefore, the choice of their treatment should be based on a specific system, especially the electrostatics treatment (19).
In this paper, we studied the nonbond interaction between chains of nine polymers and graphene nanosheet (GNS), as well as those between the chains themselves with force field. The effect of cut-off distance on the interaction was also investigated.
2 Computational methods
The nine types of flexible polymers considered were PE, PP, poly(acrylonitrile) (PAN), PMMA, PS, poly(ethylene oxide) (PEO), poly(l-lactide) (PLLA), poly(caprolactone) (PCL) and nylon 6. The polymers were all built in a head-to-tail and isotactic configuration, and there were 20 monomers in each chain of the polymers. Although the number of atoms of these polymers was <104, we still considered them as polymers in this paper. They can be regarded as small parts of the corresponding “long” polymers, and their interactions with GNS and with themselves can be exploited to understand the primary behaviors of the long polymers. Because the longest chain was 174.49 Å, we used the GNS, whose length was 178.96 Å. Carbon atoms at the edges of the GNS were saturated with hydrogen atoms to make the whole GNS segment neutral and to enhance the stability of GNS (20).
MD simulations were carried out with the Discover module in Accelrys Materials Studio v. 3.2 (Accelrys Software, Inc., San Diego, CA, USA) and the atomic force field was chosen using the Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies (COMPASS) (21). The COMPASS force field has been successfully used in the investigation of organic and inorganic materials (22–24). We used the COMPASS forced field to simulate the interactions between PE/PP/PS/PPA/PPV and SWCNTs (8).
The force field potential can be represented as follows (25):
where Evalence is the valence energy, Ecross-term is the cross-term interaction and Enonbond is the nonbond interaction energy. For the polymer and carbon-based composites, many MD simulations (26, 27) were performed at the 400–500 K range because crystallization behavior occurs in that range. Therefore, we performed MD simulations for 1000 ps at 450 K, where the molecules can change their conformation rapidly. The polymer/graphene systems used the nonperiodic boundary condition. All the simulations were carried out in the NVT ensemble, and the time step was 1 fs. The Andersen algorithm was used for temperature control (28, 29).
The considered interaction energy in the present paper was the nonbond interaction energy including vdW and EI interactions. They can be calculated using the following equation:
where Aij, and Bij are the system-dependent parameters implemented in Accelrys Materials Studio, rij is the i–j atomic separation distance, q is the atomic charge and ε is the dielectric constant. The switching function was used to smoothly turn off non-bond interactions over a range of distances to avoid the discontinuities caused by direct cut-offs. The switching function S(r) can be represented as
where RS is the cut-off distance, RC is the sum of the cut-off distance and the spline width, and rij is the i–j atomic separation distance.
However, the nonbond interaction between two groups cannot be calculated with the Discover module directly because it was for the two groups and excluded the nonbond energy of the atoms in each group itself. Using the parameters in the literature that described the COMPASS forced field in detail (21), we developed a Fortran code to calculate the vdW and EI energies. It should be noted that our code cannot perform dynamics calculation. All the dynamics calculations were carried out with the Discover module, and the coordinates, atomic types and charges used in the code were read from a text-format file, which was transferred from the trajectory file of Accelrys Materials Studio. In the code, our algorithm works by calculating a pair of vdW or EI energy using the information from two atoms, which are from different groups. Therefore, the vdW and EI energy between atoms in one group were not included.
3 Results and discussion
3.1 Interaction between GNS and polymers
Among the nine considered polymer chains, PE, PEO, PCL and nylon 6 have carbon, oxygen or nitrogen atoms in the backbone and no pendant groups, while the others have carbon atoms in the backbone but with various pendant groups, except for PLLA, which has carbon and oxygen atoms in the backbone with side groups of –CH3. A MD simulation for 1000 ps was carried out to obtain a relaxed structure. Figure 1A shows the snapshots of PE, PP, PAN, PMMA, PS, PEO, PLLA, PCL and nylon 6 adsorbed on the GNS surface at 450 K from a simulation time of 1000 ps. Initially, the polymers were almost straight chains. With the time increasing, the polymers changed their conformation by rotating around the σ-bonds of the backbone to obtain an appropriate structure with minimum energy. Apart from the PE chain, the other polymer chains became crimp conformations.
(A) Snapshots of (a) PE, (b) PP, (c) PAN, (d) PMMA, (e) PS, (f) PEO, (g) PLLA, (h) PCL and (i) nylon 6 adsorbed at the GNS surface at 1000 ps at 450 K; (a–i) top view; (a1–i1) side view. (B) EI energy and (C) vdW energy curves of the nine polymers interacting with GNS at 450 K at 1000 ps of simulation time.
We focused our investigations on the EI energies first. Using MD simulation and our package for nonbond energy calculation, we obtained the EI energies between the chains and GNS. As shown in Figure 1B, all the EI energies are close to zero. The result is not surprising because the carbon atoms of pure GNS have no net charge and the charged atoms on the edge are saturated with H atoms.
The time duration of vdW energies between the atoms of polymers and the atoms of GNS is shown in Figure 1C. It is obvious that nylon 6 has the strongest interaction with GNS, while PE is the weakest. Figure 1C also shows that the vdW energies of polymers change considerably before 500 ps, so we calculated the average energies of polymers in the last 500 ps. The nine types of polymer chains have not only various atomic types but also different numbers of atoms. Therefore, the <vdW/atom> is a fair comparison. The results are presented in Table 1. For PE, PP and PS, the interaction energy was -68.762, -79.468 and -168.655 kcal/mol at 450 K, respectively. Compare these values to -25, -30 and -55 kcal/mol of CNT composites (PE, PP and PS have 10 monomers) at 400 K, respectively, the present results are stronger (8). This may be due to the different curvature of CNT and GNS and to the temperature between the systems. As we can see in Table 1, the average vdW energies of the polymer chains are as follows: PE<PP<PAN<PLLA<PEO<PMMA<PS<PCL<nylon 6. The atom numbers of these polymers are as follows: PE<PAN<PEO<PP<PLLA <PMMA<PS<PCL<nylon 6, which results in a different order for the average vdW energy per atom as follows: PMMA<PP<PLLA<PS<PE<PAN<PCL<PEO<nylon 6. The results are in agreement with those of a study of the interaction between CNT and polymers (8). It should be noted that the chain of nylon 6 has the maximum energy of both per atom and the whole chain, which implies that it can be adsorbed more easily by GNS.
Average vdW energies of polymers and average energies per atom between polymers and GNS at 450 K between 500 and 1000 ps.
| Polymer | Polymer group (a) | Polymer group (b) | |||||||
|---|---|---|---|---|---|---|---|---|---|
| PE | PEO | PCL | nylon 6 | PP | PAN | PMMA | PS | PLLA | |
| EvdW | -68.745 | -94.649 | -227.790 | -283.985 | -79.465 | -88.496 | -104.618 | -168.609 | -89.488 |
| Eaverage | -0.5635 | -0.6619 | -0.6275 | -0.7415 | -0.4786 | -0.6232 | -0.3464 | -0.5236 | -0.4890 |
Group (a) is the chains without pendant groups, and group (b) is the chains with pendant groups. Values are in kilocalorie per mole.
In Table 1, two groups were used to understand the vdW interaction energies for the polymer chains with or without pendant groups. Group (a) is for the chains without pendant groups. It can be seen in the table that the average vdW interaction energies per atom of these polymer chains increased from PE to PEO to nylon 6. The difference is probably from the different atoms on the backbone because there are no pendant groups on these chains. There are C, O and N atoms in the backbone. The chain of nylon 6 includes nitrogen atoms that have a deeper well depth of nonbond coefficient than common C atom. However, it is not the sole factor that contributes to the interaction energy because the geometries of chains also play a role in the vdW interaction energies. Group (b) collects the vdW interaction energies for the polymer chains with various pendant groups. The average energy per atom of PP is smaller than that of PLLA with oxygen atoms in the backbone. PAN has the largest average energy per atom in the group. Like in the case of chains without pendant groups, the polymers with C, O or N atoms generally have larger vdW interaction energies. However, we found that PMMA with oxygen atoms in the pendant group –CH3 COOCH3 has a smaller average vdW interaction than PP that only contains carbon and hydrogen with a simple side group –CH3, which implies that the complexity of the pendant group is also one of the factors that contribute to the vdW interaction energies. In the present case, the more complicated the pendant groups are, the smaller the interaction energy is. The probable reason is that the complex pendant groups obstruct the backbone close to the atoms of the GNS. This can also be seen in Figure 1A.
3.2 Interchain interactions of polymer
We also simulated two same polymer chains absorbed on the surface of the GNS. Figure 2A shows the snapshots of two polymer chains of PE, PP, PAN, PMMA, PS, PEO, PLLA, PCL and nylon 6 adsorbed at the GNS surface at 450 K from a simulation time of 1000 ps. Initially, the polymer chains were straight and parallel. After 1000 ps, the polymer chains changed only slightly, as can be seen in Figure 2A. The conformation of the polymer chains cannot change freely compared with a single chain, due to inter-chain interactions.
(A) Snapshots of (a) PE, (b) PP, (c) PAN, (d) PMMA, (e) PS, (f) PEO, (g) PLLA, (h) PCL and (i) nylon 6 adsorbed at the GNS surface at 1000 ps at 450 K; (a–i) top view; (a1–i1) side view. (B) EI energy and (C) vdW energy of two same polymer chains vs. time.
Hence, we care more about the inter-chain interactions of polymers in this simulation. Figure 2B presents the EI energies of two polymer chains. It can clearly be observed that the EI energies of nylon 6 and PAN are stronger than those of other polymers. The PE and PP are nearly a straight line around zero. As in the case of GNS, we also calculated the average EI energies of the polymer chains and average energies per atom between 500 and 1000 ps, and the results are presented in Table 2. The average EI energies were as follows: PE<PP<PEO<PS<PLLA<PMMA<PCL<PAN<nylon 6, while the average energies per atom display were as follows: PE<PP<PEO<PS<PCL<PMMA<PLLA<nylon 6<PAN. Obviously, PAN and nylon 6 have stronger interactions than the other polymers. The likely reason is that they have more charge for each atom and a shorter distance. We will omit further discussions here because EI is significantly influenced by cut-off distance; detailed discussions will be presented in the next section instead.
Average EI and vdW energies of polymer chains and average energies per atom at 450 K between 500 and 1000 ps.
| Polymer | Polymer group (a) | Polymer group (b) | |||||||
|---|---|---|---|---|---|---|---|---|---|
| PE | PEO | PCL | nylon 6 | PP | PAN | PMMA | PS | PLLA | |
| EEI | -0.0008 | -0.385 | -4.220 | -41.904 | -0.003 | -37.116 | 3.739 | -2.109 | -2.368 |
| Eaverage EI | 0.000007 | -0.0027 | -0.0116 | -0.1094 | 0.00002 | -0.2614 | 0.0124 | -0.0065 | -0.0129 |
| EvdW | -4.783 | -9.608 | -27.078 | -43.267 | -5.281 | -11.495 | -15.529 | -29.740 | -15.830 |
| Eaverage vdW | 0.0392 | -0.0672 | -0.0746 | -0.1130 | -0.0290 | -0.0810 | -0.0514 | -0.0924 | -0.0865 |
Group (a) is the chains without pendant groups, and group (b) is the chains with pendant groups. Values are in kilocalorie per mole.
Figure 2C shows the vdW interaction energy of two polymer chains. Moreover, Table 2 presents the average vdW energies of polymer chains and average energies per atom between 500 and 1000 ps. As can be seen in Table 2, the interaction of PE chains is -4.784 kcal/mol, and the average of the interchain distances of PE is >4.5 Å. Yang et al. (30) presented the interactions vs. the interchain distance between two PE monomers adsorbed on graphene, which, for the stacked arrangements, was about -0.25 eV (-11.5 kcal/mol for 40 monomers) at 4.5 Å. The average vdW energies of polymers were as follows: PE<PP<PEO<PAN<PMMA<PLLA<PCL<PS<nylon 6, and the average energies per atom followed this order: PP<PE<PMMA<PEO<PCL<PAN<PLLA<PS<nylon 6. Similar to the case of GNS, the vdW energy between two chains of nylon 6 is the largest value. Generally, the chains that have nitrogen and oxygen atoms in the backbone or have pendant groups have larger vdW energies. In the complicated pendant groups of PS, the result of two chains was arranged in a more parallel manner than that of PP, which gave the former larger vdW energies.
It was found that the vdW energies between two same chains of polymer were smaller than the energies between the corresponding chain and GNS, which implies that the polymers were easily adsorbed by the GNS when they approached the latter. The result for this is that the crystallization process induced by GNS was performed in two steps: the adsorption step and the orientation step, which are highly cooperative (31).
3.3 Effect of cut-off distance on the interaction energies
The effect of cut-off distances will be investigated in this section. Using the final geometrical structures in the MD simulations and our package for calculating the nonbond energy, we calculated the nonbond interaction energies between two polymers at various cut-off distances.
Table 3 presents the vdW energies between two chains of the polymers. From the table, we can see that there is a slight increase if the cut-off distance is >9.5 Å. Beyond 25.5 Å, the vdW energy has nearly no changes. For all the considered polymers, the cut-off distance of 12.5 Å is enough to obtain a reasonable vdW energy in MD simulations. The cut-off distance of 9.5 Å (25) is also acceptable in calculations for a huge system.
Van der Waals energies (in kcal/mol) between two chains of PE, PP, PAN, PMMA, PS, PEO, PLLA, PCL and nylon 6.
| Polymer | PE | PP | PAN | PMMA | PS | PEO | PLLA | PCL | nylon 6 |
|---|---|---|---|---|---|---|---|---|---|
| 9.5 Å | -2.436 | -9.160 | -7.806 | -12.432 | -25.911 | -8.546 | -13.666 | -24.419 | -39.943 |
| 12.5 Å | -2.544 | -9.930 | -8.962 | -15.332 | -29.098 | -9.041 | -14.954 | -25.973 | -41.629 |
| 15.5 Å | -2.577 | -10.124 | -9.136 | -16.067 | -30.037 | -9.214 | -15.223 | -26.216 | -41.971 |
| 20.5 Å | -2.596 | -10.190 | -9.181 | -16.256 | -30.374 | -9.257 | -15.284 | -26.290 | -42.055 |
| 25.5 Å | -2.596 | -10.199 | -9.188 | -16.288 | -30.422 | -9.263 | -15.293 | -26.303 | -42.072 |
| 30.5 Å | -2.597 | -10.200 | -9.190 | -16.292 | -30.430 | -9.264 | -15.295 | -26.307 | -42.076 |
| 35.5 Å | -2.597 | -10.200 | -9.190 | -16.293 | -30.432 | -9.265 | -15.295 | -26.308 | -42.078 |
| 40.5 Å | -2.598 | -10.200 | -9.191 | -16.293 | -30.432 | -9.265 | -15.295 | -26.309 | -42.079 |
| 45.5 Å | -2.598 | -10.200 | -9.191 | -16.293 | -30.432 | -9.265 | -15.295 | -26.309 | -42.079 |
| 50.5 Å | -2.598 | -10.200 | -9.191 | -16.293 | -30.432 | -9.265 | -15.295 | -26.309 | -42.079 |
| 100 Å | -2.598 | -10.200 | -9.191 | -16.293 | -30.432 | -9.265 | -15.295 | -26.309 | -42.079 |
| No cut-off | -2.598 | -10.200 | -9.191 | -16.293 | -30.432 | -9.265 | -15.295 | -26.309 | -42.079 |
Table 4 shows the EI energies between two same chains for all the considered chains. When the cut-off distance increased, the change in EI energy was obvious, especially for PAN, PMMA, PLLA and nylon 6. For polymer chains with only a carbon in the backbone, like PE and PP, the EI energies between the chains of the polymer are small and have only a slight change when the cut-off distance is >9.5 Å. Therefore, 9.5 Å is an acceptable cut-off distance for these polymers. However, the values of the EI energies of PAN, PMMA, PEO, PLLA, PCL and nylon 6 when the cut-off was 9.5 Å deviated from the no cut-off results by 40.856, 60.984, 23.847, 10.085, 12.195 and 65.346 kcal/mol, respectively. For these polymers that have oxygen and nitrogen atoms in their chains, when the cut-off distance reached 30.5 Å, the EI energies of polymers changed around the limit value of the EI energy obtained with no cutoff distance. For nylon 6, PAN and PCL, Table 4 shows that the EIs have no significant changes between 50.5 and 100 Å. To examine the EI in more details, we also computed more cut-off distances in the previously mentioned range and found that the real convergent cut-off distance was about 80.0 Å. As is known, if the cut-off distance is larger, the CPU time is longer. Garemyr et al. (16) reported that CPU times whose cut-off distance is 14 Å are about three times longer than those whose truncation is 8 Å. However, a large cut-off distance is necessary to obtain a reasonable result for nylon 6, PAN and PCL. Generally, in consideration of the accuracy and efficiency, a cut-off distance of 50.5 Å can give a reasonable EI energy for all the considered polymers. Moreover, an adequate EI energy can be obtained with a cut-off distance of 9.5 Å for PE and PP and 20.5 Å for PS.
EI energies (in kcal/mol) between two chains of PE, PP, PAN, PMMA, PS, PEO, PLLA, PCL and nylon 6.
| Polymer | PE | PP | PAN | PMMA | PS | PEO | PLLA | PCL | nylon 6 |
|---|---|---|---|---|---|---|---|---|---|
| 9.5 Å | -0.658 | -1.136 | 4.312 | 65.336 | 19.962 | -24.367 | 7.567 | 8.385 | 20.858 |
| 12.5 Å | 2.153 | -4.276 | -106.844 | 86.492 | -39.625 | -26.137 | 76.800 | 53.259 | 340.379 |
| 15.5 Å | 0.406 | -2.837 | -56.913 | -83.132 | 28.061 | 1.610 | -61.490 | -15.725 | -44.460 |
| 20.5 Å | -0.478 | 1.992 | -37.311 | 12.318 | 0.783 | -1.771 | 1.920 | -24.606 | -16.264 |
| 25.5 Å | -0.002 | 0.219 | -48.296 | 19.919 | -0.238 | -6.193 | 24.627 | 4.647 | -42.911 |
| 30.5 Å | 0.005 | -0.768 | -36.681 | 8.695 | -3.739 | 0.066 | -3.636 | -2.499 | -34.961 |
| 35.5 Å | -0.415 | -0.195 | -35.284 | -8.342 | -2.239 | -0.605 | 5.958 | -2.584 | -58.120 |
| 40.5 Å | 0.437 | -0.009 | -39.278 | 4.528 | -3.265 | -2.685 | -5.048 | -4.541 | -48.622 |
| 45.5 Å | -0.005 | -0.009 | -36.544 | 4.352 | -1.397 | 1.561 | -2.518 | 0.533 | -28.370 |
| 50.5 Å | 0.001 | -0.009 | -36.544 | 4.352 | -2.180 | -2.303 | -2.518 | -8.112 | -45.383 |
| 100 Å | 0.001 | -0.009 | -36.544 | 4.352 | -1.242 | -0.520 | -2.518 | -3.450 | -44.794 |
| No cut-off | 0.001 | -0.009 | -36.544 | 4.352 | -1.242 | -0.520 | -2.518 | -3.810 | -44.488 |
As is known, most simulation works use a cut-off distance for nonbond interactions of between 0.9 and 1.4 nm. According to our results, they are appropriate for calculating vdW energy based on COMPASS force field. However, they are not enough to obtain adequate EI energy for polymers that have O and N atoms. Hence, an improper cut-off value could lead to unphysical results and, therefore, one should consider first the cut-off distance before performing a MD simulation.
4 Conclusions
The interactions of nine types of polymers with GNS or with themselves were investigated using a force field. The results show that, for the interactions between the GNS and the polymers, the vdW energy plays the main role and the EI energy can be ignored. However, for the interaction between two same polymer chains, the EI energy of nylon 6 is close to the corresponding vdW energy, whereas that of PAN is even about three times larger than the corresponding vdW energy. It implies that EI will significantly affect the melting behavior of two polymers. It was also found that nitrogen and oxygen atoms in polymers will strengthen the interactions, while the complex degree of polymer pendant groups will weaken the interaction. The vdW energies between the same chains are smaller than the corresponding energies between the chain and the GNS. A cut-off distance of 15.5 Å is enough for all the considered chains to obtain adequate vdW energy. Unfortunately, the significant role of cut-off distance on EI energy is rather broad. For most considered polymers, reasonable results can only be obtained when the cut-off distance is >50.5 Å, although it will increase too much the computing time for large systems. These findings can provide helpful information for designing a scheme for polymer simulations.
Acknowledgments
This work was supported by the National Science Foundation of China under grant nos. NSFC-11174117 and NSFC-11374132.
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