The interfacial mechanical properties of functionalized graphene–polymer nanocomposites

Abstract

The interfacial mechanical properties between graphene (GR) and a polymer matrix play a key role in load transfer capability for GR/polymer nanocomposites. Grafting of polymer molecular chains on GR can improve the dispersion of the GR in a polymer matrix and change the interfacial mechanical properties between the GR and the polymer matrix. In this work, we investigated the interfacial mechanical properties between GR functionalized with polymer molecular chains and a polyethylene (PE) matrix using molecular dynamics simulations. The influences of grafting density and chain length on the interfacial mechanical properties were analyzed. The results show that grafting of short PE molecular chains on GR can significantly improve the interfacial shear strength and interfacial Mode-II fracture toughness in functionalized GR/PE nanocomposites.

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1. Introduction

Graphene (GR) is a two-dimensional single layer carbon atom material.¹ Its extraordinary mechanical, electronic and thermal properties^2–7 make it a potential reinforcement candidate in polymer composites. The mechanical properties of a polymer can be improved significantly by the addition of GR, such as the Young's modulus, ultimate tensile strength, fracture toughness, ductility,^8–10 even at a low weight fraction of 0.1 ± 0.02%.¹¹

It is difficult to homogeneously disperse pristine GR in organic polymers, especially for a higher volume fraction of GR due to strong van der Waals (vdW) forces between graphene sheets (GRs).¹² In general, surface modification is an efficient way to improve the compatibility between GR and polymer matrix since functional groups on GRs may effectively prevent GR aggregations in polymer matrices. For instance, Ramanathan et al.¹³ found that functionalized GRs can be dispersed very well and interact intimately with polar polymers. Moreover, some researchers found that the aggregations of nanostructured materials functionalized with polymer chains can be effectively alleviated, e.g., single-walled carbon nanotubes grafted with polystyrene (PS) in the polymer matrix as identified by Chadwick et al.¹⁴ With various grafted polymer chains, it was found that various carbon nanofillers can work much better as a reinforcement phase for improving the mechanical properties of various polymer matrices. For instance, multiwall carbon nanotubes (MWCNTs) grafted with poly(caprolactone) significantly improved the Young's modulus, tensile strength, toughness and ductility of poly(vinyl chloride) composites.¹⁵ Similar results were also obtained for polypropylene composites by the addition of GR grafted with alkyl chains¹⁶ and for PS composites reinforced by GR grafted with PS chains.¹⁷ Comparing with graphene oxide, GR grafted with diglycidyl ether of bisphenol-A significantly enhanced the Young's modulus, tensile strength and fracture toughness of epoxy composites.¹⁸

Although there have been some pioneer experimental works as described above, there is no systematical theoretical or numerical investigation on the improvement mechanisms of reinforcement carbon nanofillers grafted by polymer chain in polymer matrix. It is well-known that the performance of composites is critically controlled by the interfacial characteristics between the reinforcement phase and the matrix.¹⁹ In the present work, we conducted molecular dynamics simulations to investigate the interfacial mechanical properties between GR grafted with polyethylene (PE) chains and PE matrix. The effects of grafting density and chain length on the interfacial mechanical properties were analyzed. It was found that GR grafted with short PE molecular chains can significantly enhance the interfacial shear strength and Mode-II fracture toughness of GR/PE nanocomposites.

2. Simulation method

2.1. Simulation force fields

The molecular dynamics simulations were carried out using LAMMPS molecular dynamics package²⁰ and the interatomic interactions were described by ab initio polymer consistent force field (PCFF),²¹ which can effectively evaluate the mechanical^22,23 and thermal properties²⁴ of nanocomposites. The PCFF force field has the same functional form with COMPASS (Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies) force field, the detailed expressions for both force fields can be found elsewhere.^22,25,26 The vdW and Coulomb interactions were used to describe the atomistic interactions between GR and PE matrix.

2.2. Simulation model

Our previous work shows that the molecular chain length of polymer matrix has a little effect on the interfacial mechanical properties of GR/PE nanocomposites.²² In this work, we chosen 40 PE repeat units (–CH₂–CH₂–) for the molecular chain length of PE matrix. It was long enough to obtain stable results.²² The initial cell dimensions were 62 × 34 × 35 ų for the simulation system containing a GR with the width (W): 28.4 Å and length (L): 56.6 Å, respectively. To eliminate the unsaturated boundary effect, hydrogen atoms were added to the edges of the GR. The simulation model can be seen in Fig. 1.

Fig. 1 Schematic of simulation model.

All the simulation models were relaxed in isothermal–isobaric (NPT) ensemble at temperature of 300 K and pressure of 1 atm for 500 ps using 1 fs time step. The canonical (NVT) ensemble was followed by using the same temperature and time step for another 500 ps. Additionally, 100 ps NVT ensemble was further run with the same conditions and 5 configurations were obtained every 20 ps for calculating the average interfacial mechanical properties. The cut-off distance for both the vdW and Coulomb interactions was 1.0 nm in all simulations.

3. Results and discussion

3.1. The effect of matrix size

To eliminate the thickness effect of PE layer (i.e., z-axis direction in Fig. 1) on the interfacial mechanical properties, four PE matrix layers with different thicknesses were constructed. Periodic boundary conditions (PBCs) were used except in the z-axis direction. The relaxation processes were the same as that in Section 2.2. To decrease the computational time, a portion of the PE layer at the top 5 Å was constrained after the relaxation process (see Fig. 1). Firstly, normal separation or opening simulations, i.e., Mode-I, were carried out to obtain the interfacial mechanical properties by applying the z-axis displacement of 0.01 Å to the GR at every time step. The variation of pull-out force during the opening simulation is shown in Fig. 2. The numbers of atom in the PE layers, thicknesses, maximum pull-out forces and interfacial cohesive strengths for four different simulation models are listed in Table 1. The computational results reveal that 30 Å for the thickness of PE layer with a relatively small number of atoms would be enough to yield an accurate value. Therefore, this thickness of the PE layer was set for all simulations.

Fig. 2 Variation of pull-out force for opening simulation.

Table 1 The model parameters and computational results for simulation models with different sizes

Model	Number of atom for PE layer	Thickness of PE layer (Å)	Maximum pull-out force (nN)	Interfacial cohesive stress (MPa)
1	5566	25	10.57	657.6
2	7018	30	11.70	727.9
3	8712	35	11.34	705.5
4	11 132	40	11.87	738.4

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Fig. 2 and Table 1 show that the maximum pull-out force is 11.70 nN and the corresponding interfacial cohesive stress is 727.9 MPa, respectively, which can be calculated from the following equation.

where σ_max is the interfacial cohesive stress, F_z-max is the maximum pull-out force in the z-axis direction, and A_c is the contact area of GR with PE layer, i.e., W × L.

This interfacial cohesive stress (i.e., 727.9 MPa) between the GR and the polymer matrix is much larger compared with that in carbon nanotube (CNT) based polymer nanocomposites, e.g., 479 MPa,²⁷ indicating that the GR has a better reinforcement effect than CNT in polymer matrix.

For the transversal interfacial mechanical properties, i.e., Mode-II, to the best of our knowledge, most of previous investigations of the interfacial mechanical properties between the rectangle GR and the polymer matrix were based on the pull-out simulations in the transversal direction (x-axis or y-axis shown in Fig. 1).^22,23 The applied pull-out direction may somehow have a minor impact on the results. In this work, we investigated the transversal interfacial mechanical properties by using sliding simulations, i.e., the GRs were pulled out from the PE matrix in transversal direction (y-axis). For this purpose, both the z- and the y-axes of simulation box were applied by non-PBCs. The relaxation processes of systems were the same as that in Section 2.2. The transversal pull-out velocity of 0.01 Å fs⁻¹ in y-axis was applied to the carbon atoms of GR while the atoms of PE layer at the top 5 Å were constrained (see Fig. 1). Fig. 3 shows that a typical curve of pull-out force versus displacement, and its tendency is similar to that in our previous study for the GR pulled out from PE matrix in the x-axis direction.²² As shown in Fig. 3, the pull-out force increases sharply first and then drops dramatically. Afterwards, the pull-out force fluctuates at an average value, and then gradually decreases to zero as the GR is completely pulled out from PE matrix.

Fig. 3 Variation of transversal pull-out force for sliding simulation.

Using eqn (2) as shown in the following, the maximum interfacial shear stress of the y-axis direction was evaluated as 127.1 MPa, compared with that in x-axis in our previous work (121 MPa),²² the pull-out direction has only a small impact on the interfacial mechanical properties.

where τ_max is the interfacial shear tress, and F_y-max is the maximum pull-out force in the y-axis direction.

Based on the obtained results, we can find that the interfacial shear strength is much lower than the interfacial cohesive strength, implying that the Model-II failure may be much easier compared with the Model-I.

3.2. Effects of grafting density and chain length on the interfacial mechanical properties

Some previous investigations found that the molecular weight of graft polymer has a great impact on the grafting configuration. For a polymer chain with large molecular weight, the longer the polymer chain is, the more difficult to graft uniformly to GR.²⁸ Some authors also stated that the same graft polymer on nanofillers as matrix works well in nanocomposite systems. For instance, Smith et al. found²⁹ that nanoparticles grafted with matrix molecular chains can improve the dispersion in the polymer matrix. Based on the facts, the PE molecular chains, i.e., the same as the matrix, were also used to functionalize the GR and the chain length was not longer than 15 repeat units. Moreover, Wang et al.²⁴ found that the increase of grafting density and chain length can improve the interfacial thermal transport between GR grafted by PE chains and PE matrix significantly. Therefore, it is reasonable to assume that the grafting density and chain length may affect the interfacial mechanical properties. For the first time, in this work, the effects of grafting density and chain length of GR on the interfacial mechanical properties were investigated using both the opening and sliding simulations. For analyzing conveniently, some chemical groups were ignored, which bridged the carbon atoms of GR and PE chains. Fig. 4 shows the schematic of a GR grafted with 4 PE molecular chains whose chain length is 5 repeat units.

Fig. 4 Schematic of graphene grafted with 4 PE molecular chains.

The grafting density can be defined as¹⁴

where N is the number of polymer molecular chains, A_g is the area of GR and A_g = W × L.

For the case of Mode-I in opening simulations, the interfacial cohesive stress decreases as the grafting density and the chain length increase (see in Fig. 5). Compared with pristine GR, the weak interfacial cohesive strength between the functionalized GR and the PE matrix indicates that the interatomic interactions between a pristine GR and PE chains are stronger than those among the PE chains.³⁰ Therefore, larger grafting density and longer chain length of graft PE molecular will result in weaker interfacial cohesive strength, due to a larger and thicker PE layer grafted onto the GR plane will be formed, which can effectively separate the GR from the PE matrix. Consequently, the PE matrix chains cannot directly interact with GR atoms, leading to weaker interfacial cohesive strength.

Fig. 5 Interfacial cohesive stress at different grafting density (Mode-I).

In contrast with the above opening simulations, as shown in Fig. 6, the interfacial shear strength obtained by using sliding simulations can be improved significantly as the grafting density increases, especially for the case of shorter graft PE chains (5 repeat units). In this case, the interfacial shear strength of functionalized GR/PE nanocomposites increases by 83.1% for the small grafting density of 0.0025 and by 330.7% for the large grafting density of 0.01. For the case of long graft chains (chain length of 10 repeat units), the interfacial shear strength increases only slightly when the grafting density is larger than 0.0025, and then it decreases when the grafting density exceeds 0.075. For the longest graft PE chains (15 repeat units), from the grafting density of 0.0025, the interfacial shear strength fluctuates around the average value of ∼440 MPa (see the blue dashed line in Fig. 6). Fig. 7 shows the configurations of graft PE molecules after sufficient relaxation and the green dashed lines represent the horizontal base line. For long graft PE chains, as shown in Fig. 7(b) and (c), most of the graft PE atoms are on the horizontal base line, due to the stiffness of PE molecular chains is low, which makes the PE molecules be easily adsorbed to the GR and parallel to the GR plane. The effective enhancement of interfacial shear strength by the GR grafted with PE chains are attributed to interlocking between the graft PE chains and the chains of PE matrix. Especially for short graft PE chains, due to the high stiffness of the PE molecular chains, the GR plane becomes rougher, which can be seen in Fig. 7(a) that a large portion of the graft PE atoms significantly deviate from the horizontal base line. Fig. 8 shows that the short graft PE molecular chains (see Fig. 8(a)) penetrate more deeply into the PE matrix compared with the long grafted PE molecular chains (see Fig. 8(b) and (c)), which means that short PE molecular chains can be interlocked more effectively with surrounding PE matrix molecules. On the contrary, the long graft PE chains cannot effectively interlock with PE matrix molecular chains, which results in relatively weak interfacial shear strength. Comparing with pristine GR, the functionalized GR is of significantly high interfacial shear strength, implying that the interlocking between the graft PE chains and the chains of PE matrix dominates the interfacial shear strength, instead of the vdW interactions.

Fig. 6 Interfacial shear stress at different grafting density (Mode-II).

Fig. 7 Schematic of GR grafted with 4 PE molecular chains with different chain lengths after sufficiently relaxed (a) 5 repeat units, (b) 10 repeat units, (c) 15 repeat units.

Fig. 8 Interfacial models after sufficiently relaxed, 4 graft PE molecular chains with different chain lengths (a) 5 repeat units, (b) 10 repeat units, (c) 15 repeat units.

3.3. The fracture toughness of functionalized graphene

To date, the fracture toughness of CNT based polymer nanocomposites has been studied extensively.^31–33 However, the interfacial fracture toughness between functionalized GR and polymer matrix has not been reported. Here, we used both the opening and sliding simulations to evaluate this property.

According to the theory of fracture mechanics, the energy release rate can be obtained as³⁴

and

U = U_e − U_s (5)

where U is the system potential energy, U_e is the external work, U_s is the strain energy and A is crack area. In this case U_s ≈ 0, therefore U can be obtained by evaluating the total work done by the external force, and U_e can be obtained using eqn (6), which is equal to the integration of the area under the curve of pull-out force versus displacement.where F_p is the pull-out force and the s is the pull-out displacement of GR.

Moreover, A is assumed to be equal to the initial contact area between GR and PE layer, i.e., A = A_c.

For the opening simulations, the interfacial fracture toughness of pristine GR based nanocomposites is 0.187 J m⁻² (shown by the blue dashed line in Fig. 9), which is larger than the interfacial cohesive energy of 0.107 J m⁻² for CNT based nanocomposites computed by cohesive law.²⁷ It suggests that the interfacial fracture toughness of GR/polymer nanocomposites is stronger than that of CNT/polymer nanocomposites. Using the sliding simulations, the interfacial fracture toughness of pristine GR/polymer nanocomposites is 0.180 J m⁻², which is almost equal to the opening simulation result (Model-I).

Fig. 9 Variation of fracture toughness at different grafting density (Mode-I).

Some previous experimental results show that the Mode-II interfacial fracture energy is within a range of 0.054–0.80 J m⁻² for double-walled CNTs pulled out from poly(methyl methacrylate) (PMMA),³⁵ and 0.05–0.25 J m⁻² for pristine MWCNT and Epon 828 interface.³⁶ It indicates that the present results are reasonable. One exception is that the interfacial fracture energy obtained by Barber et al.³⁷ ranged from 4.0 J m⁻² to 70.0 J m⁻² for the interface between CNT and thermoplastic polymer matrix. This result is one order of magnitude higher than other results. One possible reason is that there are covalent bonds between CNTs and the polymer matrix.

For opening simulations, Fig. 9 shows that the Mode-I interfacial fracture toughness G_I firstly increases and then decreases with the increase of grafting density. Also, the interfacial fracture toughness increases with the graft chain length. Compared with non-functionalized GR (see the blue dashed line in Fig. 9), GR grafted with long PE chains (chain length of 15 repeat units) is of significantly higher interfacial fracture toughness. Especially at the grafting density of 0.005, the interfacial fracture toughness increases by 42.2%. The reason may be that the atomic interactions between the graft PE chains and the matrix chains are dominated by vdW force instead of interlocking. Therefore, a longer separation or opening displacement is needed for the longer graft chain, i.e., larger system potential energy required.

Fig. 10 shows the results of the sliding simulations. In this figure, the Mode-II interfacial fracture toughness G_II of functionalized GR/PE nanocomposites is much larger than that of non-functionalized GR/PE nanocomposites, i.e., 0.180 J m⁻². Especially for the short graft PE chains (5 repeat units), the G_II is improved by ∼258% at the grafting density of 0.005. The reason may be that many PE matrix chains were pulled out with the GR grafted short PE chains due to rough plane (see Fig. 7(a)) and strong interlocking effect (see Fig. 8(a)), leading to larger work consumed. However, for the case of the chain length of 15, the fracture toughness decreases sharply with the increase of grafting density and then fluctuates as shown in Fig. 10. This implies that the interlocking effect of the long graft chain may remarkably decrease when the grafting density increases. Moreover, comparing with Fig. 9, for functionalized GR based PE nanocomposites, the interfacial fracture toughness of Mode-II is much larger than that of Mode-I, i.e., the interfacial crack propagation of Model-I is much easier than that of Model-II. This is very different with the case of non-functionalized or pristine GR/PE nanocomposites, where the interfacial fracture toughness of Mode-II is almost equal to that of Mode-I.

Fig. 10 Variation of fracture toughness at different grafting density (Mode-II).

4. Conclusions

The interfacial mechanical properties between GR and polymer matrix play a key role in load transfer capability of GR/polymer nanocomposite systems. GR grafted with polymer chains can effectively enhance the dispersion, and change the interfacial mechanical properties between the GR and polymer matrix. In this work, we conducted both the opening and sliding molecular dynamics simulations to investigate the interfacial mechanical properties between the GR grafted with PE chains and PE matrix. The effects of grafting density and chain length of PE molecular were investigated. We found that the grafting density and the chain length have significant impacts on the interfacial mechanical properties. The interfacial cohesive strength (Mode-I) decreases with the grafting density and chain length, as large grafting density and long chain length of graft PE molecular can evidently separate the GR from the PE matrix and the vdW interactions between PE molecules are weaker than those between the GR and PE molecules. However, the interfacial shear strength (Mode-II) increases with the grafting density for short graft PE chains due to the increase of roughness and interlocking effect, meanwhile the interfacial shear strength decreases with the chain length as long graft PE chains will reduce the interlocking effect. The interfacial fracture toughness of Mode-I for long graft PE chains can be enhanced significantly. Compared with pristine GR, the GR grafted by PE chains, especially short graft molecular chains, is of significantly higher interfacial fracture toughness in Mode-II. For functionalized GR/PE nanocomposites, the interfacial crack propagation of Model-I is much easier than that of Model-II.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11372104, No. 11372363 and No. 11332013 No. 75121543), and Zhejiang Provincial Natural Science Foundation (No. LZ12E06001).

References

1. K. S. Novoselov, A. K. Geim and S. V. Morozov, et al., Electric Field Effect in Atomically Thin Carbon Films, Science, 2004, 306(5696), 666–669.
2. C. Lee, X. Wei, J. W. Kysar and J. Hone, Measurement of the elastic properties and intrinsic strength of monolayer graphene, Science, 2008, 321(5887), 385–388.
3. A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys., 2009, 81(1), 109–162.
4. A. A. Balandin, S. Ghosh and W. Bao, et al., Superior Thermal Conductivity of Single-Layer Graphene, Nano Lett., 2008, 8(3), 902–907.
5. A. A. Balandin, Thermal properties of graphene and nanostructured carbon materials, Nat. Mater., 2011, 10(8), 569–581.
6. Y. Wei, B. Wang, J. Wu, R. Yang and M. L. Dunn, Bending rigidity and Gaussian bending stiffness of single-layered graphene, Nano Lett., 2013, 13(1), 26–30.
7. E. Zaminpayma and P. Nayebi, Mechanical and electrical properties of functionalized graphene nanoribbon: a study of reactive molecular dynamic simulation and density functional tight-binding theory, Phys. B, 2015, 459, 29–35.
8. M. A. Rafiee, W. Lu and A. V. Thomas, et al., Graphene nanoribbon composites, ACS Nano, 2010, 4(12), 7415–7420.
9. M. A. Rafiee, J. Rafiee and I. Srivastava, et al., Fracture and fatigue in graphene nanocomposites, Small, 2010, 6(2), 179–183.
10. L. S. Walker, V. R. Marotto, M. A. Rafiee, N. Koratkar and E. L. Corral, Toughening in graphene ceramic composites, ACS Nano, 2011, 5(4), 3182–3190.
11. M. A. Rafiee, J. Rafiee, Z. Wang, H. Song, Z. Z. Yu and N. Koratkar, Enhanced mechanical properties of nanocomposites at low graphene content, ACS Nano, 2009, 3(12), 3884–3890.
12. M. J. McAllister, J.-L. Li and D. H. Adamson, et al., Single sheet functionalized graphene by oxidation and thermal expansion of graphite, Chem. Mater., 2007, 19(18), 4396–4404.
13. T. Ramanathan, A. A. Abdala and S. Stankovich, et al., Functionalized graphene sheets for polymer nanocomposites, Nat. Nanotechnol., 2008, 3(6), 327–331.
14. R. C. Chadwick, U. Khan, J. N. Coleman and A. Adronov, Polymer Grafting to Single-Walled Carbon Nanotubes: Effect of Chain Length on Solubility, Graft Density and Mechanical Properties of Macroscopic Structures, Small, 2013, 9(4), 552–560.
15. W. Guojiann, W. Lijuan, Z. Mei and C. Zhengmian, Reinforcement and toughening of poly(vinyl chloride) with poly(caprolactone) grafted carbon nanotubes, Composites, Part A, 2009, 40(9), 1476–1481.
16. X. Yang, X. Wang, J. Yang, J. Li and L. Wan, Functionalization of graphene using trimethoxysilanes and its reinforcement on polypropylene nanocomposites, Chem. Phys. Lett., 2013, 570, 125–131.
17. M. Fang, K. Wang, H. Lu, Y. Yang and S. Nutt, Covalent polymer functionalization of graphene nanosheets and mechanical properties of composites, J. Mater. Chem., 2009, 19(38), 7098.
18. A. Bagri, S. P. Kim, R. S. Ruoff and V. B. Shenoy, Thermal transport across twin grain boundaries in polycrystalline graphene from nonequilibrium molecular dynamics simulations, Nano Lett., 2011, 11(9), 3917–3921.
19. K. Liao and S. Li, Interfacial characteristics of a carbon nanotube–polystyrene composite system, Appl. Phys. Lett., 2001, 79(25), 4225.
20. S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys., 1995, 117(1), 1–19.
21. H. Sun, Ab initio calculations and force field development for computer simulation of polysilanes, Macromolecules, 1995, 28(3), 701–712.
22. F. Liu, N. Hu, H. Ning, Y. Liu, Y. Li and L. Wu, Molecular dynamics simulation on interfacial mechanical properties of polymer nanocomposites with wrinkled graphene, Comput. Mater. Sci., 2015, 108, 160–167.
23. M. C. Wang, Z. B. Lai, D. Galpaya, C. Yan, N. Hu and L. M. Zhou, Atomistic simulation of surface functionalization on the interfacial properties of graphene–polymer nanocomposites, J. Appl. Phys., 2014, 115(12), 123520.
24. M. Wang, N. Hu, L. Zhou and C. Yan, Enhanced interfacial thermal transport across graphene–polymer interfaces by grafting polymer chains, Carbon, 2015, 85, 414–421.
25. H. Sun, COMPASS: an ab initio force-field optimized for condensed-phase applications overview with details on alkane and benzene compounds, J. Phys. Chem. B, 1998, 102(38), 7338–7364.
26. Q. Zheng, D. Xia, Q. Xue, K. Yan, X. Gao and Q. Li, Computational analysis of effect of modification on the interfacial characteristics of a carbon nanotube–polyethylene composite system, Appl. Surf. Sci., 2009, 255(6), 3534–3543.
27. H. Tan, L. Y. Jiang, Y. Huang, B. Liu and K. C. Hwang, The effect of van der Waals-based interface cohesive law on carbon nanotube-reinforced composite materials, Compos. Sci. Technol., 2007, 67(14), 2941–2946.
28. M. Fang, K. Wang, H. Lu, Y. Yang and S. Nutt, Single-layer graphene nanosheets with controlled grafting of polymer chains, J. Mater. Chem., 2010, 20(10), 1982.
29. G. D. Smith and D. Bedrov, Dispersing nanoparticles in a polymer matrix: are long, dense polymer tethers really necessary?, Langmuir, 2009, 25(19), 11239–11243.
30. C. Li, A. R. Browning, S. Christensen and A. Strachan, Atomistic simulations on multilayer graphene reinforced epoxy composites, Composites, Part A, 2012, 43(8), 1293–1300.
31. M. A. Rafiee, W. Lu and A. V. Thomas, et al., Graphene nanoribbon composites, ACS Nano, 2010, 4(12), 7415–7420.
32. V. V. Mokashi, D. Qian and Y. Liu, A study on the tensile response and fracture in carbon nanotube-based composites using molecular mechanics, Compos. Sci. Technol., 2007, 67(3–4), 530–540.
33. T. Ozkan, Q. Chen and I. Chasiotis, Interfacial strength and fracture energy of individual carbon nanofibers in epoxy matrix as a function of surface conditions, Compos. Sci. Technol., 2012, 72(9), 965–975.
34. J. Williams, On the calculation of energy release rates for cracked laminates, Int. J. Fract., 1988, 36(2), 101–119.
35. X. Chen, M. Zheng, C. Park and C. Ke, Direct measurements of the mechanical strength of carbon nanotube–poly(methyl methacrylate) interfaces, Small, 2013, 9(19), 3345–3351.
36. Y. Ganesan, C. Peng and Y. Lu, et al., Interface toughness of carbon nanotube reinforced epoxy composites, ACS Appl. Mater. Interfaces, 2011, 3(2), 129–134.
37. A. H. Barber, S. R. Cohen, S. Kenig and H. D. Wagner, Interfacial fracture energy measurements for multi-walled carbon nanotubes pulled from a polymer matrix, Compos. Sci. Technol., 2004, 64(15), 2283–2289.

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