The mechanical properties of the SWCNT/polyimide nanocomposites were characterized using multi-scale simulation. An equivalent cylindrical solid was proposed to model the atomistic structure of SWCNTs, and the corresponding properties were determined from the molecular mechanics in conjunction with the energy equivalence concept. The level of atomistic interaction between the SWCNTs and the surrounding polyimide polymer was modeled by an effective interphase, the properties of which were obtained from the non-bonded energy as well as the non-bonded gap as determined from the MD simulation.
With the properties of the equivalent solid cylinder, effective interphase, and polyimide polymer, the mechanical properties of SWCNTs nanocomposites can be predicted using the three-phase continuum micromechanical model. For comparison purposes, the two-phase micromechanical model (Mori–Tanaka model) was also adopted for the predictions. A comparison between the micromechanical results and the MD results indicates that the longitudinal moduli of the SWCNT nanocomposites can be precisely predicted using the two-phase micromechanical model together with the equivalent cylinder properties of the SWCNTs. However, in the transverse direction, the three-phase model can provide improved results over the two-phase micromechanical model because the atomistic interactions between the SWCNTs and polyimide polymer become essential for such conditions.
In addition, the relationship of the load transfer efficiency of SWCNT/polyimide nanocomposites to the three different interfacial adhesions, vdW interactions, SWCNTs surface modifications and covalent bonds, was investigated in this study. Both the Lutsko stress formulation and the derivative of the potential function approach were employed to calculate the stress distribution of the SWCNTs embedded in polyimide nanocomposites.
Results indicate that when the intensities of vdW interactions increase, the load transfer efficiency and the moduli of the nanocomposites increase accordingly. In addition, the surface
41
modification of SWCNTs showed to be an effective way to improve the load transfer efficiency as well as the mechanical properties of nanocomposites. Due to the limited size of the molecular structures, the effect of covalent bonds is almost the same as that of the surface modification. Further study is required to understand the influence of the covalent bonds on the mechanical properties of nanocomposites.
42
References
1. Iijima S. Helical microtubules of graphitic carbon. Nature. 1991; 354(6348): 56-58.
2. Gibsion RF. Principles of composite material mechanics. 1994, New York:
McGraw-Hall Inc.
3. Lau KT, Gu C, Hui D. A critical review on nanotube and nanotube/nanoclay related polymer composite materials. Compos Part B-Eng. 2006; 37(6): 425-436.
4. Thostenson ET, Ren Z, Chou TW. Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol. 2001; 61(13):
1899-1912.
5. Han Y, Elliott J. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comp Mater Sci. 2007; 39(2): 315-323.
6. Griebel M, Hamaekers J. Molecular dynamics simulations of the elastic moduli of polymer-carbon nanotube composites. Comput Methods Appl Mech Eng. 2004;
193(17-20): 1773-1788.
7. Zhu R, Pan E, Roy AK. Molecular dynamics study of the stress-strain behavior of carbon-nanotube reinforced Epon 862 composites. Mater Sci Eng A. 2007; 447(1-2):
51-57.
8. Liu YJ, Chen XL. Evaluations of the effective material properties of carbon nanotube-based composites using a nanoscale representative volume element. Mech Mater. 2003; 35(14): 69-81.
9. Luo D, Wang WX, Takao Y. Effects of the distribution and geometry of carbon nanotubes on the macroscopic stiffness and microscopic stresses of nanocomposites. Compos Sci Technol. 2007; 67(14): 2947-2958.
10. Selmi A, Friebel C, Doghri I, Hassis H. Prediction of the elastic properties of single walled carbon nanotube reinforced polymers: A comparative study of several micromechanical models. Compos Sci Technol. 2007; 67(10): 2071-2084.
11. Hammerand DC, Seidel GD, Lagoudas DC. Computational micromechanics of clustering and interphase effects in carbon nanotube composites. Mech Adv Mater Struct.
2004; 14(4): 277-294.
12. Gates TS, Odegard GM, Frankland SJV, Clancy TC. Computational materials:
multi-scale modeling and simulation of nanostructured materials. Compos Sci Technol.
2005; 65(15-16): 2416-2434.
43
13. Shim BS, Zhu J, Jan E, Critchley K, Ho S, Podsiadlo P, Sun K, Kotov NA.
Multiparameter structural optimization of single-walled carbon nanotube composites:
toward record strength, stiffness, and toughness. ACS Nano. 2009; 3(7): 1711-1722.
14. Mokashi VV, Qian D, Liu Y. 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.
15. Yavin B, Gallis HE, Scherf J, Eitan A, Wagner HD. Continuous monitoring of the fragmentation phenomenon in single fiber composite materials. Polym Composite. 1991;
12(6): 436-446.
16. Andersons J, Joffe R, Hojo M, Ochiai S. Fibre fragment distribution in a single-fibre composite tension test. Compos Part B-Eng. 2001; 32(4):323-332.
17. Valadez-Gonzalez A, Cervantes-Uc JM, Olayo R, Herrera-Franco PJ. Effect of fiber surface treatment on the fiber-matrix bond strength of natural fiber reinforced composites.
Compos Part B-Eng. 1999; 30(3): 309-320.
18. Moon CK. The effect of interfacial microstructure on the interfacial strength of glass fiber/polypropylene resin composites. J Appl Polym Sci. 1994; 54(1): 73-82.
19. Piggott MR, Dai SR. Fiber pull out experiments with thermoplastics. Polym Eng Sci.
1991; 31(17):1246-1249.
20. Al-Haik M, Hussaini MY, Garmestani H. Adhesion energy in carbon nanotube-polyethylene composite: Effect of chirality. J Appl Phys. 2005; 97(7): 074306.
21. Namilae S, Chandra N. Multiscale model to study the effect of interfaces in carbon nanotube-based composites. J Eng Mater-T ASME. 2005; 127(2): 222-232.
22. Gou J, Minaie B, Wang B, Liang Z, Zhang C. Computational and experimental study of interfacial bonding of single-walled nanotube reinforced composites. Comp Mater Sci.
2004; 31(3-4): 225-236.
23. Lordi V, Yao N. Molecular mechanics of binding in carbon-nanotube polymer composites. J Mater Res. 2000; 15(12): 2770-2779.
24. Frankland SJV, Caglar A, Brenner DW, Griebel M. Molecular simulation of the influence of chemical cross-links on the shear strength of carbon nanotube-polymer interfaces. J Phys Chem B. 2002; 106(12): 3046-3048.
44
25. Chowdhury SC, Okabe T. Computer simulation of carbon nanotube pull-out from polymer by the molecular dynamics method. Compos Part A-Appl S. 2007; 38(3):
747-754.
26. Zheng Q, Xia D, Xue Q, Yan K, Gao X, Li Q. 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. Li C, Chou TW. Multiscale modeling of carbon nanotube reinforced polymer composites.
J Nanosci Nanotechno. 2003; 3(5): 423-430.
28. Lutsko JF. Stress and elastic constants in anisotropic solids: Molecular dynamics techniques. J Appl Phys. 1988; 64(3): 1152-1154.
29. Cormier J, Rickaman JM, Delph TJ. Stress calculation in atomistic simulations perfect and imperfect solids. J Appl Phys. 2001; 89(1): 99-104.
30. Suzuki K, Nomura S. On elastic properties of single-walled carbon nanotubes as composite reinforcing fillers. J Compos Mater. 2007; 41(9): 1123-1135.
31. Lau KT, Chipara M, Ling HY, Hui D. On the effective elastic moduli of carbon nanotubes for nanocomposite structures. Composites B. 2004; 35(2): 95-101.
32. Rappe AK, Casewit CJ. Molecular mechanics across chemistry. Sausalito, California:
University Science Books; 1997.
33. Li C, Chou TW. A structural mechanics approach for the analysis of carbon nanotubes.
Int J Solids Struct. 2003; 40(10): 2487-2499.
34. Cornell WD, Cieplak P, Bayly CI, Gould IR, Merz Jr KM, Ferguson DM, Spellmeyer DC, Fox T, Caldwell JW, Kollman PA. A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J Am Chem Soc. 1995; 117(19):
5179-5197.
35. Smith W, Forester TR. DLPOLY-2.13 manual. 2001.
36. Mayo SL, Olafson BD, Goddard III WA. DREIDING: a generic force field for molecular simulations. J Phys Chem. 1990; 94(26): 8897-8909.
37. Xiao JR, Gama BA, Gillespie Jr. JW. An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes. Int J Solids Struct. 2005;
42(11-12): 3075-3092.
38. Agrawal PM, Sudalayandi BS, Raff LM, Komanduri R. A comparison of different methods of Young’s modulus determination for single-wall carbon nanotubes (SWCNT) using molecular dynamics (MD) simulations. Comp Mater Sci. 2006; 38(2): 271-281.
45
39. Chen WH, Cheng HC, Hsu YC. Mechanical properties of carbon nanotubes using molecular dynamics simulations with the inlayer van der waals interactions. CMCS.
2007; 20(2): 123-145.
40. Krishnan A, Dujarin E, Ebbesen TW, Yianilos PN, Treacy MMJ. Young’s modulus of single-walled nanotubes. Phys Rev B. 1998;58(20):14013-14019.
41. Battezzatti L, Pisani C, Ricca F. Equilibrium conformation and surface motion of hydrocarbon moleculars physisorbed on graphite. J Chem Soc. 1975; 71: 1629-1639.
42. Tserpes KI, Papanikos P. Finite element modeling of single-walled carbon nanotubes.
Composites B. 2005; 36(5): 468–477.
43. Wang Y, Wang XX, Ni X. Atomistic simulation of the torsion deformation of carbon nanotubes. Modell Simul Mater Sci Eng. 2004; 12(6): 1099-1107.
44 Gere JM. Mechanics of materials. 5th ed. Cheltenham, UK: Nelson Thornes; 2001.
45. Melchionna S, Ciccotti G, Holian BL. Hoover NPT dynamics for systems varying in shape and size. Mol Phys. 1993; 78(3): 533-544.
46. Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 1973; 21(5): 571-574.
47. Allen MP, Tildesley DJ. Computer simulation of liquids. Oxford: Clarendon Press; 1987.
48. Adnan A, Sun CT, Mahfuz H. A molecular dynamics simulation study to investigate the effect of filler size on elastic properties of polymer nanocomposites. Compos Sci Technol. 2007; 67(3–4):3 48–56.
49. Ashby MF, Jones DR. Engineering materials 1: an introduction to their properties and applications. Oxford: Butterworth; 1996.
50. Gedde, UW. Polymer physics. 1995, Chapman and Hall, London.
51. Shames IH, Cozzarelli FA. Elastic and Inelastic Stress Analysis. 1992. Prentice-Hall, New Jersey.
52. Dunn ML, Ledbetter H. Elastic moduli of composites reinforced by multiphase particles.
J Appl Mech 1995;62(4):1023-1028.
53. Hori M, Nemat-Nasser S. Double-inclusion model and overall moduli of multiphase composites. Mech Mater 1993;14(3):189-206.
54. Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond Ser A. 1957;241(1226):376-396.
46
55. Qiu YP, Weng GJ. On the application of Mori-Tanaka’s theory involving transversely isotropic spheroidal inclusions. Int J Eng Sci. 1990; 28(11): 1121-1137.
56. Cadek M, Coleman JN, Ryan KP, Nicolosi V, Bister G, Fonseca A, Nagy JB, Szostak K, Béguin F, Blau WJ. Reinforcement of polymers with carbon nanotubes: the role of nanotube surface area. Nano Lett. 2002;4(2):353–356.
57. Sun ZH, Wang XX, Soh AK, Wu HA. On stress calculations in atomistic simulations.
Modelling Simul Mater Sci Eng. 2006; 14(3): 423-431.
58. Basinski ZS, Duesbery MS, Taylor R. Influence of shear stress on screw dislocations in a model sodium lattice. Can J Phys. 1971;49(16):2160-2180.
47
Table 2-1. The effect of van der Waals force on axial Young’s modulus of the SWCNTs with three different radii.
Radius(Å) Young’s modulus with van der
Waals force (GPa)
Young’s modulus without van der Waals force (GPa)
3.9 790.56 794.11
5.5 791.71 793.84
7.1 790.45 792.09
Table 2-2. Mechanical properties of equivalent solid cylinder with three different CNT radii.
Radius(Å) E1 (GPa) G12 (GPa) ν12 E2 (GPa) ν23
3.9 1382.5 1120 0.272 645 0.2
5.5 981.5 779.2 0.27 504 0.2
7.1 759.9 596.3 0.27 425 0.2
Table 2-3. The transverse properties of equivalent solid cylinder with three different CNTs radii.
Radius(Å) E2(GPa) ν23
3.9 645 0.2
483 0.4
5.5 504 0.2
377 0.4
7.1 425 0.2
319 0.4
48
Table 2-4. Two different mechanical property groups of equivalent solid cylinder.
E1 (GPa) G12 (GPa) ν12 E2 (GPa) ν23
Material 1 1382.5 1120 0.272 645 0.2
Material 2 1382.5 1120 0.272 483 0.4
Table 2-5. Mechanical properties of nanocomposites based on different mechanical property groups of equivalent solid cylinder.
E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) ν12
Composites 1 113.98 4.54 1.79 1.9 0.3
Composites 2 113.98 4.51 1.8 1.9 0.3
Table 3-1. Non-bond gap and normalized non-bonded energy in SWCNTs/polyimide nanocomposites with various CNTs radii.
Radius(Å) Non-bond gap (Å) Normalized non-bonded energy (j/m2)
3.9 3.333 0.3560
5.5 3.236 0.3269
7.1 3.158 0.3142
49
Table 3-2. The elements in the stiffness matrix for the SWCNTs/polyimide nanocomposites calculated from MD simulation.
Radius(Å) C11(GPa) C22(GPa) C33(GPa) C12(GPa) C13(GPa) C23(GPa)
3.9 95.47 10.39 11.14 6.645 6.45 6.95
5.5 72.79 10.47 10.83 6.745 6.965 7.235
7.1 59.14 10.15 10.39 6.75 6.52 7.10
Table 3-3. The elements in the stiffness matrix for the neat polyimide polymer calculated from MD simulation.
C11(GPa) C22(GPa) C33(GPa) C12(GPa) C13(GPa) C23(GPa)
Polyimide 8.89 8.92 9.54 6.245 6.375 6.175
Table 3-4. Material properties of neat polyimide polymer.
E(GPa) ν
Polyimide 4.01 0.407
50
Table 3-5. Comparing of longitudinal moduli of SWCNTs/polyimide nanocomposites obtained from MD simulation, Mori-Tanaka model and three-phase model.
Radius(Å) Volume fraction
(%)
MD simulation (GPa)
Mori-Tanaka Model (GPa)
Three-phase model (GPa)
3.9 6.28 90.6 90.6 92.9
5.5 6.43 56.6 66.9 54.7
7.1 6.67 54.2 54.7 55.8
Table 3-6. Comparing of transverse moduli of SWCNTs/polyimide nanocomposites obtained from MD simulation, Mori-Tanaka model and three-phase model.
Radius(Å) Volume fraction
(%)
MD simulation (GPa)
Mori-Tanaka Model (GPa)
Three-phase model (GPa)
3.9 6.28 6.22 5.28 6.32
5.5 6.43 5.86 5.27 5.90
7.1 6.67 5.29 5.28 5.74
Table 4-1. Comparison of longitudinal moduli of CNTs/polyimide nanocomposites with different degrees of vdW interactions.
Degrees of vdW interaction Longitudinal modulus (GPa)
0.01 vdW 4.196
0.1 vdW 4.212
1 vdW 4.295
5 vdW 4.91
51
Table 4-2. Comparison of longitudinal moduli of CNTs/polyimide nanocomposites with different interfacial adhesions.
Interfacial adhesions Longitudinal modulus (GPa)
vdW interaction 4.3
Surface modification 4.52
Covalent bond 4.41
52
av
1av
2θ C v
ha
1n v
a
2m v
Figure 1-1. 2D graphene sheet with nanotube parameter.
53
(a)
(b)
(c)
Figure 1-2. Illusion of the atomic structure of an (a) an armchair, (b) a zigzag and (c) a chiral nanotube.
54
o
r
o
r
ro ro
(a) (b)
Figure 2-1. Schematic of SWCNTs cross-section (a) cylindrical shell model (b) cylindrical solid model
1
2
3 4.26Å
85.2Å
Figure 2-2. Schematic of zigzag type (18,0) SWCNTs unit cell.
55
Figure 2-3. A schematic representation of the inter-atomic potential (a) bond stretch, (b) angle potential, (c) dihedral potential, and (d) inversion potential.
φ (degree)
Figure 2-4. Comparing the torsional potential (Eq. 2.4) with Dreiding torsional potential (Eq.
2.5).
56 Energy equivalent
1
2
3 Fixed boundary
Axial deformation
Fixed boundary Axial deformation
Figure 2-5. Axial deformation was applied to the SWCNTs atomistic structures and the equivalent solid cylinder.
1
2
3 Fixed boundary
Twist angle φ
Fixed boundary Energy equivalent
Twist angle φ
Figure 2-6. Twist deformation was applied to the SWCNTs atomistic structures and the equivalent solid cylinder.
57
Radial displacement
Energy equivalent Radial displacement
Figure 2-7. Radial deformation was applied to the SWCNTs atomistic stricture and the equivalent solid cylinder.
Equivalent solid cylinder PI matrix
Effective interphase
m
Γ Ω
(a) (b)
Figure 3-1. Schematic representation of simulation process: (a) SWCNTs/polyimide molecular structure and (b) three-phase micromechanical model.
58
Figure 3-2. Sketch of polyimide monomer unit.
Simulation time (ps)
Energy(Kcal/mole)
0 25 50 75 100
-4000 -3000 -2000 -1000 0 1000
Figure 3-3. Variation of potential energy in NPT ensemble.
59
Simulation time (ps)
Temperature(K)
0 25 50 75 100
0 10 20 30 40 50
Figure 3-4. Variation of temperature in NPT ensemble.
Figure 3-5. Molecular structure of polyimide polymer.
60
r r+dr
V
rFigure 3-6. Evaluation of density distribution of polyimide polymer in the radial direction.
61
Figure 3-7. Density distribution of the SWCNTs and polyimide as well as order parameter distribution of polyimide in the radial direction with three different CNTs radii: (a) 3.9Å, (b)
5.5Å and (c) 7.1Å.
62
Figure 3-8. A schematic representation of the sub vector in the polyimide polymer.
Δz z
x
y Δθ
Figure 3-9. Schematic of the radial volume element for the non-bonded gap (Δθ = 10o, Δz = 10Å).
63
Figure 3-10. The maximum radial distance for SWCNTs atoms and the minimum radial distance for polyimide atoms within the radial volume element with three different CNT radii:
(a) 3.9Å, (b) 5.5Å and (c) 7.1Å.
64
Figure 4-1. Molecular structure of SWCNTs embedded within polyimide nanocomposites.
6.566Å
Figure 4-2. SWCNTs with surface modification of polyethylene chains.
65
Simulation time (ps)
Energy(Kcal/mole)
0 50 100 150
-4000 -2000 0 2000 4000 6000 8000 10000 12000 14000
Figure 4-3. Variation of potential energy in NPT ensemble.
Simulation time (ps)
Temperature(K)
0 50 100 150
0 20 40 60 80 100 120 140 160
Figure 4-4. Variation of temperature in NPT ensemble.
66 0.01 GPa
0.01 GPa
Z
X Y
Figure 4-5. SWCNTs/polyimide nanocomposites subjected to uniaxial loading.
Simulation time (ps)
Stress(Pa)
0 50 100 150 200
-0.004 -0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016
σxx σyy σzz
Figure 4-6. Variation of stress in modify NPT ensemble.
67
Radius distance(Å)
Polymerdensity(g/cc)
5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0.01 vdW 0.1 vdW 1 vdW 5 vdW
1.31 g/cc
Figure 4-7. Density distribution of polyimide in the radial direction associated with four different intensities of vdW interactions.
LLutsko
Average volume
t
D0
Figure 4-8. Schematic of average volume for Lutsko stress(LLtusko=10Å, t=3.4 Å).
68 Fixed boundary
Displacement
Z
X
Y
Figure 4-9. Axial deformation of SWCNTs.
imaginary plane
Z
X
Y
imaginary plane
Z
X
Y Z
X
Y
Figure 4-10. Imaginary plane assumed on the SWCNTs.
69
Z position (Å)
Axialstress(GPa)
-20 -10 0 10 20
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Potential function BDT stress Lutsko stress
Figure 4-11. Calculation of axial stress distribution along the axial (Z) direction with stress-free state.
Z position (Å)
Axialstress(GPa)
-20 -10 0 10 20
0.2 0.4 0.6 0.8 1 1.2
Potential function BDT stress Lutsko stress
Figure 4-12. Calculation of axial stress distribution along the axial (Z) direction with uniaxial loading state.
70
Z position (Å)
Axialstress(GPa)
-20 -10 0 10 20
-2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8
Potential function Lutsko stress
Figure 4-13. Axial stress distribution in the SWCNTs with 1 time vdW interaction at the stress-free state.
Z position (Å)
Axialstress(GPa)
-20 -10 0 10 20
0 0.02 0.04 0.06 0.08 0.1
Potential function Lutsko stress
Figure 4-14. Axial stress distribution in the SWCNTs with 1 time vdW interaction at the uniaxial loading state.
71
Figure 4-15. The external force distribution of SWCNTs with 1 time vdW interaction at the stress-free state.
Figure 4-16. The external force distribution of SWCNTs with 1 time vdW interaction at the uniaxial loading state.
72
Z position (Å)
Axialstress(GPa)
-20 -10 0 10 20
0 0.02 0.04 0.06 0.08 0.1
5 vdW 1 vdW 0.1 vdW 0.01 vdW
Figure 4-17. Axial stress distribution in the SWCNTs associated with four different vdW interactions (Lutsko stress formulation).
Z position (Å)
Axialstress(GPa)
-20 -10 0 10 20
0 0.02 0.04 0.06 0.08 0.1
5 vdW 1 vdW 0.1 vdW 0.01 vdW
Figure 4-18. Axial stress distribution in the SWCNTs associated with four different vdW interactions (Potential function).
73
Z position (Å)
Axialstress(GPa)
-20 -10 0 10 20
0 0.03 0.06 0.09 0.12
vdW interaction Surface modification Covalent bond
Figure 4-19. Comparison of axial stress distribution of SWCNTs with three different interfacial adhesions (vdW interaction, surface modification and covalent bond).
Z position (Å)
Axialstress(GPa)
-20 -10 0 10 20
0 0.02 0.04 0.06 0.08 0.1 0.12
vdW interaction Surface modification Covalent bond
Figure 4-20. Comparison of axial stress distribution of SWCNTs with three different interfacial adhesions (vdW interaction, surface modification and covalent bond).