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Multiwalled carbon nanotube reinforced polymer composites A. Pantano a,b,∗ , G. Modica a , F. Cappello a a

b

Universit`a degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy Massachusetts Institute of Technology, 77 Massachusetts avenue, 02139-4307 Cambridge, MA, USA Received 4 June 2007; received in revised form 29 August 2007; accepted 30 August 2007

Abstract Due to their high stiffness and strength, as well as their electrical conductivity, carbon nanotubes are under intense investigation as fillers in polymer matrix composites. The nature of the carbon nanotube/polymer bonding and the curvature of the carbon nanotubes within the polymer have arisen as particular factors in the efficacy of the carbon nanotubes to actually provide any enhanced stiffness or strength to the composite. Here the effects of carbon nanotube curvature and interface interaction with the matrix on the composite stiffness are investigated using micromechanical analysis. In particular, the effects of poor bonding and thus poor shear lag load transfer to the carbon nanotubes are studied. In the case of poor bonding, carbon nanotubes waviness is shown to enhance the composite stiffness. © 2007 Elsevier B.V. All rights reserved. Keywords: Carbon nanotube; Modeling; Composite; Finite element method

1. Introduction The sp2 carbon–carbon bond in the basal plane of graphene is the stiffest and strongest in nature. Carbon nanotubes (CNTs) possess an ideal arrangement of these bonds in their cylindrical and nearly defect-free structures, and hence approach the maximum theoretical tensile stiffness and strength. Both experiments and atomistic simulations have confirmed that CNTs meet expectations for an extremely high modulus, e.g., refs. [1–4]. These extraordinary mechanical properties, together with high ratios of geometric aspect, stiffness-to-weight, and strengthto-weight, all point to carbon nanotubes as potentially ideal reinforcing agents in advanced composites. The potential of carbon nanotubes for application as traditional structural reinforcements, however, depends critically on the ability (a) to disperse the CNTs homogeneously throughout the matrix and (b) to transfer mechanical load from the matrix to the CNTs. Even for well-dispersed CNTs, if the interfacial bonding between the CNT and matrix is weak, load is not optimally transmitted from the matrix to the CNTs, either through end∗

Corresponding author at: Dipartimento di Meccanica, Universit`a degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy. Tel.: +39 091 6657132; fax: +39 091 484334. E-mail addresses: [email protected], [email protected] (A. Pantano). 0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.08.078

tensile or lateral-shear traction, so the CNTs are substantially hindered in stiffening or strengthening the composite through traditional shear lag-based mechanisms. Such CNTs effectively act as nanostructured holes or matrix flaws, and many potential benefits of the extraordinary CNT properties remain unrealized in the composite. These difficulties dramatically impact manufacturing of CNT-based composites of high mechanical performance. There are two main kinds of nanotubes, e.g., refs. [1–4]: single-walled nanotubes (SWNTs), individual cylinders of 1–2 nm in diameter, which are actually a single molecule and multiwalled nanotubes (MWCNTs), which are a collection of several concentric graphene cylinders, a “Russian doll” structure, where weak Waals forces bind the tubes together. Here we focus our attention on MWCNTs-based composites. Solution-processed MWCNT composites generally exhibit good dispersion and superior stiffening per unit MWCNT volume fraction, as typified by early results on polystryrene (PS)/MWCNT thin films [5]. In contrast, for melt-processed bulk MWCNT composites, most experiments have demonstrated only modest improvement in composite stiffness and/or strength after incorporation of MWCNTs into polymers, e.g., refs. [6–8]. For example, Andrews et al. [6] used shear mixing to produce PS/MWCNT composites, obtaining for a 5% volume fraction of MWCNTs, only 15% increase in the tensile modulus, Ecomp , compared to that of the neat matrix, Ematrix . Xia et al. [7] prepared a polypropylene (PP)/MWCNT composite in

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which good dispersion was obtained using a novel solid-state mechanochemical pulverizing process. A modest 8.8% increase in Ecomp was observed in a composite with a 3 wt% loading of MWCNTs. Similar results were found by Song et al. [8] in epoxy composites containing different dispersions of MWCNTs: a 17% increase in Ecomp was found for 1.5 wt% MWCNTs. Such cases of limited stiffness improvement, despite success in dispersing the MWCNTs, are likely a result of weak bonding between the MWCNTs and the matrix. Several numerical models, e.g., refs. [9–12] have been developed in attempts to improve the understanding of the stiffening effects of MWCNTs in a polymer matrix. These studies are based on micromechanical models, since atomistic/molecular models are too computationally demanding for direct application to MWCNT composites. We emphasize that all of these continuum micromechanical models either explicitly [10–12] or effectively [9] adopt an assumption of an ideally bonded interface between the compliant polymer matrix and the stiff MWCNTs, and often lead to predictions of overall composite stiffness that are quite optimistic compared to experimental results. In such cases, factors tending to soften the predictions, including MWCNT bundle clustering, waviness of the MWCNTs [10–12], and reduced effective length of a reinforcing MWCNT segment [9], are noted, and their effects parametrically reconsidered within the adopted modeling framework. For example, Odegard et al. [9] developed a small cylindrical representative volume element (RVE) of a straight SWNT segment surrounded by matrix polymer, using an equivalentcontinuum model (truss element based) of SWNT and polymer based on the equilibrium molecular structure obtained from MD simulations. In turn, the cylindrical RVE was modeled as a homogeneous, transversely isotropic equivalent-continuum effective fiber whose stiffnesses were determined by matching the average response of the equivalent-truss RVE under corresponding boundary conditions. Brinson et al. [10] analyzed the effects of MWCNT waviness on the composite reinforcing effects of ideally bonded MWCNTs. Their finite element (FE) simulations modeled MWCNT tubes as embedded sinusoidally shaped linear elastic solids perfectly bonded to the matrix. They determined the reduction in composite stiffening due to MWCNT waviness, in comparison to the level of reinforcement provided when straight. Elsewhere [11], Brinson’s group modeled a wavy MWCNT segment as an infinitely long, ideally bonded sinusoidal-shaped fiber, and the dilute strain concentration tensor for the fiber domain was extracted directly from FE solutions. These studies showed that the effective axial strain in the bonded wavy MWCNT increases dramatically with increases in the ratio of sinusoidal amplitude over wavelength, leading to correspondingly dramatic reductions in stiffening. In a related study, Shi et al. [12] applied the MoriTanaka model to a linear elastic polymer matrix reinforced by a distribution of dispersed MWCNTs, modeling each MWCNT as a long fiber with transversely isotropic elastic properties. It must be noted, however, that such Eshelby-based treatments of piecewise homogeneous strain along a curved fiber-like heterogeneity cannot account for any bending deformation induced in the MWCNTs.

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These micromechanics models each assumed perfect bonding between the stiff MWCNTs and the compliant polymer matrices, leading to predictions of the overall mechanical properties of the composites that are highly optimistic with respect to experimental results. Usually, the interface between the MWCNTs and the matrix polymer is characterized by only a non-bonded van der Waals interaction capable of producing only modest cohesive strength and almost null shear strength, as investigated by Frankland et al. [13]. However, since MD simulations are limited to extremely small time frame (10−12 –10−9 s), their estimates may be affected by high loading rates resulting in a shear strength higher than the real one. The MD simulations of Frankland et al. [13] also showed that interfacial shear strengths could be enhanced by over an order of magnitude through MWCNT functionalization and formation of MWCNT/matrix covalent bonds at less than 1% of the MWCNT carbon atoms. Coleman et al. [14] note that recent studies on functionalized MWCNT composites have shown considerable promise, and this topic remains at the forefront of research. For the present, however, we will not consider functionalized MWCNTs, whose price is usually one order of magnitude higher than already expensive not functionalized ones, instead focusing on the weakened bonding of untreated interfaces. Here, the effects of carbon nanotube waviness and of matrix/MWCNT interfacial interaction on composite stiffness are investigated using micromechanical analysis. We recognize and acknowledge the possibility of weak matrix/MWCNT bonding, and thus poor shear lag load transfer to the MWCNT, as severely limiting the enhancement of composite stiffness. However, even in the case of weak interfacial bonding, transverse shear load can still be transmitted to wavy MWCNTs through lateral normal interactions with the matrix, and, as in beam theory, these transverse forces within the MWCNT will generate locally varying bending moments along the MWCNTs during macroscopic deformation of the composite. Strain energy associated with local bending of wavy MWCNTs, both within the MWCNTs and in the surrounding matrix, can thus provide a novel mechanism for enhancing polymer composite stiffness, even in the presence of weak interfacial bonding. 2. Modeling procedures The micromechanical model requires a spatial realization of an RVE of the composite microstructure, constitutive descriptions of both the MWCNTs and the polymer matrix, a mechanical characterization of the bonding between MWCNTs and matrix, and proper boundary conditions on RVE. The commercial program ABAQUS was used to create and analyze three-dimensional FE models of several RVEs. The spatial arrangement of MWCNTs within the RVEs is here simplified to a regular distributed form of staggered arrays containing isolated, and generally aligned MWCNTs. Such staggered arrays capture major effects of dispersed particle/matrix interactions better than do so-called stacked arrays [15], whose main problem is their strong “series” non-uniformity: unreinforced layers of pure matrix alternate along the tensile loading direction with layers of high particle concentration, resulting in strongly

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Fig. 1. Finite element model of RVE. For the particular model shown, a/d = 15 and a/λ = 0.227; t = 44 nm, H = 1800 nm, W = 1400 nm, a = 300 nm, λ = 1320 nm; and Vf = 0.0132. (a) Frontal view, (b) three-dimensional view, and (c) see-through view.

heterogeneous deformation within the RVE that is uncharacteristic of a macroscopically homogeneous composite. The adopted RVE, when periodically replicated in space, produces a staggered arrangement of overlapping MWCNTs in their alignment direction 2. Each FE model is one-eighth part of the identified RVE, of total volume V = tHW, and contains four sinusoidalshaped half-MWCNTs. Fig. 1(a–c) shows one of the model geometries employed. The sinusoidal waviness of the MWCNTs embedded in the model can be characterized by two ratios, a/λ and a/d, where d is the MWCNT diameter, λ is the wavelength, and a/2 is the amplitude. Three orthogonal faces of the FE model are subjected to symmetry boundary conditions of zero shear traction and zero normal displacement. The remaining faces are shear-free but are constrained to remain planar, and only rotation about the three axes is allowed at the shell nodes of the half-MWCNTs that are on RVE boundaries parallel to the 1–2 plane. Prescribed normal axial displacement, δ, in the 2 direction drives the RVE strain ε = δ/H, and the work-conjugate reaction force, F, was used to extract macroscopic stress as σ = F/tW, leading to modulus E = σ/ε. A previously developed FE procedure for modeling mechanical behaviour of carbon nanotubes is used [16–19]. The approach realizes the extreme computational savings necessary to effectively model MWCNTs embedded within the matrix without losing significant accuracy with respect to atomistic methods. Individual tubes are successfully modeled using shell finite elements with a specific pairing of elastic properties and effective mechanical thickness of the tube wall. The effects of non-bonded forces are simulated with special interaction elements that are crucial in maintaining the interwall separation. Further details and documentation of this approach to mechanical modeling of MWCNTs are available in refs. [16–19]. The polymer matrix is modeled as an isotropic linear elastic continuum. Several combinations of matrix Young’s modulus, Ematrix , and Poisson ratio, νmatrix , have been considered.

As investigated by Frankland et al. [13] using MD simulations, only weak non-bonded interactions exist between MWCNTs and several polymers, leading to low fiber–matrix shear stiffness and strength. In the present calculations, neither tensile nor shear bonding between the MWCNT and the polymer are considered; however, contact resistance to interpenetration is retained, so the results should represent a lower limit to reinforcing properties of the MWCNTs in the considered models. Assuming perfect bonding, the addition of dispersed MWCNTs could always be expected to stiffen the polymer composites. But if only negligibly small (or zero) tangential traction is transmitted across the MWCNT/polymer interface, then local sliding of matrix with respect to MWCNTs would occur in response to loading. In such cases, the polymer composite can become stiffer, more compliant, or show little difference, depending on the bending rigidity and initial curvature of the MWCNTs in comparison to matrix stiffness, as well as the MWCNT aspect ratio and volume fraction. 3. Numerical results Here we first utilize the modeling technique to validate the approach against the MWCNT laboratory experiments of Xia et al. [7], Song et al. [8], and Andrews et al. [6]. Then the effects of MWCNTs waviness, outer diameter, volume fraction and of the matrix stiffness on the reinforcing capabilities of MWCNTs is investigated. Song et al. [8] prepared an epoxy composite with different volume fractions of MWCNTs whose outer diameters were around 20 nm. A high resolution image of the fracture surface for the prepared composites, Song et al. [8, Fig. 8b], as been used to imaging a representative portion of the material and developing a proper waviness distribution function characterizing the magnitude and pervasiveness of the MWCNT waviness, leading to an appropriate multiphase numerical model. In Table 1, the

A. Pantano et al. / Materials Science and Engineering A 486 (2008) 222–227 Table 1 Geometrical parameters of 10 different MWCNTs seen in Song et al. [8, Fig. 8b] MWCNTs

λ (nm)

a (nm)

douter (nm)

a/λ

λ/d

a/d

1 2 3 4 5 6 7 8 9 10

698.6 489.0 383.6 215.1 328.8 246.6 402.7 445.2 321.9 242.5

178.1 137.0 95.9 49.3 100.0 41.1 57.5 105.5 71.2 46.6

32.9 23.3 15.1 19.2 13.7 16.4 23.3 19.2 19.2 19.2

0.25 0.28 0.25 0.23 0.30 0.17 0.14 0.24 0.22 0.19

21.25 21.00 25.45 11.21 24.00 15.00 17.29 23.21 16.79 12.64

5.42 5.88 6.36 2.57 7.30 2.50 2.47 5.50 3.71 2.43

Average

377.4

88.2

20

0.23

19

4.4

geometry of 10 different MWCNTs seen in Song et al. [8, Fig. 8b], are reported. Starting from an average a/d = 4.4, the several RVEs of the composite have been modeled. The geometry of the models is reported in Table A1 of Appendix A. Three different values of the outer diameter have been considered since the MWCNTs used for the composite have an outer diameter of 20 nm only on average, but the individual value can vary as can be seen in Song et al. [8, Fig. 8b]. Numerical simulations and experimental results are compared in terms of resulting stiffening effect of the MWCNTs in the polymer matrix. Fig. 2 shows the predicted elastic modulus of the composite Ecomp at the three different volume fractions used by Song et al. [8] in preparing the composites. Numerical results underestimate experimental ones, as expected from assuming zero shear lag load transfer from the polymer to the MWCNT, but agreement is rather good. The second experimental work that has been considered investigates the accuracy of the numerical approach is that of Xia et al. [7]. They prepared a polypropylene (PP)/MWCNT composite with MWCNTs whose outer diameter varied from 20 to 30 nm and the inner diameter was from 5 to 10 nm. The polymer matrix has a Young’s modulus of 837 MPa and the volume fraction of the MWCNTs present in the composite is equal to 3%. There are no figures in the work of Xia et al. [7] that allow

Fig. 2. Experimental vs. numerical results for the composites studied by Song et al. [8].

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Table 2 Experimental vs. numerical results for the composites studied by Xia et al. [7] Waviness

Ecomp (MPa)

Error (%)

a/d = 5 a/d = 10 a/d = 15

886.56 911.05 830.09

−2.7 0.0 −8.9

for the determination of an appropriate average waviness distribution. Thus, while maintaining the volume fraction at 3%, three models with three different ratios a/d = 5, a/d = 10 and a/d = 15 are considered. Table A2 in Appendix A, illustrate the geometrical parameters of composite models used in the calculations. Mechanical tests performed by Xia et al. [7] on the composite under investigation, determined a Young’s modulus of 911 MPa that is very close to the outcomes of the numerical approach, see Table 2. After establishing the proposed numerical model against experimental results, we explore the effects of the main parameters that determine the reinforcing capabilities of MWCNTs in polymer matrices. Due to the large number of parameters involved in the study, we focus our attention only on a limited number of cases. We consider two MWCNTs outer diameters, the volume fraction, and the properties of the polymer matrix are fixed while the waviness of the MWCNTs vary from almost straight MWCNTs to a significant curled ones. Here we show the results for a polymer matrix with a Young’s modulus Ematrix = 68 MPa, MWCNTs outer diameter of 25 and 40 nm, and a volume fraction of 5%. The complete geometry of the models is described in Table A3 in Appendix A. Fig. 3 shows how the stiffening effect of the MWCNTs changes with a/λ in terms of resulting Young’s modulus of the composite Ecomp . It can be noticed that Ecomp first grows until a max is reached then it gradually decreases. The inclination of the contact surface between MWCNT wall and matrix depends on the MWCNT shape, for a MWCNT with sinusoidal variation of its shape increasing the amplitude and the frequency of the wave (large a/λ) should determine a stiffer RVE. This is because of the larger contact areas with less inclination with respect to the direction of deformation of the matrix. But this cannot be generalized since the MWCNT is not completely rigid, long

Fig. 3. Young’s modulus of the composite as function of the MWCNTs waviness and outer diameter with Ematrix = 68 MPa.

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Acknowledgements This research was funded by the Ministero dell’Universit`a e della Ricerca “Rientro dei Cervelli” funding. Appendix A See Tables A1–A3. Table A1 Geometrical parameters of the FE models reproducing the composite from Song et al. [8] Fig. 4. Young’s modulus of the composite as function of the MWCNTs waviness and outer diameter with Ematrix = 1.9 GPa.

portions of the MWCNT almost perpendicular to the matrix also means longer moment arms where the polymer applies its forces. The second observation regards how the change in the outer diameter of the MWCNTs affect the variation of Ecomp with the volume fraction. As expected the MWCNTs with larger diameter due to their higher bending stiffness have better reinforcing effects. Fig. 3 also shows that the maximum in Ecomp is reached for a lower volume fraction in the case of MWCNTs of smaller diameter, a/λ = 0.15 for d = 40 nm and a/λ = 0.11 for d = 25 nm. This happens since the lower bending stiffness of the smaller diameter MWCNTs determine a smaller value of waviness at which the MWCNT becomes weak enough to be deformed by the loading on polymer matrix. Finally, the effect of the Young’s modulus of the polymer matrix on the reinforcing capabilities of MWCNTs is explored by replicating the numerical calculations reported (Fig. 3) with a stiffer polymer matrix, Ematrix = 1.9 GPa. Results are illustrated in Fig. 4. Observations made for the weaker matrix regarding the presence of a maximum and the effect of the outer diameter is still valid. Figs. 3 and 4 prove that the advantage of adding MWCNTs is more significant in polymer matrices with lower elastic modulus. The overall composite stiffness found for the model having MWCNT 25 nm in diameter and a/λ = 0.05 is slightly lower than the pure polymer. This shows that if at the MWCNT/polymer interface there is very small tangential friction or no friction, MWCNTs effectively act as nanostructured holes or matrix flaws. In presence of zero shear lag load transfer, the RVE will results stronger as a consequence of the insertion of MWCNTs if the weakening effect due to the discontinuity in the polymer material is overcompensated by the presence of a stiff inclusion hard to deform. If the traditional assumption of perfect bonding between MWCNTs and polymer matrices is implemented in the models, the improvement in the elastic modulus of the composite is one order of magnitude, far beyond any experimental results. We can conclude that in the presence of weak bonding, and thus poor shear lag load transfer to the MWCNTs, the composite stiffness enhancement can be best achieved through the bending energy of the MWCNT rather than through the axial stiffness and energy of the MWCNTs.

douter (nm)

Vf (%)

a (nm)

λ (nm)

a/λ

λ/d

H (nm)

W (nm)

t (nm)

15 15 15 20 20 20 25 25 25

0.5 1 1.5 0.5 1 1.5 0.5 1 1.5

66 66 66 88 88 88 110 110 110

1320 1320 1320 1320 1320 1320 1320 1320 1320

0.05 0.05 0.05 0.1 0.1 0.1 0.08 0.08 0.08

88 88 88 66 66 66 52.8 52.8 52.8

1800 1700 1700 2200 1770 1800 2600 2100 1800

1500 800 550 1700 1350 900 2000 1600 1400

45 42 42 58 44 44 65 50 45

Table A2 Geometrical parameters of the FE models reproducing the composite from Xia et al. [7] douter (nm)

dinner (nm)

Vf (%)

a (nm)

λ (nm)

a/λ

a/d

H (nm)

W (nm)

t (nm)

25 25 25

10 10 10

3 3 3

125 250 375

1320 1320 1320

0.09 0.19 0.28

5 10 15

1700 1700 1700

1000 1200 1700

33 34 30

Table A3 Geometrical parameters of the FE models for the parametric study douter (nm)

dinner (nm)

Vf (%)

a (nm)

λ (nm)

a/λ

λ/d

H (nm)

W (nm)

t (nm)

40 40 40 40 40 25 25 25 25 25

20 20 20 20 20 10 10 10 10 10

5 5 5 5 5 5 5 5 5 5

330 264 200 132 80 66 145 198 264 330

1320 1320 1320 1320 1320 1320 1320 1320 1320 1320

0.25 0.20 0.15 0.10 0.06 0.05 0.1098 0.15 0.2 0.25

33 33 33 33 33 52.8 52.8 52.8 52.8 52.8

1770 1800 1770 1750 1700 1770 1770 1700 1700 1700

1400 1350 920 1300 1200 550 650 900 800 1000

50 45 50 40 40 35 30 28 32 32

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