Composites: Part B 54 (2013) 133–137

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Composites: Part B

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Numerical analysis of synergistic reinforcing effect of silicananoparticle–MWCNT hybrid on epoxy-based composites

1359-8368/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compositesb.2013.04.002

⇑ Corresponding author. Fax: +86 10 64412084.E-mail address: yangxp@mail.buct.edu.cn (X. Yang).

Xiaolong Jia a, Baiyang Liu a, Lan Huang a, David Hui b, Xiaoping Yang a,⇑a State Key Laboratory of Organic-Inorganic Composites, Key Laboratory of Carbon Fiber and Functional Polymers, Ministry of Education, College of Material Science andEngineering, Beijing University of Chemical Technology, Beijing 100029, PR Chinab Department of Mechanical Engineering, University of New Orleans, New Orleans, LA 70148, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 February 2013Accepted 2 April 2013Available online 3 May 2013

Keywords:A. Particle-reinforcementA. Polymer–matrix composites (PMCs)B. Mechanical propertiesC. Numerical analysis

A novel nanostructural hybrid (SiO2–MWCNTs) composed of zero-dimensional silica nanoparticles (SiO2)and one-dimensional multi-walled carbon nanotubes (MWCNTs) were successfully prepared by multi-step functionalization. Synergistic reinforcing effect of SiO2–MWCNTs on epoxy-based composites wasinvestigated using various theoretical methods. Specifically, a novel finite element method (FEM) basedon nanoscale representative volume element (RVE) model was built up to describe the irregular geometryand reinforcing effect of SiO2–MWCNTs. In contrast with the corrected rule of mixtures and correctedHalpin–Tsai equations, the established FEM based on nanoscale RVE model was highly accurate andeffective in predicting mechanical properties for polymer-based composites reinforced by SiO2–MWCNTs.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Carbon nanotubes (CNTs) have been increasingly attractingnumerous interests in both academic and industrial fields, due totheir advantages of high specific surface and excellent mechanicalstrength, etc. [1–3]. Nevertheless, practical applications of CNTs re-mained limited because of disadvantages of the distinctively poorinterfacial adhesion and the spontaneously entangled aggregation[4,5]. Generally, chemical functionalization of CNTs surface hasbeen considered as one of effective ways to enhance the interfacialadhesion and positively affect the dispersibility of CNTs in thecomposites [6–10]. Furthermore, with the rapid development ofmaterial science and technology, the nanostructural hybrids com-posed of CNTs and other nano-reinforcements were prepared andincorporated into the polymer matrix to obtain a new generationof multiscale, multifunctional composites with high performance[11–15], as the resulting composites were synchronously rein-forced and functionalized with such nanostructural hybrids. There-fore, it was necessary to investigate the reinforcing mechanism ofthese hybrids, which provided theoretical foundation to design andfabricate high performance composites.

A great deal of theoretical works have been carried out with theaim of modeling the mechanical properties of nanofiller reinforcedcomposites [16–20], since the experimental investigation was veryexpensive due to nanoscale dimensions involved. Among these

theoretical predictions, the rule of mixtures and Halpin–Tsai equa-tions were two of the most common theories [16,17]. By usingthese two theories, mechanical properties of composites reinforcedby CNTs could be predicted. However, the reinforcement of nano-structural hybrids on composites could not be accurately predictedat all, since structural characteristics of such hybrids like nanoscaleirregular geometry was not taken into account in these traditionaltheories. Noticeably, finite element method (FEM) was an efficientand economic way for predicting the reinforcing effect of variousnanofillers on composites, since it permitted the continuum repre-sentation of nanofiller structure and matrix phase. However, to thebest of our knowledge, there was no report on predicting mechan-ical properties of irregular nanostructural hybrid reinforced com-posites using FEM up to date.

In our previous works [21,22], a novel irregular nanostructuralhybrid (SiO2–MWCNTs) composed of zero-dimensional silicananoparticles (SiO2) and one-dimensional MWCNTs was prepared.Silica nanoparticle based functionalization was known as a favor-able method for CNT reinforcement on polymer composites[23,24], since silica nanoparticles enlarged interfacial reinforcingareas of CNTs and showed the characteristics of fine compatibilitywith polymer chains of the matrix, which was beneficial toenhancing interfacial adhesion and dispersion degree of CNTs incomposites. Furthermore, zero-dimensional silica nanoparticlesshowed isotropic nanoscale reinforcing ability on the matrix andnaturally offset the shortage that the reinforcing effect of CNTswas only generated along their axis direction. Thus, this SiO2–MWCNT hybrid showed prominent reinforcement on polymer

134 X. Jia et al. / Composites: Part B 54 (2013) 133–137

matrix rather than pure MWCNTs as reported in our previousworks [21,22], which exhibited a great potential in preparing anew generation of high performance composites. In the presentstudy, several numerical methods were used to predict mechanicalproperties of epoxy composites reinforced with SiO2–MWCNTs.Specifically, to accurately describe the irregular geometry andreinforcing mechanism of SiO2–MWCNTs, the traditional theoriesof the rule of mixtures and Halpin–Tsai equations were properlycorrected and a novel FEM based on nanoscale representativevolume element (RVE) model was built up. To better understandsynergistic reinforcing effect of MWCNTs and silica nanoparticlesin SiO2–MWCNTs, the tensile modulus for SiO2–MWCNTs/epoxycomposites predicted by corrected rule of mixtures, correctedHalpin–Tsai equations and FEM, respectively, were compared withexperimental values.

2. Experimental section

The nanofillers, carboxylic multi-walled carbon nanotubes(MWCNTs) (purity P95%, diameter 15–30 nm, length 8–12 lm)and colloidal silica suspension (LUDOX AS-30) containing silicananoparticles (diameter 10–20 nm) were produced by ShenzhenNanotech Port Co., Ltd., China and Sigma–Aldrich Co., USA, respec-tively. The epoxy resin, 4,5-epoxycyclohexane-1,2-dicarboxylicacid diglycidyl ester was produced by Tianjin Jindong ChemicalPlant, China. The SiO2–MWCNT hybrid and SiO2–MWCNTs/epoxycomposites were prepared as following our previous works [21].The schematic procedure of SiO2–MWCNTs preparation is shownin Fig. 1. Morphologies of SiO2–MWCNTs were observed by TEM(JEM100CX) and SEM (S 4700), respectively. Tensile properties ofcomposites were measured by mechanical testing machine (IN-STRON 1121) according to ASTM D 638.

3. Numerical analysis of reinforcing effect of SiO2–MWCNTs oncomposites

To investigate reinforcing effect of SiO2–MWCNTs, (1) the ruleof mixtures, (2) Halpin–Tsai equations and (3) finite element meth-od (FEM) were used to predict mechanical properties of epoxycomposites reinforced with SiO2–MWCNTs in our present study.With the aim of accurately describing the characteristics of irregu-lar geometry of SiO2–MWCNTs, the traditional theories of the rule

Fig. 1. Schematic procedure of S

of mixtures and Halpin–Tsai equations were properly correctedand FEM was built up based on nanoscale representative volumeelement (RVE) model.

3.1. The rule of mixtures

The length correction factor and orientation correction factorwere used to correct the rule of mixtures [25,26]. For epoxy com-posites with randomly orientated SiO2–MWCNTs, the tensile mod-ulus of the composite could be calculated from the corrected ruleof mixtures expressed as Eqs. (1)–(3):

EC=M ¼ ðgLg0EC � EMÞVC þ EM ð1Þ

gL ¼ 1�tanhða � LC

DCÞ

a � LCDC

ð2Þ

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� 3EM

2ECInVC

sð3Þ

where EC/M, EC and EM denote the Young’s modulus of MWCNTs/epoxy composites, MWCNTs and epoxy matrix, respectively, as wellas VC, gL, g0, LC and DC represent the volume fraction of MWCNTs,the length correction factor, the orientation correction factor, theaverage length of MWCNTs and the average diameter of MWCNTs,accordingly. The measured EM and VC values of 3.3 GPa and 0.005as well as the estimated EC and g0 values of 1300 GPa and 0.2 wereused in the calculation.

3.2. Halpin–Tsai equations

The efficiency factor and orientation correction factor were usedto correct Halpin–Tsai equations [27,28]. For epoxy compositeswith randomly orientated SiO2–MWCNTs, the tensile modulus ofthe composites could be calculated from the corrected Halpin–Tsaiequations expressed as follows :

C ¼ 2 � LC

DCð4Þ

g ¼a � EC

EM� 1

a � ECEMþ C

ð5Þ

iO2–MWCNTs preparation.

Fig. 2. Simulation diagram of finite element analysis of SiO2–MWCNTs/epoxycomposites: (a) RVE model, (b) grid division of RVE model and (c) high resolution ofgrid division of RVE model. In these images, the cylinder part (blue), sphere part(red), matrix part (purple) and mesh line (white) denote MWCNTs, silica nanopar-ticles, resin matrix and grid division, respectively. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

Fig. 3. Stress distribution field under various strains in RVE model of SiO2–MWCNTs/epoxy composites: (a) X axis, (b) Y axis and (c) Z axis. In these images, thecylinder part, sphere part and matrix part denote MWCNTs, silica nanoparticles andresin matrix, respectively.

X. Jia et al. / Composites: Part B 54 (2013) 133–137 135

EC=M ¼ EM �ð1þ CgVCÞð1� gVCÞ

ð6Þ

where g and a represent the efficiency factor and orientation cor-rection factor, respectively. In this study, the estimated EC, EM anda of 1300 GPa, 3.27 GPa and 1/6, respectively, were used in the cal-culation, associated with the measured LC and DC of 10 lm and15 nm correspondingly. The density of MWCNTs (qC) was estimatedby assuming that the graphitic layers of the tube shell had the samedensity as that of the fully dense graphite structure (qg = 2.25 g/cm3) and the density of the epoxy matrix (qM) was determined tobe �1.2 g/cm3 [27]. The density and volume fraction of MWCNTscould be calculated by Eqs. 7 and 8 [26], respectively:

qC ¼qg d2

C � ðdC � 2tÞ2� �

d2C

ð7Þ

VC ¼qMWC

qC � qCWC þ qMWCð8Þ

where WC and t are the weight fraction of MWCNTs and averagewall thickness of MWCNTs. In this study, the measured t value of4 nm was used in the calculation.

3.3. Finite element method (FEM)

With the aim of predicting the modulus of resin matrix rein-forced with irregular nano-reinforcement of SiO2–MWCNTs, FEMbased on nanoscale RVE model was established in this study andexpected to describe the real combined geometry of MWCNTsand silica nanoparticles. This RVE model was considered to accordwith hexagonal fiber micromechanics [29,30]. For the numericalanalysis, the characteristic parameters of compositional materialswere determined and the related assumed conditions were setup. Assuming that mass contents of silica nanoparticles and

MWCNTs in the hybrid nano-reinforcement were equivalent, thusthe volume of silica nanoparticle was half of that of MWCNTs,which could be also coarsely judged from the SEM and TEM imagesof SiO2–MWCNTs (shown in Fig. 4). In order to simplify thecalculation, the thickness of each RVE model along the directionof carbon nanotube was selected to be 100 nm. Thus, referring tothe images of SiO2–MWCNTs (shown in Fig. 4), four silica nanopar-ticles were supposed to symmetrically distribute onto every100 nm length of carbon nanotube in this RVE model. Noticeably,such symmetrical distribution was beneficial to adjusting the vol-ume fraction of silica nanoparticle by changing the thickness ofRVE. Additionally, the space between silica nanoparticles andMWCNTs was assumed to be filled with the resin matrix. Sincethe computational complexity of finite element analysis increasedsharply with the number of unit, one eighth of finite element unitwas established in order to further simplify the computation [31],as shown in Fig. 2a. And the grid distribution and high resolution ofgrid division of SiO2–MWCNTs in RVE model were shown in Fig. 3band c, respectively.

Fig. 3 shows the stress distribution in RVE model of SiO2–MWCNTs/epoxy composites. Based on the stress distributionshown in Fig. 3, the macro-scale modulus of epoxy compositescould be obtained by calculating the corresponding stiffness ma-trix. In order to accurately describe the randomly oriented anduncontinuous distribution of SiO2–MWCNTs in the matrix, thelength correction factor and orientation correction factor wereused to calibrate the initial modulus of MWCNTs. The length cor-rection factor and orientation correction factor of MWCNTs couldbe obtained from Eqs. 2 and 3, respectively. On the basis of theabove assumptions and results, the load applied in RVE was givenby Eqs. 9 and 10 [32,33]:

�r1

�r2

�r3

�r4

�r5

�r6

9>>>>>>>>=>>>>>>>>;

8>>>>>>>><>>>>>>>>:

¼

C11 C12 C12 0 0 0C12 C22 C23 0 0 0C12 C23 C22 0 0 00 0 0 1

2 ðC22 � C23Þ 0 00 0 0 0 C66 00 0 0 0 0 C66

2666666664

3777777775

�e1

�e2

�e3

�c4

�c5

�c6

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

ð9Þ

Cij ¼ �ri ¼1V

ZVriðx1; x2; x3ÞdV ; e0

j ¼ 1 ð10Þ

Fig. 4. (a and b) SEM and (c and d) TEM images of silica nanoparticle–MWCNT hybrid, (b) and (d) were the high-magnification images.

Table 1Experimental tensile modulus of neat epoxy, MWCNTs/epoxy composites and SiO2–MWCNTs/epoxy composites and predicted tensile modulus of SiO2–MWCNTs/epoxycomposites.

Neatresin

MWCNTs/epoxycomposites

SiO2–MWCNTs/epoxycomposites

Predicted value by the rule ofmixtures

Predicted value by Halpin–Tsaiequations

Predicted valueby FEM

Tensilemodulus/MPa

3270 ± 32 3420 ± 85 3955 ± 58 4399 4219 3916

136 X. Jia et al. / Composites: Part B 54 (2013) 133–137

where �ri, Cij, �ei and �ci are the volume-average stress components,stiffness tensor, volume-average engineering strain componentsand volume-average shear strain components, respectively. In orderto determine the components Ci1 with i = 1, 2, 3, the following strainwas applied to stretch the RVE along the fiber direction (xl-direction)

e01 ¼ 1 e0

2 ¼ e03 ¼ c0

4 ¼ c05 ¼ c0

6 ¼ 0 ð11Þ

The components Ca2, with a = 1, 2, 3, were determined bysetting:

e02 ¼ 1 e0

1 ¼ e03 ¼ c0

4 ¼ c05 ¼ c0

6 ¼ 0 ð12Þ

The components Ca3, with a = 1, 2, 3, could be found by applyingthe following strain:

e03 ¼ 1 e0

1 ¼ e02 ¼ c0

4 ¼ c05 ¼ c0

6 ¼ 0 ð13Þ

Thus, the required components of Cij were determined by aver-aging the stress field as shown in Eq. (10).

Then, the stress was generated with the deformation of RVE andthe stress distribution field under various strains in the compositeswas shown in Fig. 4. Finally, the macro-scale modulus of SiO2–

MWCNTs/epoxy composites could be calculated by Eq. (14)[33,34]:

EC=M ¼ C11 �2C2

12

C22 þ C23ð14Þ

4. Results and discussion

4.1. Characterizations of silica nanoparticle–MWCNT hybrid

Fig. 4 shows SEM and TEM images of silica nanoparticle–MWCNT hybrid (SiO2–MWCNTs). As seen from SEM images, silicananoparticles were firmly adhered on the outer wall of MWCNTsalong their axis direction. Such microstructure was also observedin TEM images. Measuring from SEM and TEM images in Fig. 2,the average diameter of silica nanoparticles was around 30 nm,close to that of MWCNTs. All these images demonstrated that silicananoparticles were grafted onto the surface of MWCNTs and theaimed product of SiO2–MWCNTs was successfully obtained.Noticeably, it could be coarsely counted from SEM and TEM images

X. Jia et al. / Composites: Part B 54 (2013) 133–137 137

that four silica nanoparticles distributed onto every 100 nm lengthof carbon nanotube, which was consistent with the above assump-tion condition in RVE model of finite element method (FEM).

4.2. Comparison of experimental and predicted results

Table 1 shows experimental tensile modulus of neat epoxy,MWCNTs/epoxy composites and SiO2–MWCNTs/epoxy compositesand predicted tensile modulus of SiO2–MWCNTs/epoxy compos-ites. As shown in Table 1, tensile modulus of MWCNTs/epoxy com-posites was apparently higher than those of neat epoxy, showingthe reinforcing effect of MWCNTs. Moreover, compared toMWCNTs/epoxy composites, tensile modulus was distinctively en-hanced by 15.6% for SiO2–MWCNTs/epoxy composites, furthershowing the synergistic reinforcing effect of MWCNTs and silicananoparticles. In addition, tensile modulus of SiO2–MWCNTs/epoxy composites predicted by the corrected rule of mixtures, cor-rected Halpin–Tsai equations and FEM were 4399, 4219 and3916 MPa, respectively. It could be found that the result predictedby FEM was very close to experimental value of 3955 ± 58 MPawith the deviation of 1.0%, whereas the deviations were 11.2 and6.7% for the results predicted by the corrected rule of mixturesand corrected Halpin–Tsai equations. Therefore, the establishedFEM based on nanoscale RVE model was highly accurate and effec-tive in predicting mechanical properties for polymer-based com-posites reinforced by SiO2–MWCNTs.

5. Conclusions

A novel nanostructural hybrid (SiO2–MWCNTs) composed of sil-ica nanoparticles and multi-walled carbon nanotubes were suc-cessfully prepared. Compared with the predicted values by thecorrected rule of mixtures and corrected Halpin–Tsai equations,the predicted tensile modulus of SiO2–MWCNTs/epoxy compositesusing finite element method (FEM) based on representative vol-ume element (RVE) model was much closer to experimental value.Therefore, the established FEM based on RVE model provided a fea-sible way to predict the reinforcing effect of nano-reinforcementwith irregular geometry on polymer-based composites.

Acknowledgements

The authors are very pleased to acknowledge financial supportfrom National Natural Science Foundation of China (Grant No.50873010), the National High Technology Research and Develop-ment Program of China (Grant No. 2012AA03A203,) and Programfor Changjiang Scholars and Innovative Research Team in Univer-sity (PCSIRT).

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