Computational Materials Science 79 (2013) 368–376

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Computational Materials Science

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Understanding the effect of CNT characteristics on the tensile modulusof CNT reinforced polypropylene using finite element analysis

0927-0256/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.commatsci.2013.06.046

⇑ Corresponding author at: George Woodruff School of Mechanical Engineering,Georgia Institute of Technology, Atlanta, GA 30332, USA. Tel.: +1 404 385 3446; fax:+1 404 894 9342.

E-mail address: kyriaki.kalaitzidou@me.gatech.edu (K. Kalaitzidou).

Md A. Bhuiyan a, Raghuram V. Pucha a, Johnny Worthy b, Mehdi Karevan a, Kyriaki Kalaitzidou a,c,⇑a George Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USAb School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USAc School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

a r t i c l e i n f o

Article history:Received 30 January 2013Received in revised form 2 May 2013Accepted 19 June 2013Available online 17 July 2013

Keywords:CNT/PP nanocompositesCNT/PP interphaseFinite element analysisDistribution functionsTensile modulus

a b s t r a c t

The focus of this study is to understand the reinforcing efficiency of carbon nanotubes (CNT) in polymersthrough finite element modeling. The novelty of our work is that the probability distribution functions ofCNT diameter, orientation, dispersion and waviness, determined through image analysis, are incorpo-rated in the finite element model allowing thus for fundamental understanding of how the CNT charac-teristics affect the tensile modulus of CNT reinforced polypropylene (PP) composites. The presence ofinterphase, confirmed by atomic force microscopy, is also accounted for in our model. The image analysisapproach utilizes scanning electron microscopy images of the CNT/PP composites made by melt mixingand injection molding. Model predictions are compared with the data obtained experimentally accordingto ASTM D638. A good agreement between the model predictions and experimental data is observed.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The remarkable properties of CNT along with their high aspectratio and low density suggest that CNT are excellent candidatesfor reinforcing a variety of materials including polymers [1]. How-ever, the reinforcing efficiency of CNT depends on a variety of fac-tors such as their homogeneous dispersion, orientation andwaviness within the polymer [2]. Non homogeneous dispersionmeans presence of agglomerates which due to their reduced aspectratio and the relative slippage of CNT within the agglomerate leadto significant reduction in strength [3] due to insufficient loadtransfer at the CNT–polymer interface [4]. Furthermore, CNT wav-iness and random orientation within polymer reduce their effec-tive length along the applied load direction and thus thereinforcing efficiency [2,5,6].

These difficulties impede the manufacture of CNT-based com-posites with enhanced mechanical properties. As a consequence,despite some encouraging results [7], many experiments havedemonstrated only modest improvement in the mechanical prop-erties upon addition of CNT to polymers [8–10]. Therefore, it isnecessary to investigate the impact that the CNT parameters haveon the macroscopic properties of CNT-based composites. However,

determining the distribution of these parameters within the poly-mer and their relative effect on the composites effective modulusthrough experimental means is extremely difficult even with themodern nanotechnology tools [11–13]. Computational approachessuch as image analysis of electron microscopy (EM) images and fi-nite element analysis (FEA) can provide the solution to this prob-lem. Analysis of scanning electron microscopy (SEM),transmission electron microscopy (TEM) and atomic force micros-copy (AFM) images to determine CNT diameter and length distri-bution in polymers is reported in [14–17]. SEM images of shortmicro-size fiber reinforced polymer composites have also beenused to evaluate the three dimensional (3D) fiber orientation dis-tributions in the polymer [18]. Continuum mechanics based FEAis suitable for large scale analysis and thus capable of modelingnanocomposites at macroscale [19,20]. Several studies have alsodemonstrated that FEA is an adequate tool to investigate themechanical behavior of CNT composites [19,21–23].

In this study, we develop and employ a combined image analy-sis approach and finite element method (FEM) to determine theeffective tensile modulus of CNT reinforced polypropylene (PP)composites. The distribution function of the CNT parameters aredetermined using SEM images captured from the polished andetched surfaces of CNT/PP composites fabricated through meltmixing and injection molding. To the best of our knowledge thereis no prior study on FEA of nanocomposites’ tensile modulus using2-D SEM images to extract 3-D information and determine thedistribution functions of the nano-reinforcements’ parameters

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including dispersion, waviness and orientation. The FE model usedconsists of multiple CNT embedded within the PP matrix and thecomputational results are compared with experimental ones. Thisimage analysis approach coupled with FEA can be used for variouscomposite systems providing up-front guidelines for manufactur-ing of nanocomposites with engineered properties.

2. Experimental

2.1. Materials used and fabrication of composites

Polypropylene powder with the trade name Pro-fax 6301 and12 g/10 min, melt flow index (purchased from Basell, Lansing MI,USA) is used in this study as the matrix. Multi walled CNT (pur-chased from Cheap Tubes, VT, USA) with outer diameter (OD)20–30 nm, inner diameter (ID) 5–10 nm, length 10–30 lm, purity>95 wt%, and ash <1.5 wt% are used as reinforcements. CNT/PPcomposites are fabricated by coating the PP powder with CNT fol-lowed by melt mixing and injection molding. A DSM Micro 15 ccCompounder, (vertical, co-rotating twin-screw micro extruder)and 10 cc injection molding machine are used for melt mixingand injection molding respectively. The details of the fabricationcan be found in [24].

2.2. Characterization of composites

The stress–strain relation including tensile modulus of CNT/PPcomposites is determined as a function of CNT content for CNTconcentration of 0, 1 and 4 wt% according to ASTM D638 usingan Instron 33R 4466 apparatus. A 500 N load cell and an extensom-eter (Instron 2630-101) with a gage length of 10 mm are used. Adisplacement control with a velocity of 2.54 mm/s is applied. Eachdata point reported is an average of five repetitions.

CNT/PP interphase is characterized in terms of its thickness andmodulus using a Veeco AFM with Nanoscope V controller, operatedin tapping mode using an aluminum coated cantilever (length225 lm, spring constant 45 N/m, resonance frequency 190 kHz, sil-icon tip radius 2 nm) by Nanoscience Instruments Inc. Phoenix, AZ.No special sample preparation of the CNT/PP composites is em-ployed prior to AFM.

The distribution of CNT diameter (used as a measure of CNTagglomeratation), spatial distribution, orientation and wavinesswithin the PP matrix are assessed by studying the polished andetched surfaces of CNT/PP composites using SEM. A Zeiss Ultra60FE-SEM, operated at an accelerating voltage of 5 kV is used. Allspecimens are cleaned with isopropyl alcohol and coated with athin gold layer to avoid charging. Prior to the SEM study the com-posite specimens are polished to 0.5 lm diamond finish, dried for24 h at ambient temperature and etched using 1.0 wt% solution ofpotassium permanganate in a 2:1 mixture of concentrated sulphu-ric acid and phosphoric acid in an ultrasonic bath at ambient tem-perature [25,26] for 1 h. After etching, the specimens are washedsequentially in dilute sulphuric acid, distilled water and acetoneand are dried at ambient temperature. In order to obtain a statisti-cally meaningful distribution of the CNT orientation, diameter, dis-tribution and waviness a set of about 15 SEM images are capturedfrom various locations of the composites surface. RepresentativeSEM images of polished and etched surfaces of 4 wt% CNT/PP com-posites normal and parallel to the injection molding/applied loaddirection are shown in Fig. 1a and 1b respectively.

2.3. Determination of the distribution functions of CNT parameters

The distribution of the CNT orientation is determined by usingthe CNT footprint and geometry just below the polished surface

of the SEM images. CNT intersect the plane (polished surface) asan ellipse or circular disk depending on their inclination belowthe surface. By measuring the elliptical parameters, one can obtaintwo angles h and u that specify the 3D orientation of each CNT. Thegeometric parameters that are measured from each elliptical fea-ture, shown in Fig. 2a, are the center of the ellipse (xc, yc), the major(2a) and minor (2b) axes and the in plane angle (u). In Fig. 2a, out-of-plane angle h is defined as the angle that the CNT form with theinjection/applied load direction (z axis), while u is defined as theangle that the ellipses major axis make with the x-axis in the x–yplane. Fig. 2a (inset) also shows a typical elliptical footprint of aCNT intersecting the polished plane, and indicates that h is givenby the inverse cosine of the ratio of the semi-minor (b) to semi-ma-jor (a) axis of each elliptical image. The determination of 2a, 2b, h,u and the center of ellipse from each elliptical image can be deter-mined through an image analysis algorithm. However, using justthe elliptical marks creates ambiguity in the determination of thein-plane angle u. As indicated in the schematic of Fig. 2a therecan be two possible orientation configurations, (h,u) and(h,u + p), for the same elliptical cross section in the x–y plane thatcannot be distinguishable.

This ambiguity is resolved by using the dark regions, henceforthreferred to as ‘‘etch marks’’, around the CNT cross sections shownin the SEM images. These etch marks are revealed or amplifiedby chemical etching of the polished surface, which removes a thinpolymer layer exposing some portion of the CNT underneath thesurface being studied. These etch marks next to the major axis ofthe elliptical cross sections of the inclined CNT, shown in theSEM images, Fig. 2b, are used to correct the in-plane-angle. Theway the etch marks are related to the CNT orientation and the cor-rection methodology for the in plane angle are also shown inFig. 2b. If the etch mark is located at the opposite side with respectto the measured in plane angle then no correction is required (toright of Fig. 2b). A correction of u = u + p is applied when the etchmark is in the same side with the measured in plane angle (bottomright of Fig. 2b).

The CNT diameter is usually determined using SEM and TEM[14,15]. Such techniques however, are either not automated toanalyze large number of CNT or cannot account for CNT agglomer-ates and therefore are not capable of measuring CNT diameter dis-tribution function. In this work, these limitations are overcome byusing the major (2a) and minor (2b) axes, see Fig. 2a, of CNT cross-sections in the SEM images. The CNT diameter is calculated usingthe following three criteria: (i) if the major and minor axes areequal for a CNT cross section, 2a = 2b = CNT diameter; (ii) if the ma-jor axis is larger than 2 � 2b, then CNT diameter = 2b; (iii) for allother cases, CNT diameter = (2a + 2b)/2.

To determine the spatial distribution of CNT within the poly-mer, a method used for determining the spatial distribution of fi-bers is adopted [27]. The spatial position of the CNT is describedin a polar coordinate system using two parameters R and a, shownin Fig. 3a, which are chosen in a convenient relation to the coordi-nates of the CNT centroids. Specifically, R is the distance of the CNTcross-section centroid (xc, yc) from the origin and a is the angle be-tween the x axis and R. In this schematic, circular cross-sectionsrepresent CNT perpendicular to the plane while the ellipticalcross-sections represent CNT at an angle with the plane.

Fig. 3b shows the CNT parameters measured, to determine CNTswaviness distribution, from the SEM images captured from the sur-face that are parallel to the injection/applied load direction of CNT/PP composites (see Fig. 1b). These parameters are the distance be-tween the endpoints of a CNT (Lep) and the length of the CNT (LCNT)and are measured using a modified image analysis algorithm orig-inally developed for extracting the network geometry of threedimensional (3D) collagen gels [28]. The algorithm is based on aprinciple of nucleation and local maxima points. Using the distance

Fig. 1. SEM images of 4 wt% CNT/PP composites (a) normal and (b) parallel to the injection/applied load direction.

Fig. 2. (a) Definition of in-plane (u) and out-of-plane (h) angles, geometrical parameters measured from the elliptical cross-sections: coordinates of the center of the ellipse(xc, yc), minor axis (2b), major axis (2a), and in-plane angle (u) and ambiguity in CNT in-plane orientation angle; (b) etch marks next to the CNT cross sections and correctionof CNT in-plane angle.

Fig. 3. (a) CNT cross sections extracted from the SEM image through binary conversion, R: Distance of the CNT center from the origin, R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

c þ y2c

p); and a: angular position

of CNT center with respect to the x axis, a = tan�1 (yc/xc); (b) parameters used to determine CNT waviness characteristics.

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function and parameters set by the user; the program finds nucle-ation points and maps the CNT network through the local maximapoints. The concept is then extended to output both the length andthe endpoint to endpoint distance of the CNT. The straightnessparameter (Ps) [29] defined by Eq. (1) is used to characterize theCNTs waviness. Ps value is bounded from 0 to 1, with Ps � 1 indicat-ing a totally straight CNT whereas Ps � 0 indicates high degree ofwaviness.

Ps ¼ Lep=LCNT ð1Þ

3. Modeling of CNT/PP composites

3.1. Generation of 3D RVE geometric models

3D representative volume element (RVE) that consists of multi-ple CNTs embedded in PP matrix and CNT/PP interphase are gener-ated automatically using a programming code written in MATLAB

�.

The critical CNT length used for the FEA is 1500 nm. The details ofthe selection of CNT critical length can be found in our previouswork [24]. Thus an RVE length of 1700 nm is selected to completelyembed the CNT within the RVE. The other two dimensions of theRVE are chosen �1000 nm (W) X �1000 nm (H) considering thecomputational composites. While generating the RVE it is madesure that the CNT are not overlapped with each other using a con-tact algorithm. The volume fraction is updated when a new CNT isadded and the process continues until the desired volume fractionis achieved.

The mass centers of the CNTs within the RVE are selected basedon the spatial distribution function obtained from the SEM images.Similarly, the CNT 3D orientation is also modeled by using the dis-tribution functions obtained for two angles, h and u, that describethe orientation of the CNT axis with respect to the axial loadingdirection (z-axis) and with respect to a defined plane (x–z), respec-tively. To make the model more realistic individual CNT as well asCNT agglomerates of different sizes are modeled within the RVE

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using the diameter distribution function. The CNT waviness, asdetermined by SEM image analysis is also integrated into the mod-el. To generate the wavy CNT geometry a discretization method de-scribed in [30] is used in addition to the distribution function.Fig. 4a shows the shape of the generated wavy CNTs. The distribu-tion functions of CNT characteristics are defined by using parame-ters such as mean and standard deviation. These quantities arethen used to define the distribution of CNT parameters withinthe RVE with the help of a programming code written in MATLAB

�.

The three dimensional RVE, with CNT characteristics distributedaccording to the obtained distribution functions, is then generatedin Autodex Inventor before importing the geometry to ANSYS forFEA.

In this study CNT are represented as effective (solid) fibers withdiameter equal to the outer diameter of the CNT [31–34], as shownin Fig. 4c. The modulus of the effective fiber is calculated using thefollowing equation, which is derived based on the assumption thatan applied external force (F) on the CNT and the effective fiber will(Fig. 5c) result in an iso-strain condition [31].

Eeff ¼r2

f � ðr2f � tÞ2

r2f

� ECNT ð2Þ

where rf (=d/2) is the radius of effective fiber, t (= �0.34 nm) is thethickness of CNT outer layer and ECNT (=1250 GPa) is the averagemodulus of the CNT as provided by the supplier.

3.2. FEA models

Commercially available software ANSYS 14.0 is used to analyze3D RVE models described in Section 3.1. The RVE is subjected touniform extension within the linear regime of the stress–straincurve as determined experimentally. The loading and boundaryconditions applied to the RVE for FEA is presented in Fig. 5. Foreach model, the z = 0 end (Fig. 5a) is constrained in the axial direc-tion (z-direction) and free to move in the lateral directions. Thefree edges (Fig. 5b) are constrained to their respective normaldirections in order to allow contraction of the RVE due to tension.An axial displacement, equivalent to the experimental strain, is ap-plied to all nodes on the end surface (z = L) (Fig. 5c), where L is thelength of the RVE. Periodic boundary conditions are used on theother faces. A higher order 3D structural brick element (SO-LID185-8-node quadratic element with three degrees of freedomper node) is used for all phases. It is noted that the interphase isconsidered in the FEA in order to capture the load transfer between

Fig. 4. (a) RVE containing wavy CNTs and (b) schematic of the CNT and effectiv

the CNT and the PP matrix. From AFM observations (see Fig. 6), theexistence of voids at the CNT/PP interface is observed, which re-sults in inefficient load transfer from CNT to polymer. Therefore,an interphase of constant thickness and modulus lower than neatPP is considered in the FEA to capture the inefficient load transferbetween the CNT and the polymer. More details discussion on whya soft interphase is considered can be found in our prior work [35].The assumptions made are that there is perfect bonding at theinterface [36–38], and that all the phases are homogeneous, isotro-pic, and linearly elastic. An optimum element size (mesh density)that leads to a fully converged solution with minimum computa-tional time is determined based on a FEA parametric study. Thedimensions and mechanical properties of all the phases [35], usedin this analysis, are presented in Table 1. A representative FE meshis shown in Fig. 6. The mesh density in each phase is systematicallycontrolled with less number of elements in PP and large number ofelements in CNT/PP interphase as shown in Fig. 6. The effectivemodulus of the composite is calculated as the ratio of the averagestress generated at z = 0 to the applied strain (DL/L) on the RVEmodel.

4. Results and discussion

4.1. Tensile modulus of CNT/PP composites

The modulus calculated as the initial slope of the tensile stress(r)–strain (e) curves, obtained from the tensile experiment, andthe corresponding standard deviation for CNT/PP composites with0, 1 and 4 CNT wt% are presented in Table 2. Although the modulusincreases upon addition of CNT it is still much lower than the mod-ulus predicted by the theoretical micromechanical models such asHalpin-Tsai [39]. Many factors may contribute to this discrepancy,such as existence of imperfect contact at the CNT/PP interface,presence of agglomerates and the random CNT orientation andwaviness within the polymer.

4.2. CNT/PP interphase characterization

In this study, the CNT/PP interphase is characterized in terms ofits’ thickness and modulus using AFM phase images obtained inthe AFM tapping mode. Phase imaging measures the phase lag inoscillation frequency when the AFM tip interacts with areas of dif-ferent material properties. A representative phase image of the topsurface of 4 wt% CNT/PP composite is presented in Fig. 7a, where aclear transition area, the CNT/PP interphase, is observed between

e fiber used to evaluate the effective elastic modulus of CNT in composites.

Fig. 5. Loading and boundary conditions applied to the different surfaces of the 3D RVE.

Fig. 6. Illustration of detailed meshes used in the finite element analysis for all the phases.

Table 1Mechanical properties and geometric characteristics of the phases considered in theFEA.

Young’s modulus(GPa)

Poisson’sratio

Characteristiclength

CNT 70 (Eq. (2)) 0.3 [35] Diameter = 25 nmCNT agglomerates Vary with diameter

(Eq. (2))0.3 [35] CNTs diameter

distributionPP 1.38 (tensile

testing)0.36 [35]

CNT/PP interphase 0.7 (FEA) 0.36 [35] Thickness = 20 nm(AFM)

Table 2Tensile modulus of CNT/PP composites obtained from tensile experiment.

CNT (wt%) Modulus (GPa)

0 1.38 ± 0.041 1.55 ± 0.034 1.75 ± 0.08

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the CNT and the PP matrix. The corresponding property gradientprofile across the interphase is also presented in Fig. 7b. UsingAFM phase images taken at various locations (five in an average)on the CNT/PP composites (0.1 wt%, 1 wt% and 4 wt%) surfacesand using the method described in [24,35], the average thicknessand modulus of the CNT/PP interphase measured are �20 nmand 0.7 GPa (soft interphase with modulus equal to �50% of thePP modulus), respectively.

4.3. Determination of the distribution functions of CNT parameters

In this study a MATLAB�

algorithm, shown in Fig. 8a, is devel-oped and implemented to determine the distribution functions ofCNT orientation, diameter and spatial distribution using SEMimages. Specifically, each SEM image is converted to a gray scaleimage first and the location of the etch marks, see Fig. 2b, is deter-mined and extracted with a binary conversion factor. The imagecontaining the extracted region is then stored. Using another bin-ary conversion, the CNT cross sections are then extracted fromthe original image. CNT cross-sections extracted from the grayscale image are presented in Fig. 8b. Five geometrical parametersare measured for each CNT cross-section: the location (xc, yc) ofthe centroid of the cross-section, the major (2a) and minor axes(2b), and the orientation angles, h (out-of-plane) and u (in-plane).Fig. 8 shows the resulting histograms for the CNT orientation (hand u) distribution obtained from a set of fifteen SEM images ofthe 4 wt% CNT/PP composite. It can be seen that the CNT out-of-plane angle, Fig. 9a, follows a normal distribution with peak nearto 45� (standard deviation 15�) whereas the in plane angle,Fig. 9b, follows an extreme value distribution function with peaknear to 250� (standard deviation 50�).

The CNT diameter distribution follows lognormal distributionfunction with the peak close to �50 nm as shown in Fig. 10 accord-ing to which the minimum and maximum diameter is �20 nm(which is in agreement with the supplier data) and �180 nm(which is actually CNT agglomerates), respectively. AFM phase im-age shown in Fig. 7(a) also supports the presence of CNT agglomer-ates of diameter �150–200 nm, thus validating the obtaineddiameter distribution. The CNT agglomerates, a result of strongvan der Waals interactions among the CNT, result to poor mechan-ical properties due to insufficient stress transfer. The modulus ofthe agglomerates, calculated using Eq. (2), is less compared to thatof individual CNT and decreases with the diameter of the agglom-

Fig. 7. (a) AFM phase image and (b) property gradient profile used to determine the CNT/PP interphase thickness and modulus.

Fig. 8. (a) Image analysis algorithm implemented to determine the distribution of CNT orientation, diameter and spatial distribution in polymers and (b) extraction of CNTcross sections from the binary image.

Fig. 9. CNT (a) out of plane and (b) in-plane orientation distribution within the PP matrix.

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Fig. 10. CNT diameter distribution.Fig. 12. CNTs waviness distribution obtained from SEM image analysis.

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erates. The main reason for the decrease of the modulus is that CNTagglomerates do not behave as rigid structures because of the slip-page (relative motion) of the CNTs with respect to each other with-in the agglomerate.

The spatial distribution of CNT is described in terms of theparameters R and a which are calculated as shown in Fig. 3a andare shown in Fig. 11a and b respectively. The histogram for theCNT waviness distribution, along with the distribution curve is pre-sented in Fig. 12. It can be seen that the distribution of CNT wavi-ness in PP matrix follows an extreme value distribution with mean(l) = 0.9512 and standard deviation (r) = 0.0513.

4.4. Modeling tensile modulus of CNT/PP composites

FEA of uniaxial loading (within the elastic regime) of 4 wt%CNT/PP composite is performed as a case study in order to deter-mine the effect of the CNT parameters’ distribution functions, asdetermined by the SEM image analysis, on the composite’s tensilemodulus. A representative 3D RVE model used for FEA to predictthe tensile modulus of CNT/PP composites are presented in Fig. 13.

Fig. 14 shows the tensile modulus of 4 wt% CNT/PP compositesobtained using the present approach and experiments. The hori-zontal dotted line shown in Fig. 14 represents the tensile modulusof 4 wt% CNT/PP composites obtained from experiment The modu-

Fig. 11. Spatial distribution of CNT in PP extracted fro

lus obtained in our earlier work considering unit cell (single CNTRVE) model [35] and multi-CNT RVE models with random distribu-tion (RD) of CNT parameters [24] are also presented for compari-son. The corresponding modulus values and standard deviationare also presented in Table 3. The simulations are repeated sixtimes for multi-CNT RVE models with random distribution ofCNT various parameters [24] and models considering distributionfunctions (present approach). As shown in Fig. 14 and Table 3,when single CNT RVE is considered, the FEA prediction is muchhigher than the modulus obtained experimentally. This is expectedsince no CNT parameters are incorporated in these models. How-ever, when the CNT parameters are included in the FEA, using ran-dom and pdf (statistical distribution of CNT parameters obtainedfrom SEM image analysis) approach, the predicted modulus variesfor each FEA simulation. As can be seen, when random distributionof CNT parameters is considered, FEA predictions shows large var-iation (shown as triangles in Fig. 14) with a standard deviation of0.13 (see Table 3, column 3). When the distribution of CNT param-eters obtained from SEM image analysis is considered, the FEA pre-dictions show less variation (shown as diamonds in Fig. 14) with astandard deviation of 0.024 (see Table 3, column 4). The FEA pre-dictions obtained from present approach also falls within the rangeof experimental error bar as shown in Fig. 14. Therefore, it can be

m the SEM images of 4 wt% CNT/PP composites.

Fig. 13. 3D RVE model for 4 wt% CNT/PP composite considering the distribution functions of CNTs various parameters as obtained from SEM image analysis.

Fig. 14. FEA predictions for tensile modulus of 4 wt% CNT/PP composites: valida-tion of the proposed integrated modeling technique (RD: random distribution ofCNT parameters; and pdf: statistical distribution of CNT parameters in PP).

Table 3Tensile modulus of 4 wt% CNT/PP composites: Experimental vs. FEA predictions.

Tensile modulus of 4 wt% CNT/PP composites (GPa)

Experimental FEA (singleCNT RVE)[35]

FEA (multiCNT RVEwith RD)[24]

FEA (multiCNT RVEwith pdf)

1.67 2.7 1.85 (run 1) 1.77 (run 1)1.68 1.96 (run 2) 1.79 (run 2)1.82 1.92 (run 3) 1.74 (run 3)1.74 1.76 (run 4) 1.8 (run 4)1.83 1.63 (run 5) 1.8 (run 5)

1.70 (run 6) 1.76 (run 6)

Average ± Stdev 1.75 ± 0.08 2.7 ± 0 1.8 ± 0.13 1.77 ± 0.024

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concluded that the proposed integrated images analysis and FEAsupported by experiments can be successfully applied to extractthe information on orientation, agglomeration, spatial distributionand waviness of CNTs to model the composites. The strongdependence of composites modulus on the distribution of theseparameters justifies the need an integrated approach to accuratelymodel these parameters in order to develop CNT-based compositeswith engineered properties.

A systematic study to understand the effect of each individualparameter on the tensile modulus is also performed. It is foundthat the modulus of the composites is more sensitive to presenceof CNT agglomerates. A reduction of tensile modulus of approxi-mately 16% is observed when presence of CNT agglomerates is con-sidered in the analysis. The effect of other parameters such as CNT

non-homogeneous distribution, random orientation, waviness andCNT/PP interphase on the modulus is approximately same, that isin the range of � 6–8%. When all the parameters are consideredin the modeling at the same time, the reduction in tensile modulusis approximately 40%. A more detailed discussion on sensitivity ofindividual parameters can also be found in our previous paper [24].

5. Conclusions

The elastic properties of CNT based composites, for given CNTtype; grade and wt.%, are particularly sensitive to various parame-ters such as CNT/polymer interfacial interactions, CNT orientation,dispersion, distribution, waviness and accurate use of these param-eters in terms of their geometry/distributions within the compos-ites while modeling RVE for FEA are crucial for exact prediction ofthe composites’ effective modulus. In this study, an image analysistechnique coupled with FEA is presented in detail that is able pre-dict the effective tensile modulus of CNT/PP composites by quanti-fying various CNT parameters within the composites using 2D SEMand AFM images of the composites and by incorporating them intoFE models.

The distribution function for CNT orientation within PP ob-tained from the SEM images indicates that most of the CNT makean angle of 45� with the applied load direction. This is expectedfor the CNT/PP composite presented in this work, as the injectionpressure forces the CNT to align along the injection/applied loaddirection. The CNT diameter distribution shows that majority ofthe CNT have a diameter of �50 nm or less indicating the presenceof small CNT agglomerates. This is the effect of coating and melt-mixing used at the early stage of fabrication of CNT/PP composites.A realistic spatial distribution of the CNT within the PP matrix ismodeled for FEA by using the spatial distribution function (param-eters R and a) obtained from the SEM images. Finally from the dis-tribution of CNT waviness, it can be observed that all CNT exhibit ahigh degree of waviness. It is noted that for measuring CNT wavi-ness, in this study we have used the length of the CNT in the 2DSEM images assuming that the rest of the CNT below the imagedsurface follow the same waviness distribution. Study of AFM phaseimages supports the presence of voids at the CNT/PP contact area.Therefore, the CNT/PP interphase considered in this study is a softinterphase with a modulus lower than the neat PP.

3D FE models with CNT/PP interphase and the quantified distri-bution functions for CNT orientation, diameter, spatial distributionand waviness are generated for CNT/PP composites and are ana-lyzed for their tensile modulus. A good agreement between themodel predictions and the experimental data for the same compos-ite system is observed. Therefore, it can be concluded that the ran-dom orientation and distribution of wavy CNT within the polymersas well as presence of CNT agglomerates within the polymer andvoids at the CNT/PP contact area are the main factors that reducethe reinforcing efficiency of the CNT when added to the polymers.It is noted that an error in the SEM image analysis is introduced bydefining the distribution curves that describe the histograms. The

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range of FEA predictions is narrow when distribution functions forCNT parameters are used in the FE models instead of using averagevalue for these parameters. The integrated approach of image anal-ysis coupled with FEA supported by experimental characterizationis therefore possesses large potential for the design and develop-ment of CNT based polymer composites with engineered proper-ties for targeted applications.

Acknowledgements

The authors thank Professor J. Colton from the Woodruff Schoolof Mechanical Engineering at Georgia Institute of Technology forhelping with the tensile testing. Special thanks to Dr. VirendraSingh, post doctoral fellow in the Woodruff School of MechanicalEngineering, for helping with the chemical etching of the compos-ite samples.

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