Influence of Carbon Nanotube Defects on the Elastic
Modulus of Nanocomposite: Multiscale Simulation

Krishnendu Bhowmik, Niloy Khutia, Mostapha Tarfaoui, Ayan Basu, Shamim Akhtar, Swati Dey, Amit Roy Chowdhury

► To cite this version:

Krishnendu Bhowmik, Niloy Khutia, Mostapha Tarfaoui, Ayan Basu, Shamim Akhtar, et al.. Influence of Carbon Nanotube Defects on the Elastic Modulus of Nanocomposite: Multiscale Simulation. Journal of Materials Engineering and Performance, 2023, 32, pp.2356-2369. 10.1007/s11665-022-07287-1. hal-03808882

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<figure><img src="image_1.png" /></figure>
Influence of Carbon Nanotube Defects on the Elastic Modulus of
Nanocomposite: Multiscale Simulation

Krishnendu Bhowmik, Niloy Khutia, Mostapha Tarfaoui, Ayan Basu, Shamim Akhtar, Swati Dey, and Amit Roy Chowdhury

In the present study, atomistic-based continuum modeling of single-walled carbon nanotube (SWCNT)-reinforced composite had been
performed to predict the effective modulus of the nanocomposite through multiscale modeling. SWCNT of armchair type (10,10) was
modeled with a three-dimensional beam element to estimate equivalent axial Young's modulus using finite element analysis (FEA) for
pristine and defective CNTs. The equivalent modulus of CNT with 5% vacancy defects has been reduced by 35.35% in com-parison to the
pristine CNTs. In addition, molecular dynamics (MD) simulations were conducted on standalone CNT for pristine and defective CNTs.
Further, long and short CNT-reinforced nanocomposites were modeled through interphase FE modeling using Representative Volume
Element (RVE) which is comprised of three phases, i.e., SWCNT, interphase, and matrix. Finally, the effects of CNT defects, interphase
property, modulus ratio, height fraction, and volume fraction on the overall elastic response of the nanocomposite were demonstrated by
statistical analysis. The analytical expressions of effective elastic modulus of both short and long CNT-based nanocomposites have been
derived through regression analysis.

**Keywords** CNT vacancy defects, effective Young's modulus, finite element analysis, molecular dynamics, nanocomposite, regression analysis

**1. Introduction**

The carbon nanotubes (CNTs) are being widely used as reinforcement in nanocomposites due to their superior proper ties in low density, higher modulus, strength and stiffness, fracture toughness, and better thermal and electrical properties (Ref 1 4). Kan et al. (Ref 1) worked on single walled carbon nanotubes (SWCNT)/polyethylene nanocomposite to show enhancement of tensile strength and modulus due to the addition of CNT as a reinforcement. Naito (Ref 2) demonstrated the influence of CNT on tensile strength, modulus, and fracture toughness of carbon fiber composites. Burkov and Eremin (Ref 3) reported that the CNT reinforcement can improve the electrical conductivity of carbon fiber reinforced nanocompos ite. Tarfaoui et al. (Ref 4) explored the application of CNT in bucky paper glass fiber composite to enhance the heating performance. Moumen et al. (Ref 5) demonstrated that CNT reinforcement in nanocomposite can be used as a good mechanical shock absorber. However, the uses of CNTs in nanocomposites are limited due to the formation of CNT agglomerations (Ref 6 8). Maghsoudlou et al. (Ref 8) inves

tigated elastic modulus of SWCNT/epoxy nanocomposite
through experiments and finite element analysis (FEA). They
observed a drop of modulus for 0.5% wt. SWCNT reinforce
ments due to agglomeration effect. The mechanical properties
of hybrid CNT/carbon fabric/epoxy woven structural compos
ites were reported in several works (Ref 9 11). A graphene
sheet is rolled down in the shape of cylindrical form to get a
carbon nanotube structure, and its properties vary with different
chirality (n, m) (Ref 12). Krishnan et al. (Ref 13) and Yu et al.
(Ref 14) measured the mean Young's modulus of SWCNT as
1250 GPa and 1002 GPa through experimental technique using
transmission electron microscope (TEM) and atomic force
microscope (AFM), respectively. Li and Chow (Ref 15)
assumed a structural mechanics approach to estimate the elastic
moduli for armchair and zigzag SWCNT with different
diameters and helicity. Shen and Li (Ref 16) predicted the
transversely isotropic properties of SWCNT using a molecular
mechanics (MM) approach. Wan and Delale (Ref 17) demon
strated the buckling behavior of SWCNT through FEA. Tserpes
and Papanikas (Ref 18) studied the effects of nanotube wall
thickness, chirality, and diameter on the elastic moduli of
SWCNTs with different chirality, through FEA. Dastmard et al.
(Ref 19) computed the effective modulus of pristine SWCNT
(10,10) using density functional theory and FE approach.
Defects in CNTs are expected to be present in the manufac
turing process, and such defects have an important role in the
effective modulus of the CNTs. In case of a vacancy defect
(missing carbon atoms), it may be a single vacancy or multiple
vacancy types. Sakharova et al. (Ref 20) demonstrated the
effect of vacancy defects of SWCNT beam on elastic properties
using FE simulations. The mechanical properties like Young's
modulus, yield, and ultimate strength of CNT structure with/
without defects were explored by Liew et al. (Ref 21) from MD
results of the nonlinear stress strain response of CNT. Yuan and
Liew (Ref 22) demonstrated the decrease in Young's modulus
of SWCNT with different vacancy defects through MD

Krishnendu Bhowmik, Niloy Khutia, Ayan Basu, Shamim Akhtar, Swati Dey, and Amit Roy Chowdhury. Department of Aerospace Engineering and Applied Mechanics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India; Mostapha Tarfaoui, ENSTA Bretagne, IRDL UMR CNRS 6027, 29200 Brest, France; and Green Energy Park (IRESEN/UM6P), km2 R206 Benguerir, Morocco. Contact e mail: niloy@aero.iiests.ac.in.
simulations. Roy et al. (Ref 23) studied the effect of structural defects of SWCNT on stress strain response to predict Young's modulus, fracture strength, and strain through MD simulations. Kundalwal and Choyal (Ref 24) performed MD simulations to estimate the elastic properties of pristine and defective SWCNTs. It was reported that the location of vacancy defects along the CNT lengths was the main prompting factor that governed the variations in the CNT properties.

The effective Young's modulus of CNT/matrix nanocomposites was estimated using FE analyses considering the Representative Volume Element (RVE) approach by many researchers (Ref 25, 26). The addition of nanotubes as a reinforcement agent in the nanocomposite has the most significant effect on strengthening the epoxy (Ref 27, 28), but the effectiveness of reinforcement also depends upon the waviness, agglomeration, and orientation of nanoparticles (Ref 29, 30). Interphase between CNT and matrix plays an important role in the overall modulus of nanocomposites (Ref 31, 32). Wan et al. (Ref 33) adopted FE approach to investigate the mechanical properties of SWCNT (10,10) reinforced polymer composite considering the anisotropic interphase with different values of modulus. Giannopoulos et al. (Ref 34) studied the elastic properties of graphene based nanocomposites via a 3 D multiscale FE approach for two and three phase models (without and with interphase). They have studied the effect of imperfection in bonding between components and represented by adhesion coefficient (k as 1.0, 0.66, and 0.33) for estimating interphase property. Similar studies pertaining to the influence of interphase on the overall elastic properties of SWCNT reinforced polymer composites were also reported in the literature (Ref 35, 36). Zhu et al. (Ref 37) proposed an extended Mori Tanaka (MT) model to estimate the equivalent elastic modulus of wavy carbon nanotube reinforced composites for various volume fractions, waviness, and interfacial damage. The mechanical properties of CNT reinforced nanocomposites extracted from molecular dynamics simulations were reported by Ionita (Ref 38), Anjana et al. (Ref 39), and Yang et al. (Ref 40). The effect of vacancy defects on the elastic behavior of CNT reinforced nanocomposites was demonstrated by many researchers using FE analysis (Ref 41, 42) and MD simulations (Ref 43, 44). Rahimian Koloor et al. (Ref 45) conducted MD simulations to extract the mechanical properties (stress strain response and

directional elastic moduli) of pristine and defective SWCNT (TSW and vacancy defects) reinforced nanocomposites.

Available literature shows that the modeling of defective SWCNT reinforced nanocomposites with functionally graded interphase between CNT and matrix was not yet explored in multiscale modeling and statistical regression analysis by previous researchers. In the present work, the effect of CNT defects has been incorporated through the random distribution of the CNT defects in the three dimensional finite element analysis. Further, the molecular dynamics simulation of the CNTs with defects has been performed to correlate the FE model with the atomistic simulation of CNT. The primary focus of this work is to derive an analytical expression of the effective modulus of SWCNT reinforced composite in terms of CNT parameters, e.g., modulus ratio ($\lambda_1$, the ratio of Young's modulus of carbon nanotube and matrix), CNT height fraction ($\lambda_3$, the ratio of carbon nanotube height and RVE length), CNT defect ($D_{CNT}$), CNT volume fraction ($\lambda_4$ or $V_{CNT}$), and different values of interphase properties (different thickness and material properties) based on the bonding coefficients. The aspect ratio of CNT ($\lambda_2$, the ratio of carbon nanotube length, and diameter) was considered for the present study as 30 (Ref 26). A detailed regression analysis was performed and the sensitivity of the parameters was demonstrated through Pareto Chart and response surface plots for long/short CNT reinforced nanocomposite.

## 2. Methodology

In molecular mechanics (MM), the motions of a CNT (a large molecule of carbon atoms) are controlled by the force field generated by the interactions of the nucleus nucleus and electron nucleus. The schematic of SWCNT as a space structure is shown in Fig. 1. The steric potential energy due to the force field is a function of the molecule made up of relative locations of the nuclei. The cumulative steric potential energy can be considered as an algebraic summation of all the energies caused by the bonded interactions as well as non bonded interactions by neglecting the electrostatic interaction (Ref 46):

<figure><img src="image_2.png" /><figcaption>**Fig. 1** Schematic of (a) SWCNT as space frame structure and (b) interatomic interaction based on molecular mechanics</figcaption></figure>
$$U = \sum U_r + \sum U_0 + \sum U_\phi + \sum U_\omega + \sum U_{vdw} \quad (\text{Eq 1})$$

where $U_r$, $U_0$, $U_\phi$, $U_\omega$, and $U_{vdw}$ are interaction due to bond stretch, bond angle caused by bending, torsion due to dihedral angle, out of plane torsion, and a van der Waals interaction, respectively, as shown in Fig. 1(b). In the cumulative steric energy for covalent systems, the most dominating contributors are the first four terms of Eq (1). By considering the deformations as a small and harmonic approximation for the calculation of the energy (Ref 47) and assuming the combined effects of the dihedral angle of torsion and the out of plane torsion, the following equations may be considered (Ref 15):

$$U_r = \frac{1}{2} k_r (r - r_0)^2 = \frac{1}{2} k_r (\Delta r)^2 = \frac{1}{2} \frac{EA}{L} (\Delta L)^2 \quad (\text{Eq 2})$$

$$U_0 = \frac{1}{2} k_0 (\theta - \theta_0)^2 = \frac{1}{2} k_0 (\Delta\theta)^2 = \frac{1}{2} \frac{EA}{L} (2\alpha)^2 \quad (\text{Eq 3})$$

$$U_{\tau} = U_{\phi} + U_{\omega} = \frac{1}{2} k_{\tau} (\Delta\phi)^2 = \frac{1}{2} \frac{GJ}{L} (\Delta\beta)^2 \quad (\text{Eq 4})$$

where $k_r$, $k_0$, and $k_\tau$ are the force constant due to bond stretching, force constant for bending due to the bond angle, and torsional resistance, respectively. The parameters $\Delta r$, $\Delta\theta$, and $\Delta\phi$ represent the variation in bond stretching, bond angle, and bond twisting angle, respectively. Here, $\Delta L$ represents the deformation due to the axial loading, $\alpha$ is the rotational angle of the beam at its ends, and the relative rotation of the beam between its ends is represented by $\Delta\beta$. For simplicity, it can be assumed that cross sections of C C bonds as identical and uniformly round (i.e., $I_x = I_y = I$) (Ref 15). It can be concluded that in Eq. (2 4), $U_r$ represents the energy due to the stretching, the bending energies are represented by $U_0$, and $U_\tau$ is denoting the torsional energies. The total change of bond angle $\Delta\theta$ is expressed as the angle $2\alpha$, $\Delta L$ relates to $\Delta r$, and $\Delta\beta$ relates $\Delta\phi$.

The axial, bending, and torsional stiffness of the CNT beam ($k_r$, $k_0$, and $k_\tau$) can be estimated by comparing Eq. (2 4), as follows (Ref 15):

$$k_r = \frac{EA}{L}; k_0 = \frac{EI}{L}; k_\tau = \frac{GJ}{L} \quad (\text{Eq 5})$$

where $EA$, $EI$, and $GJ$ represent the sectional stiffness parameters for axial, bending, and torsions, respectively. Equation (5) represents the basis for the solid mechanic's approach for the analysis of carbon nanotubes and carbon related nanostructures. With the consideration of the diameter of the circular beam section as $d$, and hence, $A = \pi d^2/4$, $I = \pi d^4/64$, and $J = \pi d^4/32$, the following relationships can be formulated:

$$d = 4 \sqrt{\frac{k_r}{k_0}}; E = \frac{k_r^2 L}{4\pi k_0}; G = \frac{k_\tau k_r^2 L}{8\pi k_0^2} \quad (\text{Eq 6})$$

For, $k_r = 6.52 \times 10^7$ N nm$^{-1}$ (Ref 48), $k_0 = 8.76 \times 10^{10}$ N nm rad$^{-2}$ (Ref 15), $k_\tau = 2.78 \times 10^{10}$ N nm rad$^{-2}$ (Ref 48) and $L = 0.1421$ nm (Ref 15), the values of $d$, $E$ and $G$ can be calculated from Eq. (6), as 0.147 nm, 5448 GPa, and 871 GPa, respectively.

<figure><img src="image_3.png" /><figcaption>Fig. 2 Typical distribution of the different percentages of CNT defects (defects are marked as red)</figcaption></figure>

When CNTs are being used as reinforcement into the matrix, they can be classified as long or continuous ($\lambda_3 = 1$) and short or discontinuous ($\lambda_3 \Delta 1$) types based on the length of CNT and RVE. The effective longitudinal Young's modulus of long CNT reinforced two phase (CNT and matrix) composite ($E_C$) with perfect particle matrix bonding was predicted using a simple analytical expression, rule of mixtures (ROM) in which weighted mean of modulus is used for different CNT volume fractions (Ref 32, 41, 42) as:

$$E_c = E_p V_p + E_m V_m \quad (\text{Eq 7})$$

where $E_p$, $E_m$ are Young's modulus of particle and matrix, whereas $V_p$ and $V_m$ are volume fraction of particle and matrix, respectively.

In the case of three phase modeling (particle, interphase, and matrix), the elastic modulus of interphase ($E_{int}$) plays a significant role in the equivalent elastic response of the nanocomposites (Ref 35, 36). In this present study, Young's modulus of interphase ($E_{int}$) is being estimated as an exponential function concerning $E_p$ and $E_m$ similar to the work reported by Pérez and Avilés as follows (Ref 35):

$$(E_{int})_{exp} = E_p e^{\left(\frac{1}{t_{int}} \ln \frac{E_p}{E_m}\right) \Delta t_k} \quad (\text{Eq 8})$$

where $t_{int}$ and $\Delta t_k$ are the thickness of interphase and distance from the outer surface of SWCNT to the mid point of the interphase. Another model suggested by Samandari and Khatibi (Ref 36) can also be used for the elastic modulus of interphase as:

$$E_{int} = \frac{1}{r_i - r_p} \int_{r_p}^{r_i} E_{int}(r) dr \quad (\text{Eq 9})$$

where

$$E_{int}(r) = E_m \left(\frac{r_i}{r}\right) + \left[\frac{r_i - r}{r_i - r_p}\right]^{\frac{\alpha}{2}} \left[E_p - E_m \left(\frac{r_i}{r_p}\right)\right] \quad (\text{Eq 10})$$

where $r_p$ and $r_i$ are the radii of fiber and outer radius of interphase, respectively. Here, $n$ is a dimensionless parameter that depends upon chemical behavior and distribution of nanoparticles in the matrix, and the value of $n$ is taken as 10 (Ref 36).
<figure><img src="image_4.png" /><figcaption>**Fig. 3** Typical boundary conditions of RVE ($V_{CNT}$ =5.0%) for (a) long CNT (b) short CNT, and (c) variations of Young's modulus of interphase layers for long CNT reinforced nanocomposite</figcaption></figure>

<table><caption>**Table 1** Parameters considered in FE simulations for long and short CNT nanocomposites for different configurations</caption><thead><tr><th>Parameters</th><th>Symbol</th><th>Range for long CNT</th><th>Range for short CNT</th></tr></thead><tbody><tr><td>Modulus ratio</td><td>λ<sub>1</sub></td><td>10 to 500</td><td>10 to 500</td></tr><tr><td>CNT height fraction</td><td>λ<sub>3</sub></td><td>...</td><td>0.4 to 0.8</td></tr><tr><td>CNT volume fraction</td><td>λ<sub>4</sub> or V<sub>CNT</sub></td><td>1.5%</td><td>1.5%</td></tr><tr><td>Interphase thickness</td><td>t</td><td>0.68 nm</td><td>0.68 nm</td></tr><tr><td>Bonding coefficient</td><td>k</td><td>0.33 to 1.0</td><td>0.33 to 1.0</td></tr><tr><td>CNT defects</td><td>D<sub>CNT</sub></td><td>1.5%</td><td>1.5%</td></tr><tr><td>Total FE simulations</td><td>...</td><td>160</td><td>315</td></tr></tbody></table>

### 3. Finite Element Modeling

First of all, SWCNT was modeled as 3 D space structure in which each C atom is connected to three other C atoms with covalent bonds. The C C bonds were modeled as beam element (B33) in ABAQUS 6.14. Figure 1(a) shows the schematic representation of the carbon nanotube as a space structure and the C C bond as a beam element. The effective Young's modulus (E) as 5488 GPa and the diameter of beam elements as 0.147 nm were obtained from Eq. (6). For the CNT

<figure><img src="image_5.png" /><figcaption>**Fig. 4** Typical armchair SWCNT (10,10) structure of (a) pristine, and (b) defective CNT used in MD simulations</figcaption></figure>

beam, the Poisson's ratio was assumed as 0.3 (Ref 17, 42). The numbers of nodes and elements for SWCNT in linear static FE analysis were 6620 and 9910, respectively, for the aspect ratio of CNT is considered as 30 (Ref 26). The boundary conditions and loading of CNTs were: all degrees of freedom restricted at one side end, and uniform displacement in axial directions at the other side corresponds to 1% strain.

The longitudinal equivalent Young's modulus of SWCNT ($E_{CNT}$) can be extracted from Eq. (11).

$$E_{CNT} = \frac{F}{A\varepsilon} \qquad (\text{Eq 11})$$
<table><caption>**Table 2** Equivalent elastic modulus in GPa of pristine SWCNTs</caption><thead><tr><th colspan="6">Equivalent elastic modulus (GPa)</th></tr></thead><tbody><tr><td>FE</td><td>1052 (Present)</td><td>1090 (Ref 19)</td><td>1083 (Ref 20)</td><td></td><td></td></tr><tr><td>MM</td><td>~ 893 1035 (Ref 15)</td><td>1060 (Ref 16)</td><td></td><td></td><td></td></tr><tr><td>MD</td><td>1057.2 (Present)</td><td>1043 (Ref 19)</td><td>~ 900 (Ref 22)</td><td>1088 (Ref 23)</td><td>1200 (Ref 24)</td></tr><tr><td>Experiment</td><td>1250 (Ref 13)</td><td>1002 (Ref 14)</td><td></td><td></td><td></td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 5** Equivalent Young's modulus variations versus CNT defects ($D_{CNT}$)</figcaption></figure>

where *F*, *A*, and *ε* are reaction forces along the axis, CNT's cross sectional area ($πDt$, where *D* is the nominal diameter of the CNT tube, *t* is assumed as the interlayer thickness (0.34 nm) of graphene sheet (Ref 15, 18) and the average strain, respectively. Both pristine (without defect) and different volume fractions of carbon vacancy defects ($D_{CNT}$ as 1, 2, 3, 4 and 5%) are considered. A code developed in C language to generate a random distribution of CNT defects. The vacancy defects are introduced in the model by randomly selecting the nodes and hence the corresponding elements connected to those nodes. Young's modulus of defective CNT elements is assigned as negligible (10¹⁰ times less than CNT bonds) compared with pristine CNTs. The typical vacancy defects with different defect % (based on % of defective elements) are shown in Fig. 2.

When CNT reinforced nanocomposite is modeled through Representative Volume Element (RVE), two types of SWCNTs are being considered; long or continuous CNT reinforced composite as well as short or discontinuous carbon nanotubes composite. The loads and boundary conditions of typical RVEs for long and short CNT (height fraction, $λ_3=0.7$) reinforced nanocomposites ($V_{CNT}=5.0\%$) are presented in Fig. 3(a and b), respectively (tie elements are not shown here). The carbon nanotube is located at the center of the RVE, and CNT is aligned along the longitudinal axis of the RVE. In the case of short nanotubes, the hemispherical caps were modeled at both ends. The RVE is taken as three phases of carbon nanotube, interphase, and matrix. Figure 3(c) shows a typical front view of half section of the long CNT reinforced RVE. The four layers of interphase are also marked in Fig. 3(c). To compare the results, the two

phases of RVE (carbon nanotube and matrix) without interphase are also used. The matrix and interphase are modeled with solid tetrahedron elements (C3D4), whereas the CNTs are modeled as beam elements (B33). The numbers of layers of interphase are varied from 1 to 4 in the analyses. The thickness of each layer of interphase is taken as 0.17 nm (half of the thickness of a graphene layer). The maximum thickness of the interphase is 0.68 nm (twice the thickness of a graphene layer and close to the nominal radius of SWCNT) (Ref 32, 33). The innermost interphase layer is tied with the carbon nanotube beams. The interactions between the outermost interphase and matrix are assumed as glued together. The typical number of nodes and elements for the CNT, matrix, including the interphase, were 37,841 and 178,824, respectively. In the case of CNT reinforced nanocomposite, similar loading and boundary conditions of standalone CNT are being applied and Eq. (11) is used to extract the equivalent modulus ($E_C$) of the nanocomposite, where *A* is the cross sectional area of the RVE. Different configurations in FE simulations are achieved by considering various parameters for long and short CNT nanocomposites as shown in Table 1.

## 3.1 Mechanical Properties of Matrix and Interphase Considered in FE Analysis

The matrix and interphase materials are considered iso tropic. Different Young's modulus values are taken to obtain different modulus ratios ($λ_1$) or aspect ratios as 10, 20, 50, 100, 200, and 500 to account for metal as well as polymer matrix composites. The modulus of the matrix is calculated based on the equivalent Young's modulus of the pristine CNT as described in section 5.1. The Poisson's ratio of the matrix is taken as 0.3 (Ref 26, 32, 35, 37). The moduli of interphase are extracted by using Eq. (8). The variations of Young's modulus of interphase are taken such that it decreases exponentially from the CNT wall (similar to (Ref 34)). The bonding coefficient, *k*=1.0, represents interphase is perfectly bonded to the surface of CNT. The Poisson's ratio of interphase is assumed as 0.3 (Ref 32, 35) in FE analyses.

## 4. Molecular Dynamics Modeling

Molecular Dynamics (MD) is a popular computer based simulation method for evaluating the physical movements of molecular and atomic structures and lattices (Ref 24). The MD simulations were performed in the present work to estimate the elastic properties of pristine as well as defective (10,10) CNTs (vacancy defect) at the atomic scale. The MD simulation allows atoms of the system to interact with each other under different forces (i.e., Coulombic interactions, van der Waals interaction, electronic repulsion to name a few) for a certain amount of time
<table><caption>Table 3 Equivalent elastic modulus (GPa) of defective SWCNTs</caption><thead><tr><th></th><th>Pristine</th><th>1% Defects</th><th>2% Defects</th><th>3% Defects</th><th>4% Defects</th><th>5% Defects</th></tr></thead><tbody><tr><td>FE simulations</td><td>1052</td><td>975.6</td><td>905.2</td><td>834.6</td><td>755.9</td><td>680.1</td></tr><tr><td>MD simulations</td><td>1057.2</td><td>974.4</td><td>949.4</td><td>876.4</td><td>736.2</td><td>632.9</td></tr><tr><td>% of error</td><td>0.49</td><td>0.12</td><td>4.88</td><td>5.01</td><td>2.61</td><td>6.94</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Fig. 6 Stress versus Strain curve for pristine and defective standalone SWCNTs obtained from MD simulations</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7 Percentage reduction in equivalent Young's modulus versus CNT Defects</figcaption></figure>

(in nano seconds) and then gives us the view of the dynamic evaluation of the system. In the case of simulation of CNT models, the main consideration is the conservation of linear momentum. The CNT model considered for simulation is created by using NanoEngineer 1 software. The goal is to model the system and get the PDB (protein database) file, which can be further used to generate the input data file for the MD simulations. The considered model is a single walled carbon nanotube (SWCNT) of armchair type (10,10). The bond length between C C of such a structure is considered to be 1.422 Å which results in the diameter of the CNT being 13.563 Å (Ref 24). The aspect ratio of CNT is taken as 30 (Ref 26) (same as FE simulations). The molecular dynamics simulations were performed with a large scale atomic/molecular massively parallel simulator (LAMMPS) (Ref 24, 49), and the interactions of molecules in CNTs are designated by Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) force fields (Ref 50). As CNTs are made out of carbon atoms we have considered the weight of atoms to be 12.01 g/mol. The design of the CNT provides us with 11,060 such atoms. For energies of the system, the considered pair style is “pair style” airebo 2.5 1 0”, i.e., the potential scale factor is considered to be 2.5 times the bond length. In molecular dynamics study, Lennard Jones (LJ) 12 6 potential has been used to model the dispersion as well as interatomic repulsion due to its mathematical simplicity and versatility in applications. In addition, this potential requires less number of adaptable parameters with reasonable accuracy (Ref 21, 50). The simulation is done under the ‘npt’ regime at 300 K equilibration temperature. The applied strain rate is 0.00005/picosecond in the longitudinal direction. The simulation is run up for the maximum strain obtained is 15%. The carbon atoms are removed from the pristine CNTs to introduce the vacancies in the CNTs. The CNT vacancy (porosity) is generated at random at the middle section of the CNT by using the “delete atoms” command in LAMMPS. The

typical pristine and defective CNT used in MD simulations is shown in Fig. 4.

# 5. Results and Discussions

## 5.1 Equivalent Young's Modulus of Pristine and Defective SWCNT

The equivalent axial Young's modulus of CNTs ($E_{CNT}$) is being calculated from Eq. (11). The finite element simulation of pristine CNT predicts $E_{CNT}$ as 1052 GPa. The equivalent moduli of SWCNT are reported by many researchers using different methods are summarized in Table 2 which confirmed that $E_{CNT}$ obtained from the present study matched well with available open literatures. The effects of defects on the equivalent modulus of CNTs with different volume fractions are shown in Fig. 5 and summarized in Table 3. For each percentage of random CNT defects, 10 sets of calculations are performed. It can be observed the equivalent modulus decreased linearly with the increase of defects in CNTs. The mean $E_{CNT}$ values for defects of 1, 2, 3, 4, and 5% are 975.6, 905.2, 834.6, 755.9, and 680.1 GPa, respectively. The FE results show that the equivalent modulus of CNT with 5% vacancy defects reduces by 35.35% in comparison to the pristine CNTs. Figure 6 shows the stress versus strain curve for pristine and defective standalone SWCNTs obtained from MD simulations considering CNT thickness as 0.34 nm. The calculated effective moduli of CNT ($E_{CNT}$) are summarized in Table 3 and found in good agreement with FE results. In order to compare the present FE and MD simulation results, percentage of reduction in effective Young's modulus versus CNT vacancy defects are compared with available literature
<figure><img src="image_9.png" /><figcaption>**Fig. 8** Effective Young's modulus variations of long CNT reinforced nanocomposite for $\lambda_1=500$ with (a) $V_{CNT}$ for $t=0$, and (b) CNT defects ($D_{CNT}$) for $V_{CNT}=5\%$</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 9** Effective Young's modulus variations with $V_{CNT}$ for different (a) interphase thickness (long pristine CNT) with $\lambda_1=500$ and (b) modulus ratio (long CNT, $t=0.34$ nm) for pristine CNT (solid line), and 5% CNT defect ($D_{CNT}$, dashed line)</figcaption></figure>

(Ref 20, 22) and all most similar trends of variation of $E_{CNT}$ can be observed (Fig. 7).

## 5.2 Effective Modulus of Nanocomposite Reinforced with Long CNT

Finite element simulations were carried out to obtain the normalized effective Young's modulus ($E_c/E_m$) for long CNT reinforced nanocomposite for 160 configurations as shown in Table 1. The $E_c/E_m$ values without consideration of interphase ($t_{int} = 0$) are being plotted against $V_{CNT}$ for long or continuous CNT ($\lambda_3 = 1$) reinforced composite for modulus ratio ($\lambda_1$) of 500 in Fig. 8(a). Different $V_{CNT}$ values are considered as 1, 2, 3, 4 and 5%. Results are shown for pristine (no defect) and carbon vacancy defective CNT ($D_{CNT}$ varied 1 5%) reinforced composites. The defective CNTs considered in nanocomposites with 1, 2, 3, 4 and 5% defects correspond to the equivalent tensile modulus of 975.2, 905.0, 835.6, 755.2 and 678.6 GPa. The variations of $E_c/E_m$ with $V_{CNT}$ are observed as linear. When SWCNTs are reinforced with defects, the $E_c/E_m$ values drop for the same % of $V_{CNT}$ (Fig. 8a). More percentage of CNT defects ($D_{CNT}$) yield lower values of the overall elastic response of the composite. The $E_c/E_m$ values for pristine and 5% CNT defects with $V_{CNT} = 1\%$ are 6.004 and 4.253, respectively. The corresponding values for $V_{CNT} = 5\%$ are 25.973 and 17.194, respectively. Figure 8(b) represents the

variations of $E_c/E_m$ with % CNT defects for $\lambda_1 = 500$ and $V_{CNT} = 5\%$. It shows that the effective elastic modulus of composite decreases linearly with CNT defects. To demonstrate the effects of the interphase thickness of the nanocomposite, the thickness values of $t_{int} = 0, 0.17, 0.34, 0.51$, and $0.68$ nm are also plotted in Fig. 8(b). With higher values of interphase thickness, the $E_c/E_m$ was observed as higher. The $E_c/E_m$ values of nanocomposite with pristine CNT for $t_{int} = 0.17, 0.34, 0.51$ and $0.68$ nm are 26.596, 29.147, 31.905 and 34.684, respectively. The corresponding values for 5% CNT defects are 18.149, 21.388, 24.511, and 27.497, respectively. The variations of $E_c/E_m$ with $V_{CNT}$ for pristine CNT reinforced composites with different interphase thickness for $\lambda_1 = 500$ are shown in Fig. 9(a). The variations of $E_c/E_m$ with $V_{CNT}$ obtained from FE analysis are linear for all interphase thickness considered. Similar linear variations of $E_c/E_m$ with $V_{CNT}$ are also reported by (Ref 31 33). The rate of enhancement of $E_c/E_m$ with $V_{CNT}$ is indicated by the slope of $E_c/E_m$ vs $V_{CNT}$ graph and observed as high in the case of the higher value of interphase thickness ($t = 0.68$ nm) because of a large volume of material imparting higher modulus (i.e., interphase) than the matrix. For $\lambda_1 = 500$ and $V_{CNT} = 5\%$, the $E_c/E_m$ values are estimated from FE analysis as 25.973 and 34.684 for $t = 0$ and $0.68$ nm, respectively. The $E_c/E_m$ values are calculated using the rule of mixtures (ROM) without interphase from Eq. (7)
and also plotted in Fig. 9(a). Results obtained from both analytical and numerical approaches are in good agreement for all $V_{CNT}$ for the nanocomposites without the presence of any interphase. Figure 9(b) shows the variation of $E_c/E_m$ values with $V_{CNT}$ at $t = 0.34$ nm for different modulus ratios ($\lambda_1$) for pristine SWCNTs. Finite element results show that the variations of $E_c/E_m$ for different CNT volume fractions are linear for all modulus ratios considered. For a high value of $\lambda_1$, the enhancement in $E_c/E_m$ values is higher with $V_{CNT}$ compared with the lower value of $\lambda_1$ (Fig. 9b). For $\lambda_1 = 500$, the $E_c/E_m$ values are estimated from FE analysis as 6.639 and 29.147 for $V_{CNT} = 1\%$ and 5%, respectively, while their corresponding values are 1.138 and 1.645 for $\lambda_1 = 10$. The percentage variation in Equivalent Elastic Modulus between pristine and 5% defective CNT reinforced nanocomposite ($\lambda_1 = 100$) is summarized in Table 4. Further, results are compared with the MD simulations reported by Sharma et al. (Ref 43). The results are reasonable comparable for $V_{CNT} = 2.0\%$. However, higher deviation has been observed for $V_{CNT} = 4.0\%$.

## 5.3 Effective Modulus of Nanocomposite Reinforced With Short CNT

In the case of short or discontinuous CNTs ($\lambda_3 < 1$), $E_c/E_m$ values were obtained from FE analyses for 315 configurations as shown in Table 1. The effects of CNT volume fraction ($V_{CNT}$) on normalized effective Young's modulus ($E_c/E_m$) without interphase ($t_{int} = 0$) for CNT height fraction ($\lambda_3$) and modulus ratio ($\lambda_1$) of 0.6 and 500 are shown in Fig. 10(a). FE results show significant enhancement in $E_c/E_m$ values with higher values of $V_{CNT}$, but the variations of $E_c/E_m$ are not linear and less compared with long CNT reinforced nanocom

posites for the same $V_{CNT}$. Figure 10(a) shows the variation of $E_c/E_m$ for pristine (no defect) as well as carbon vacancy defective CNT ($D_{CNT}$ varied from 1 to 5%) reinforced composites. When short CNTs reinforced with defects, the $E_c/E_m$ values drop for the same % of $V_{CNT}$, similar to continuous CNT reinforced composite. In comparison with the composites reinforced with long CNT, drops of $E_c/E_m$ values due to the presence of defects in CNTs are less for short CNT reinforced nanocomposites. The $E_c/E_m$ values for pristine and 5% CNT defects with $V_{CNT} = 1\%$ are 1.662 and 1.6096, respectively. The corresponding values for $V_{CNT} = 5\%$ are 2.2243 and 2.187, respectively. Figure 10(b) represents the variations of $E_c/E_m$ with % CNT defects for $\lambda_1 = 500$, $\lambda_3 = 0.6$, and $V_{CNT} = 5\%$. It shows that the decrease in $E_c/E_m$ values with CNT defects is very less compared with long CNT reinforced composites. The variations of $E_c/E_m$ values for different interphase thicknesses are also plotted in Fig. 10(b), and higher values of interphase thickness exhibit higher $E_c/E_m$. The $E_c/E_m$ values of nanocomposite with pristine CNT for $t_{int} = 0.17$, 0.34, 0.51, and 0.68 nm are 2.289, 2.344, 2.396, and 2.448, respectively. The corresponding values for 5% CNT defects are 2.254, 2.317, 2.375, and 2.431, respectively. The variations of the normalized effective Young's modulus ($E_c/E_m$) values with $V_{CNT}$ with interphase thickness, $t = 0.34$ nm, and height fraction ($\lambda_3$) of 0.8 are presented in Fig. 11(a). The $E_c/E_m$ values obtained from FE analysis for various modulus ratios of $\lambda_1 = 10, 20, 50, 100, 200$, and 500 are shown in Fig. 11(a). Further, FE results are beings compared with pristine and CNT defects of 5%. The increment of $E_c/E_m$ is higher with the increase in $V_{CNT}$ for $\lambda_3 = 0.8$ (Fig. 11a). For $\lambda_3 = 0.8$, the $E_c/E_m$ values are estimated for pristine CNT from FE analysis as 1.766 and 3.422 for $\lambda_3 = 0.8$, respectively. In the case of CNT reinforced

<table><caption>Table 4 Percentage variation in equivalent elastic modulus for long 5% defective SWCNT-reinforced nanocomposite</caption><thead><tr><th></th><th>V<sub>CNT</sub> = 1%</th><th>V<sub>CNT</sub> = 2%</th><th>V<sub>CNT</sub> = 3%</th><th>V<sub>CNT</sub> = 4%</th><th>V<sub>CNT</sub> = 5%</th></tr></thead><tbody><tr><td>FE (Present)</td><td>13.6</td><td>17.6</td><td>19.5</td><td>20.7</td><td>21.4</td></tr><tr><td>MD (Ref 43)</td><td>.</td><td>19.2</td><td>.</td><td>14.7</td><td>.</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>Fig. 10 Effective Young's modulus variations of short CNT reinforced nanocomposite for $\lambda_1=500$ and $\lambda_3=0.6$ with (a) $V_{CNT}$ for $t=0$, and (b) CNT defects ($D_{CNT}$) for $V_{CNT} = 5\%$</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 11 Effective Young's modulus variations with (a) $V_{CNT}$ for different modulus ratio (short CNT, $\lambda_3=0.8$, $t=0.34$ nm), and (b) CNT height fraction ($\lambda_3$) for different volume fraction (short CNT, $t=0.68$ nm, $\lambda_1=500$) [pristine CNT (solid line) and 5% CNT defect ($D_{CNT}$, dashed line)]</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12 Comparison of Effective Young's modulus from FE simulation with experiments for short CNT reinforced nanocomposite</figcaption></figure>

with defects $E_c/E_m$ values are consistently lower to some extent compared to the pristine CNTs. Figure 11(b) shows the variation of $E_c/E_m$ with CNT height fraction ($\lambda_3$) for $t = 0.68$ nm and $\lambda_1 = 500$. The different values of $V_{CNT}$ are considered as 1, 2, 3, 4 and 5%. It can be observed that $E_c/E_m$ value increases with $\lambda_3$ for all values of $V_{CNT}$. The maximum value of $E_c/E_m$ corresponds to $\lambda_3 = 0.8$ and $V_{CNT} = 5\%$. It can be observed that FE results for composites with defective CNTs are slightly less compared with pristine reinforced composites (Fig. 11b). The $E_c/E_m$ values obtained from FE simulation for short CNT reinforced nanocomposite is compared with experimental data as reported by Wang et al. (Ref 29) with good agreements (Fig. 12).

## 5.4 Regression Analysis for Effective Modulus of Nanocomposite Reinforced With Long/Short CNT

The regression analysis (Ref 51, 52) is performed using Minitab 18 statistical software package to predict the normalized effective Young's modulus ($E_c/E_m$) and surface response design of nanocomposite correlating all parameters considered in FE analyses (total 475 FE simulations). A linear regression model is adopted. Analysis of variance (ANOVA) (Ref 51, 52) is used to identify the significance of different variables on the overall modulus of the nanocomposite. The results of ANOVA are presented in Table 5 and 6 for long and short CNT

composite, respectively. The SE Coef represents the standard error of the coefficient, the lesser the value, the precession in the estimation of the coefficient (Coef) is higher. The P values or probability values obtained for various parameters are the indications of the significance of each factor. The factor with a lower P value has higher statistical significance, say, for example, P value of < 0.05 indicates a significant contribution of that factor (Ref 51). It can be concluded from Table 6 that CNT defects of short CNT have the least effect on $E_c/E_m$ (P value = 0.638). The variance inflation factor (VIF) is the measure of multicollinearity. The maximum VIF value is observed as 1.49 for the interphase thickness of long CNT composite (Table 5) which is quite substantial (Ref 51). The Pareto chart (P chart) with 95% confidence level ($\alpha = 0.05$) obtained from the regression analysis for long and short CNT reinforced composites is shown in Fig. 13(a) and (b), respectively. The significance of various parameters is represented by the horizontal length of the P chart. Figure 13(a) discloses that the impact of input variables on the response for long CNT/matrix is following the stated sequence: modulus ratio ($\lambda_1$) > % CNT volume ($V_{CNT}$) > interphase thickness ($t$ in nm) > bonding coefficient ($k$) > % CNT defects ($D_{CNT}$). Similarly, the corresponding sequence for short CNT/matrix is modulus ratio ($\lambda_1$) > CNT height fraction ($\lambda_3$) > % CNT volume ($V_{CNT}$) > interphase thickness ($t$ in nm) > bonding coefficient ($k$) > % CNT defects ($D_{CNT}$) as shown in Fig. 13(b). It can be concluded from Fig. 13 shows that the modulus ratio ($\lambda_1$) is the most significant variable for $E_c/E_m$ value, whereas $D_{CNT}$ has the least contribution. It can be noted that the maximum % CNT defects ($D_{CNT}$) are considered as 5.0% in the FE analysis. The empirical relationships are derived for long and short CNT reinforced nanocomposites as follows:

For long CNT/matrix ($\lambda_3 = 1$):

$$ \frac{E_c}{E_m} = -17.29 + 0.03366\lambda_1 - 0.346D_{CNT} + 9.74t + 6.62k + 2.877V_{CNT} \quad (Eq 12) $$

where $\lambda_1$, $D_{CNT}$, $t$, $k$, $V_{CNT}$ are modulus ratio, height fraction, % CNT defect, interphase thickness in nm, bonding coefficient, and % CNT volume fraction in nanocomposites, respectively.

For short CNT/matrix ($\lambda_3 < 1$):
<table><caption>**Table 5 Analysis of variance for long CNT composite**</caption><thead><tr><th>Parameters</th><th>Sym Fbol</th><th>Coef</th><th>SE Coef</th><th>T Value</th><th>P Value</th><th>VIF</th></tr></thead><tbody><tr><td>Constants</td><td></td><td>17.29</td><td>2.31</td><td>7.48</td><td>0.000</td><td></td></tr><tr><td>Modulus ratio</td><td>λ<sub>1</sub></td><td>0.03366</td><td>0.001</td><td>24.04</td><td>0.000</td><td>1.07</td></tr><tr><td>CNT volume fraction</td><td>λ<sub>4</sub> or V<sub>CNT</sub></td><td>2.877</td><td>0.205</td><td>14.01</td><td>0.000</td><td>1.05</td></tr><tr><td>Interphase thickness</td><td>t</td><td>9.74</td><td>1.580</td><td>6.15</td><td>0.000</td><td>1.49</td></tr><tr><td>Bonding coefficient</td><td>k</td><td>6.62</td><td>2.050</td><td>3.23</td><td>0.002</td><td>1.44</td></tr><tr><td>CNT defects</td><td>D<sub>CNT</sub></td><td>0.346</td><td>0.150</td><td>2.30</td><td>0.023</td><td>1.12</td></tr></tbody></table>

<table><caption>**Table 6 Analysis of variance for short CNT composite**</caption><thead><tr><th>Parameters</th><th>Symbol</th><th>Coef</th><th>SE Coef</th><th>T Value</th><th>P Value</th><th>VIF</th></tr></thead><tbody><tr><td>Constants</td><td></td><td>2.132</td><td>0.281</td><td>7.59</td><td>0.000</td><td></td></tr><tr><td>Modulus ratio</td><td>λ<sub>1</sub></td><td>0.002222</td><td>0.000</td><td>19.90</td><td>0.000</td><td>1.17</td></tr><tr><td>CNT height fraction</td><td>λ<sub>3</sub></td><td>3.192</td><td>0.187</td><td>17.10</td><td>0.000</td><td>1.22</td></tr><tr><td>CNT volume fraction</td><td>λ<sub>4</sub> or V<sub>CNT</sub></td><td>0.1855</td><td>0.015</td><td>12.29</td><td>0.000</td><td>1.03</td></tr><tr><td>Interphase thickness</td><td>t</td><td>0.9094</td><td>0.099</td><td>9.20</td><td>0.000</td><td>1.22</td></tr><tr><td>Bonding coefficient</td><td>k</td><td>0.486</td><td>0.189</td><td>2.58</td><td>0.010</td><td>1.20</td></tr><tr><td>CNT defects</td><td>D<sub>CNT</sub></td><td>0.00467</td><td>0.001</td><td>0.47</td><td>0.638</td><td>1.06</td></tr></tbody></table>

<figure><img src="image_14.png" /><figcaption>**Fig. 13 P charts with different configurations for (a) long CNT and (b) short CNT reinforced nanocomposite**</figcaption></figure>

$$ \frac{E_C}{E_m} = -2.132 + 0.002222\lambda_1 + 3.192\lambda_3 - 0.00467D_{CNT} + 0.9094t + 0.486k + 0.1855V_{CNT} \quad (Eq 13) $$

where $\lambda_1$, $\lambda_3$, $D_{CNT}$, $t$, $k$, $V_{CNT}$ are modulus ratio, height fraction, % CNT defect, interphase thickness in nm, bonding coefficient, and % CNT volume fraction in nanocomposites, respectively.

Response surface plots for long and short CNT/matrix composites are shown in Fig. 13 and 14, respectively. The maximum values of $E_C/E_m$ obtained surface response analysis are higher for long CNT composites than that of short CNT composites. In Fig. 14(a), the surface response plot shows that the influence of CNT defects ($D_{CNT}$) on the $E_C/E_m$ value is less than the interphase thickness ($t$); whereas Fig. 14(d) confirmed that $\lambda_1$ is the most influencing factor for long CNT/matrix composite. Similar trends are also observed for short CNT composites (Fig. 15).
<figure><img src="image_15.png" /><figcaption>Fig. 14 Response surface plot of long CNT reinforced nanocomposite for different configurations</figcaption></figure>

6. Conclusions

The present study was intended to determine the effective
directional Young's modulus of nanocomposite for CNT with
defects as well as different combinations of modulus ratio (ratio
of Young's modulus of carbon nanotube and matrix), CNT
height fraction (ratio of carbon nanotube height and RVE
length), CNT volume fraction, and different interphase prop
erties (different thickness and material properties) based on
different bonding coefficients.

The following are the conclusions from the present work:

* It is observed from the present FE analysis and MD simulations that the equivalent tensile moduli of pristine SWCNT (10,10) are estimated as 1052 GPa and 1057.16 GPa, respectively, which are in a good agreement with the data reported in the open literature. The equivalent modulus decreases linearly with increasing CNT defects. The FE results show that the equivalent modulus of CNT with 5% vacancy defects reduces by 35.35% in comparison to the pristine CNTs.

* The interphase between CNT and matrix plays a significant role in the overall elastic response of the nanocomposite. On increasing the interphase thickness, the

normalized effective Young's modulus increases for CNT
reinforced composites. For a constant interphase thickness,
the higher the value of the bonding coefficient, the overall
Young's modulus of composites will be higher.

* Increasing the height fraction of CNT leads to a higher value of effective Young's modulus. In the case of long as well as short CNT reinforced nanocomposites, it can be concluded that modulus ratio, height fraction, and volume fraction influence the effective modulus significantly. The effects of the presence of CNT defects are more significant on the effective Young's modulus of SWCNT reinforced nanocomposite for continuous or long CNTs in comparison with the discontinuous or short one.

* Regression analysis and Pareto chart confirmed that the effect of input variables on the response for long CNT/matrix is following the stated sequence: modulus ratio ($\lambda_1$) > % CNT volume ($V_{CNT}$) > interphase thickness ($t$ in nm) > bonding coefficient ($k$) > % CNT defects ($D_{CNT}$). Similarly, the corresponding sequence for short CNT/matrix is modulus ratio ($\lambda_1$) > CNT height fraction ($\lambda_3$) > % CNT volume ($V_{CNT}$) > interphase thickness ($t$ in nm) > bonding coefficient ($k$) > % CNT defects ($D_{CNT}$).
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 15** Response surface plot of short CNT reinforced nanocomposite for different configurations</figcaption></figure>

**Conflict of interest**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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