Effects of carbon nanotubes’ dispersion on effective mechanical properties of nanocomposites: A finite element study

Abstract

Effects of volume fraction and random dispersion of carbon nanotubes on effective mechanical properties of carbon nanotube-reinforced nanocomposites are studied at continuum level using finite element methods. Utilizing a continuum model of tubes with specific pairing of elastic properties and wall thickness for describing carbon nanotubes, representative volume elements with non-uniform distribution of uniaxial nanotubes within a base matrix are generated. Furthermore, a dispersion quantification technique is employed to quantify nanotubes’ dispersion degree. Finite element simulations are carried out in six independent loading conditions, while applying periodic boundary conditions to the models. Homogenizing the models and using the formulations of linear elasticity, equivalent mechanical properties of the models are computed and the effects of the aforementioned influential parameters as well as the modeling volume size are investigated. The results demonstrate that in volume fractions higher than 5%, the effects of carbon nanotubes’ dispersion become more significant and if increasing the volume fraction leads to bigger agglomerations and subsequently worse dispersion, the effective properties of the composite as a whole will decrease. Moreover, the pre- and post-processing procedures implemented are verified by analyzing previously studied models available in the literature and comparing the corresponding results.

Keywords

Carbon nanotubes nanocomposites effective mechanical properties finite element method

Introduction

Carbon nanotubes (CNTs) possess incredible properties that are of interest in various disciplines such as mechanics and electrics, thus their use as reinforcing elements has been a novel subject of research and application.^1,2 However, due to the presence of many influential parameters such as nanotubes’ mechanical properties, structure, size, volume fraction, orientation, and degree of dispersion, the design and optimization of CNT-reinforced materials have been limited.

Various results have been reported for CNTs’ mechanical properties evaluated through experimental observations,^3–6 atomistic simulations,^7–9 and continuum modelings^10–14 over the past two decades. The results of many molecular dynamics (MD) simulations indicate that as long as proper attention is given to the definition of wall modulus and thickness, nanotube walls could be modeled accurately based on elastic shells theory.¹² Pantano et al.¹² presented a continuum shell theory/finite element (FE) approach for modeling CNTs. Using an equivalent-continuum model, Odegard et al.¹³ developed a method for modeling a nanotube and its surrounding polymer along with their interface as an effective fiber.

One of the key elements that directly influences the effectiveness of CNTs as reinforcing materials is the state of their distribution within the base matrix. Because of their small size and the Van der Waals interactions between them, CNTs tend to stay close to one another and form agglomerations. The presence of these agglomerations reduces the interactions between CNTs and matrix and consequently their efficiency as reinforcement mechanism. Several experimental works^15–18 have been conducted on studying the effects of CNT dispersion and orientation on effective mechanical properties of CNT-reinforced composites. But, in order to analyze the effects of non-uniform dispersion, it is necessary to define a parameter that can properly represent the state of dispersion. Numerous methods have been proposed to quantify the dispersion using various concepts, such as average deviation from uniform distribution^19,20 and average inter-particle distance.^21,22 Reviewing some of the proposed methods, Khare and Burris²³ developed a simple and broadly applicable dispersion quantification technique that directly reflects the reinforcement mechanism at nanoscale. They defined the free-space length, $L_{fs}$ , as the characteristic size of the unreinforced domains of matrix. Another factor that directly affects the mechanical reinforcement of nanotubes is their orientation.²⁴ Thus, in order to enhance the properties of nanocomposites, various alignment techniques, such as mechanical stretching and magnetic and electric fields, are proposed to align CNTs.^25,26 In the current study, it is assumed that the CNTs are uniaxially dispersed through the host matrix.

Apart from experimental approaches, theoretical (e.g. Mori–Tanaka)²⁷ and numerical methods at nano- and macro-scale have been implemented to study the mechanical behavior of nanocomposites.^28–30 Sheng et al.²⁹ employed both numerical and analytical models based on the “effective clay particle” to evaluate the overall elastic modulus of amorphous and semi-crystalline polymer/clay nanocomposites. Using FE methods, Hbaieb et al.³⁰ studied the overall stiffness of 2D and 3D polymer-based nanocomposites with random distribution of nanoclay particles. They compared their results with corresponding results derived by Mori–Tanaka model and realized that it cannot accurately estimate the stiffness for randomly oriented particles and volume fractions of over 5%. Moreover, they demonstrated that 2D models of the nanocomposites cannot accurately predict the stiffness either.

Numerical approaches based on MD can be useful at nanoscale, e.g. for studying the structure of nanotubes and their interactions with surrounding matrix. But, implementing them to investigate the macroscopic properties of an entire nanocomposite is very complicated as well as computationally expensive and will restrict the simulations to small models. Adnan et al.³¹ analyzed the influence of filler size on elastic properties of polymer nanocomposites by MD simulations. Han and Elliott³² performed classical MD simulations on polymer/CNT composite models constructed by embedding a single wall CNT into two different amorphous polymer matrices with different volume fractions. Their results show that CNTs can be used to mechanically reinforce an appropriate polymer matrix, especially in the longitudinal direction of the nanotube. Continuum mechanics-based approaches are still practical for studying the interfacial interactions between CNT and the matrix and the macroscopic properties of nanocomposites; as reviewed by Rafiee et al.³³ The interaction between CNT and matrix, which is defined as an interphase region in continuum modeling, is the key element in stress transferring and thus the reinforcement phenomena. In order to study the load transfer at the interface and thus its effects on macroscopic properties at continuum level, using numerical simulations, some researchers have considered a separate material region between CNT and the matrix (e.g. Wan et al.³⁴ and Ayatollahi et al.³⁵) and some others modeled this region using variable elements such as truss, or spring; where they achieved the properties of the elements by Lennard-Jones potentials (e.g. Li and Chou³⁶ and Montazeri and Naghdabadi³⁷). Using FE analysis on square representative volume elements (RVEs), Chen and Liu³⁸ evaluated the effective mechanical properties of CNT-based composites with uniform dispersion of CNTs. Utilizing the effective fiber model,¹³ Luo et al.³⁹ investigated the effects of distribution state and geometry of CNTs (regular and staggered arrays of straight and wavy CNTs) on overall mechanical properties of nanocomposites based on multi-scale homogenization theory. Leclerc and Surville⁴⁰ studied the effects of fiber dispersion on effective elastic properties of 2D overlapping random fiber composites and observed that the clustering phenomenon undermines the reinforcement of fibers.

It can be observed that most of the simulations are carried out on models with uniform distribution of CNTs. However, random distributions must be considered in order for studies to be more realistic and practical. In the present paper, 3D RVEs with non-uniform dispersion of uniaxial CNTs within the base matrix are generated and a dispersion quantification technique²³ is employed to evaluate the dispersion state of CNTs, thus enabling the study of dispersion effects. Applying six independent load cases to the modeling volumes together with periodic boundary conditions (PBCs),⁴¹ the FE simulations are conducted. Subsequently, by homogenizing the models, effective mechanical properties of the materials are evaluated and the effects of two influential parameters, namely CNTs’ volume fraction and dispersion, are studied. Moreover, the pre- and post-processing procedures are verified by analyzing previously studied models available in the literature and comparing the corresponding results.

Modeling

The first step in numerical simulations is to define a modeling volume that could represent the whole medium under study. In the subject of studying the behavior of nanocomposites with non-uniform dispersion of nanotubes, because of the large difference between the size of the whole model and the smallest dimension, which is related to nanotubes, analyzing the whole nanocomposite is not practically possible. Therefore, it is necessary to select a volume element which could stand for the effective behavior of the medium with least deviation from it and at the same time does not cause exceeding computational cost. Moreover, in order to be able to study the nanocomposite model by analyzing an RVE, the use of PBCs is required. The primary requirement to apply PBCs to a model is to observe the constraints of periodicity in generating the RVE which means that if a particle passes through a boundary face, the remaining part of it must continue from the opposite face. Thus by repeating the modeling volume in three dimensions, the whole medium can be generated without any void or discontinuity.

Based on the above constraint, a code was written in MATLAB⁴² that could generate models of nanocomposites with random non-uniform distribution of uniaxial nanotubes within the base matrix and consequently generate the required script for modeling the RVE in Abaqus.⁴³ A typical model of the RVE studied in present research is shown in Figure 1. The nanotubes are modeled as tubes with a specific pairing of elastic properties and tube wall thickness, as modeled by Pantano et al.,¹² with length of $L_{CNT}$ and outer diameter of D and they are dispersed in a square matrix with length of L_m and thickness of t_m. Since the nanotubes are dispersed in the middle of the matrix cross section, the modeling volume has a plane of elastic symmetry perpendicular to its cross section and thus can be considered as monoclinic. In the following section the method by which the dispersion of nanotubes in base matrix is quantified is discussed.

Figure 1.

Isometric view of the RVE with a periodic structure.

Free-space length, $L_{fs}$

As it was mentioned earlier, nanoparticles play their role as reinforcing materials best when they are well dispersed within a base matrix. But they are not often dispersed regularly and tend to form agglomerations which can be very detrimental in higher volume fractions. In other words, in higher volume fractions, an increase in nanoparticles loading without considering their state of distribution would not guarantee increase in effective mechanical behavior of nanocomposite.

Primarily, a dispersion characterization technique should be applied, a technique that can evaluate a quantity for state of nanoparticles’ distribution within the base matrix. In 2010, Khare and Burris²³ developed the free-space length method for quantifying the nanocomposite dispersion. They reviewed the previously proposed methods, and stated that using “uniform distribution” and “inter-particle distance” as dispersion characterization tools, those methods were insensitive to “filler size and loading” and “dispersion quality”, respectively. Therefore, none of those methods have been widely adopted in the field. The free-space length method, on the other hand, appropriately reflects the reinforcement mechanisms at nanoscale, has a physically intuitive output, and is simple and broadly applicable. They introduced the free-space length $L_{fs}$ as a characteristic size of the unreinforced polymer domain. As the distribution state of nanoparticles becomes more uniform, the size of this domain becomes smaller and, therefore, $L_{fs}$ reduces. They defined the free-space length as the size of the largest square that, when randomly placed on the RVE, would most likely contain no particles within it.

The algorithm that was proposed by Khare and Burris²³ was used to evaluate the free-space length $L_{fs}$ for each generated model in current paper. Based on this algorithm, initially a square of length L is randomly placed on the RVE and the total area of nanotubes which is within the square is measured and the number is stored. Repeating this step statistically significant number of times (e.g. 10,000 times), the histogram of stored numbers is used to calculate the mode which is the most probable amount for the areas of nanotubes within a random square domain of length L on the RVE. Iterating on size of the square, the largest square for which the mode becomes zero can be found. The length of this square is introduced as the free-space length $L_{fs}$ .

Figure 2 illustrates the above procedure for an example RVE. It can be seen that for square lengths of 30 and 20 nm the modes are 170 and 68, respectively. In this example the free-space length was evaluated as 10 nm. It means that the mode will be zero for any square size smaller than 10 nm, but it will not become zero for larger square sizes.

Figure 2.

Applying the dispersion quantification technique,²³ to a square RVE of size 60 nm.

Governing equations

The constitutive equation for linear elastic behavior of materials is written as

σ_{i} = C_{ij} ɛ_{j}; i, j = 1, 2, \dots, 6

(1)

where σ, ɛ, and C represent stress, strain, and elastic stiffness tensors, respectively. The stiffness tensor is a symmetric tensor, i.e.

C_{ij} = C_{ji}

. Moreover, the following numbering is utilized in equation (1) to describe the stress and strain components.

σ_{1} = σ_{11}, σ_{2} = σ_{22}, σ_{3} = σ_{33} σ_{4} = σ_{23}, σ_{5} = σ_{13}, σ_{6} = σ_{12} ɛ_{1} = ɛ_{11}, ɛ_{2} = ɛ_{22}, ɛ_{3} = ɛ_{33} ɛ_{4} = 2 ɛ_{23}, ɛ_{5} = 2 ɛ_{13}, ɛ_{6} = 2 ɛ_{12}

(2)

Since the plane which cuts the thickness of the present models in half is a plane of elastic symmetry, the whole nanocomposite is then monoclinic and its stiffness tensor, having 13 independent components, has the form

[C_{ij}] = [C_{11} C_{12} C_{13} 00 C_{16} C_{12} C_{22} C_{23} 00 C_{26} C_{13} C_{23} C_{33} 00 C_{36} 000 C_{44} C_{45} 0000 C_{45} C_{55} 0 C_{16} C_{26} C_{36} 00 C_{66}]

(3)

If the material has two perpendicular planes of symmetry, it is orthotropic and its C matrix is further reduced to

[C_{ij}] = [C_{11} C_{12} C_{13} 000 C_{12} C_{22} C_{23} 000 C_{13} C_{23} C_{33} 000000 C_{44} 000000 C_{55} 000000 C_{66}]

(4)

In addition, the inverse of C matrix, compliance matrix, for orthotropic materials can be written based on the elastic constants as

[S_{ij}] = [\frac{1}{E_{1}} - \frac{ν_{21}}{E_{2}} - \frac{ν_{31}}{E_{3}} 000 - \frac{ν_{12}}{E_{1}} \frac{1}{E_{2}} - \frac{ν_{32}}{E_{3}} 000 - \frac{ν_{13}}{E_{1}} - \frac{ν_{23}}{E_{2}} \frac{1}{E_{3}} 000000 \frac{1}{G_{23}} 000000 \frac{1}{G_{13}} 000000 \frac{1}{G_{12}}]

(5)

where

S_{ij}

is the compliance matrix, E_i is the Young’s modulus along axis i,

G_{ij}

is the shear modulus in

ij

plane, and

ν_{ij}

is the Poisson’s ratio that denotes for the ratio of contraction in j direction when a unit extension is applied in i direction. The compliance matrix is used in the following sections to evaluate the effective elastic constants of nanocomposites.

FE simulations

A computer code is developed in MATLAB that generates required script to mesh each RVE in Abaqus. Using a combination of C3D8R and C3D6 elements together with an appropriate partitioning algorithm, allowed generating a regular and fine mesh that enabled applying PBCs to 3D models, which needs the nodes on opposite faces to be symmetric. Because of the small size of the nanotubes’ thickness compared to the RVE size, it is very important that elements become smaller around the nanotubes gradually and with no distortion. Moreover, perfect bonding is applied between CNTs and the base matrix by use of necessary partitioning and seeding on CNTs and their surrounding matrix. Figure 3 shows a meshed RVE studied in current paper.

Figure 3.

Isometric view of a meshed RVE.

Periodic boundary conditions

In addition to the necessity of observing periodic structure in the models, which guarantees generation of the whole medium by repeating the RVE in three dimensions with no discontinuity, proper constraint relations should be applied on the boundaries so that the models can maintain their repeatability after deformation under loadings. In other words, opposite faces are required to deform identically so that by copying the RVE in three dimensions after deformation they could match each other completely. PBCs as well as rigid body constraints are applied to the models, based on the formulation developed by Abolfathi et al.⁴¹

In order to apply PBCs to the RVE, at first it is necessary to create node sets of nodes on boundary faces and edges in which node numbers are sorted according to their coordinates; so that each two opposite nodes are mentioned in their node sets in same order as each other. For this purpose, a separate simulation is executed on each model and the coordinates and numberings of boundary nodes are reported as outputs of the simulation. Subsequently, the reported file is inserted as input data in another MATLAB code, which divides the nodes into node sets and sorts them with respect to their locations and finally generates equations of periodicity between them.

Homogenization

To evaluate the effective elastic properties associated with the homogenized models of the RVEs studied, six independent loading cases (three axial and three shear forces) along with the aforementioned boundary conditions are applied to them and FE simulations are carried out. Getting stress and strain components on the models as outputs of the analysis, their average over the entire volume is calculated using the following equations

{\bar{σ}}_{i} = \frac{1}{V} \int_{V} σ_{i} dv

(6)

{\bar{ɛ}}_{i} = \frac{1}{V} \int_{V} ɛ_{i} dv

(7)

Subsequently, the effective mechanical properties of the homogenized models are computed by employing the inverse of constitutive equation (1) which is written as

{\bar{ɛ}}_{i} = S_{ij}^{*} {\bar{σ}}_{j}; i, j = 1, 2, \dots, 6

(8)

where the superscript “*” denotes the effective properties of the homogenized models.

The loading conditions are employed in such a way that in each case, all the average stress components except one of them, the one which corresponds to that particular loading condition, become zero. Therefore, the components of each column of $S^{*}$ matrix can be obtained by the output results of its corresponding load case. Moreover, because of the symmetric nature of the compliance matrix, each non-diagonal component can be obtained from two independent load cases. It means that $S_{ij}$ will be obtained by the load case by which $σ_{j}$ becomes non-zero and $S_{ji}$ will be obtained by the load case by which $σ_{i}$ becomes non-zero. Checking the symmetry of the obtained effective compliance tensor is used as double checking criterion throughout the analysis.

Finally, after all the components of the effective compliance matrix are derived, the effective stiffness matrix can be evaluated with the following:

C^{*} = S^{* - 1}

(9)

Models with uniform CNT dispersion

In order to verify the accuracy of pre- and post-processing procedures utilized in present research, they are applied to a number of polymer composite models reinforced with regular arrays of straight CNTs. The macroscopic mechanical properties of composites are obtained employing the effective continuum fiber model, proposed by Odegard et al.,¹³ and they are compared with their counterparts in research of Luo et al.³⁹

Descriptions of squarely arranged fibers and the repeating unit cell studied here are illustrated in Figure 4.

Figure 4.

Illustrations of the regular arrays of straight fibers. a) View in $x_{2} - x_{3}$ plane, b) view in $x_{1} - x_{2}$ plane, c) view in $x_{1} - x_{3}$ plane, and d) unit cell for regular arrays of straight fibers.

Three views of the models are depicted in Figure 4(a) to (c) and an isotropic view of the simulated periodic unit cell is depicted in Figure 4(d). The diameter and length of the effective fibers are D and L_f, respectively, and parameters $2 T_{f}$ and $2 H_{f}$ denote the end gap between two coaxial fibers and parallel distance of neighboring ones, respectively. The parameters a, b, and c, shown on the unit cell are the length, height, and width of the unit cell, respectively, and can be calculated for various volume fractions of effective fibers based on their diameter, which is the basic unit in the analysis. Moreover, since fibers are squarely arranged in $x_{2} - x_{3}$ plane, the height and width of the unit cell are equal.

Nanocomposite reinforced by regular arrays of straight fibers which is simulated using the periodic RVE depicted in Figure 4(d) is approximately assumed as transversely isotropic material. The Hooke’s law for these materials is written as:

{ɛ_{1} ɛ_{2} ɛ_{3} ɛ_{4} ɛ_{5} ɛ_{6}} = [\frac{1}{E_{1}} - \frac{ν_{21}}{E_{2}} - \frac{ν_{21}}{E_{2}} 000 - \frac{ν_{12}}{E_{1}} \frac{1}{E_{2}} - \frac{ν_{23}}{E_{2}} 000 - \frac{ν_{12}}{E_{1}} - \frac{ν_{23}}{E_{2}} \frac{1}{E_{2}} 000000 \frac{1}{G_{23}} 000000 \frac{1}{G_{12}} 000000 \frac{1}{G_{12}}] {σ_{1} σ_{2} σ_{2} σ_{4} σ_{5} σ_{6}}

(10)

Note that subscripts 1, 2, and 3 in equation (10) refer to x₁, x₂, and x₃ directions of the coordinate system as shown in Figure 4(d). By applying the PBCs to the unit cells and subjecting them to four load cases (two axial forces along x₁ and x₂ directions and two shear forces in $x_{1} - x_{2}$ and $x_{2} - x_{3}$ planes) the diagonal terms of the compliance matrix in columns one, two, four, and six can be computed and, therefore, the macroscopic mechanical properties of the nanocomposite can be obtained.

Numerical results

Following the procedures explained in previous sections, the described models are studied and numerical results are derived. Primarily the results used for verifying the procedures of current paper are presented and subsequently, the main numerical results of the paper are presented.

The polymer matrix utilized throughout the present research is LaRC-SI and its mechanical properties, found in Odegard et al.,¹³ are Young’s modulus $E_{m} = 3.8 GPa$ and the Poisson’s ratio $ν_{m} = 0.4$ , which are used by Luo et al.³⁹ too. In addition, the five independent elastic constants of the effective fiber model with transverse isotropic constitutive relationship are:¹³

C_{11}^{f} = 457.6, C_{12}^{f} = 8.4, C_{22}^{f} = 14.3, C_{23}^{f} = 5.5, C_{66}^{f} = 27.0 s (GPa)

(11)

and the other elastic constants can be calculated as follows:

C_{13}^{f} = C_{12}^{f} = 8.4 C_{33}^{f} = C_{22}^{f} = 14.3, C_{44}^{f} = \frac{1}{2} (C_{22}^{f} - C_{23}^{f}) = 4.4, C_{55}^{f} = C_{66}^{f} = 27.0 (GPa)

(12)

Repeating unit cells of regular array of straight fibers with aspect ratio $L_{f} / D = 20$ are generated in different volume fractions of fibers and in three cases of $T_{f} = 0.1 H_{f}$ , $T_{f} = H_{f}$ , and $T_{f} = 10 H_{f}$ . Figure 5 shows the effects of volume fraction on four effective elastic constants ( $E_{11}$ , $E_{22}$ , $G_{12}$ , and $G_{23}$ ) of the homogenized models. It can be seen from the comparison that the presently employed procedure gives the results with proper accuracy.

Figure 5.

Variations of effective elastic constants with fibers volume fraction. Comparison of the present results (solid lines, filled markers) with previous ones from Luo et al.³⁹ (dashed lines, empty markers). a) Triangular marker, blue lines,: $T_{f} = 0.1 H_{f}$ , b) square marker, red lines: $T_{f} = H_{f}$ , and c) circular marker, green lines: $T_{f} = 10 H_{f}$ .

As it was mentioned in the “Modeling” section, the CNTs in present paper are modeled based on the continuum shell theory/FE modeling proposed by Pantano et al.¹² The properties of CNTs are modulus of elasticity $E_{wall} = 4.84 TPa$ , Poisson’s ratio $ν = 0.19$ , and wall thickness $t_{wall} = 0.075 nm$ . The nanotubes’ length and outer diameter are assumed to be 20 and 1 nm, respectively. Nanotubes are dispersed non-uniformly in the previously introduced polymer matrix. The thickness of the matrix t_m is assumed to be 2 nm, and the procedure of evaluating the proper size for the RVE is discussed in the following sections.

Firstly, a mesh sensitivity study is conducted for an example model studied in this research in order to evaluate the proper sizing of elements by which the results have reasonable accuracy with appropriate computational cost. For a square matrix length $L_{m} = 60 nm$ , CNT volume fraction $V_{CNT} = 3 %$ , and free-space length $L_{fs} = 12 nm$ , components of the effective stiffness tensor for monoclinic material are derived and presented in Figure 6 for three cases of meshing. It can be observed that the case with 98,000 elements for this model gives reasonable results. Therefore, in order to select the optimum mesh sizing, the sizes used in this case are applied in meshing the RVEs in the following.

Figure 6.

Mesh sensitivity study for an RVE with $L_{m} = 60 nm$ , $V_{CNT} = 3 %$ , and $L_{fs} = 12 nm$ .

Moreover, it is obviously noticed that the scale of the stiffness tensor components in Figure 6(b) is three to four orders smaller than the components shown in Figure 6(a), thus they can be assumed to be equal to zero. Therefore, the effective stiffness tensor of the homogenized models gets the form of orthotropic materials and the models can be approximately assumed to act as orthotropic materials with stiffness and compliance tensors in the forms of equations (4) and (5), respectively. This is because the application of PBCs in the simulation process repeats the RVEs in three dimensions infinitely to create the entire nanocomposite and thus the two orthogonal planes perpendicular to length and height of the RVEs approximately behave as two other planes of elastic symmetry for the RVEs. Consequently, in the following sections, equation (5) is used to evaluate the effective elastic properties of the homogenized nanocomposite models.

As it was mentioned earlier, proper size should be defined for RVE to perform numerical simulation.

Here, for evaluating the appropriate size for the RVE, a dimensionless parameter $λ = L_{m} / L_{CNT}$ is defined, based on the approach proposed by Jafari et al.,⁴⁴ and models of 3% volume fraction of CNTs with an equal dispersion ( $L_{fs} = 12 nm$ for all models) are generated in square RVEs ranging from 1.5 to 5 times of the CNTs’ length. Figure 7 shows the variations of effective elastic constants obtained for the models with respect to dimensionless size λ. It can be seen that after $λ = 3$ , all the effective elastic constants remain approximately constant and are not sensitive to the RVE size. Therefore, square RVE with dimensionless size $λ = 3$ is selected as a proper modeling volume which could appropriately represent a nanocomposite with non-uniform dispersion of uniaxial CNTs.

Figure 7.

Variations of effective elastic constants with respect to dimensionless size of the RVE for CNT volume fraction of 3% and free-space length of 12 nm.

In order to evaluate the effects of influential parameters on mechanical behavior of nanocomposites, random models with various CNT volume fractions ( $V_{CNT} = 1 %, 3 %, 5 %, 7 %$ ) and dispersions ( $L_{fs} = 15, 25, and 35 nm$ ) are generated. It should be pointed that for each particular case of $V_{CNT}$ and $L_{fs}$ pairing, five random models are generated and the FE analysis is performed. To illustrate the influence of the aforementioned parameters on the effective elastic constants, the ratio of each coefficient to its counterpart in the neat polymer is calculated and their variations with respect to CNT volume fraction $V_{CNT}$ are presented in Figure 8. The results are presented for the three different values of free-space length $L_{fs}$ too. The data points on the plots denote the average values for five random RVEs and the 95% confidence interval for the obtained results is shown by error bars.

Figure 8.

Variations of effective coefficients with CNT volume fraction and for $L_{fs} =$ 15 nm (blue line with circular marker), 25 nm (red line with square marker), and 35 nm (green line with triangular marker). Each data point is the average for five random RVEs. Margin of error are displayed via error bars at each point.

It is shown that as CNT volume fraction increases, all the effective elastic constants increase, but the most improvement is in Young’s modulus along CNTs direction.

Moreover, it can be seen that as free-space length $L_{fs}$ increases in constant volume fraction, which denotes decrease in uniformity of CNTs in the base matrix, the effective coefficients decrease. The influence of this parameter becomes more dominant in higher volume fractions, to the point where it becomes more important than CNT volume fraction. This phenomenon has happened in the models of current study from CNT volume fraction of 5%. It can be easily noticed from Figure 8(a) that the effective Young’s modulus along CNTs direction is higher for a nanocomposite with $V_{CNT} = 5 %$ and $L_{fs} = 15 nm$ , than a model with $V_{CNT} = 7 %$ and $L_{fs} = 25 or 35 nm$ . The reason is that as CNT volume fraction increases, bigger agglomerations can be created with increasing $L_{fs}$ which leads to inefficient nanotube reinforcement. However, the free-space length does not have that huge of an influence on other effective elastic properties of nanocomposites.

Although nanocomposites studied in current research contain uniaxially dispersed CNTs, it must be mentioned that a truly random distribution would include random orientations of nanotubes within the nanocomposite. It is known that the stress is transferred from the matrix to CNTs at their interfaces through shear stress,⁴⁵ which is a result of the difference between axial displacements of matrix and CNT. In a random orientation of a nanotube, an increase in the angle between nanotube axis and loading direction, would decrease the equivalent aspect ratio of the nanotube and thus the load transfer efficiency reduces.⁴⁶ Therefore, for a nanocomposite with aligned CNTs, much larger increase in effective mechanical properties will be achieved.

Figure 9 shows the normal stress $σ_{11}$ distribution in middle section of a deformed shape of an RVE that is subjected to a normal loading along CNTs direction. Three types of CNTs regarding their neighboring conditions are specified via letters a, b, and c that denote CNTs with close end-to-end distance, isolated CNTs, and overlapped CNTs, respectively. It can be seen from the stress distributions in the nanotubes of these three cases that the load carrying efficiency can vary due to the interactions between particles. The load carrying efficiency of CNTs with close end-to-end distance is higher than that of the two other types, which cause a significant enhancement in effective stiffness $E_{11}$ of the nanocomposite. On the other hand, the small load carrying capacity of the overlapped CNTs results in a decrease in the effective stiffness $E_{11}$ . In addition, as the CNT volume fraction increases, chances for the three aforementioned kinds of interactions between CNTs (i.e. close end-to-end distanced, isolated, and overlapped; see Figure 9) to take place increase and, thus, variations in the effective coefficients of the nanocomposite can increase in random generations of modeling volumes. The effective stiffness $E_{11}$ of the nanocomposite becomes higher, if CNT interactions within the matrix are dominated by type “a” shown in Figure 9, and lower, if interactions of type “c” predominate. This phenomenon, which is previously explained by Sheng et al.²⁹ for polymer/clay nanocomposites, can be a reason why the margin of errors for the obtained results increases in higher volume fractions of CNTs and have some overlaps for different free-space lengths (see Figure 8).

Figure 9.

Stress contour on an RVE subjected to tension in one direction and effects of CNT neighboring conditions on their load carrying capacities, a) CNTs with close end-to-end distance, b) isolated CNTs, and c) overlapped CNTs.

Conclusion

The effects of CNT volume fraction and dispersion on the effective behavior of CNT-reinforced nanocomposites are investigated using FE simulations. A MATLAB code is developed to both generate periodic RVEs of randomly distributed CNTs within a base matrix and generate the required script to model and mesh the modeling volumes in Abaqus. Quantifying the dispersion of CNTs in the base matrix using a dispersion characterization technique enabled studying the effects of dispersion on effective mechanical properties of nanocomposites in detail. PBCs are applied to the models and they are analyzed under six independent loading conditions. Homogenizing the models and computing the equivalent stiffness tensor, it is observed that because of applying PBCs to the models, they approximately act as orthotropic materials. Therefore, constitutive relations for orthotropic materials are utilized to calculate the effective elastic constants of the homogenized models.

By defining the dimensionless size λ for the periodic RVEs, the effects of RVE size are studied and it is determined that $λ = 3$ is a proper size for the modeling volume to perform numerical simulations. Furthermore, it is observed that the Young’s modulus $E_{11}$ is the elastic constant which has the most improvement due to the addition of CNTs to the base matrix. The results verified the influence of CNT dispersion on mechanical behavior of nanocomposites such that as the uniformity of CNTs within the base matrix decreases (increase in free-space length $L_{fs}$ ), the effective elastic properties decrease. This influence becomes more significant in higher volume fractions; to the point where from CNT volume fraction of 5%, a decrease in dispersion of CNTs becomes more crucial than an increase in their volume fraction. Therefore, it should be considered in nanocomposite technology that increased volume fractions of effectively dispersed CNTs would result in higher macroscopic properties of nanocomposites. However, if that increase leads to a higher agglomeration rate and thus a lower dispersion, it could have an inverse effect on the mechanical properties of the resulting composite material.

Footnotes

Conflict of interest

None declared.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Thostenson

Chou

. Nanocomposites in context. Compos Sci Technol 2005; 65: 491–516.

Hussain

Hojjati

Okamoto

. Review article: polymer-matrix nanocomposites, processing, manufacturing, and application: an overview. J Compos Mater 2006; 40: 1511–1575.

Treacy

MMJ

Ebbesen

Gibson

. Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 1996; 38: 678–680.

Krishnan

Dujardin

Ebbesen

. Young’s modulus of single-walled nanotubes. Phys Rev Lett 1998; 58: 14013–14019.

Poncharal

Wang

Ugarte

. Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 1999; 283: 1513–1516.

Wong

Sheehan

Lieber

. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes and carbon nanotubes. Science 1997; 277: 1971–1973.

Yakobson

Brabec

Bernholc

. Nanomechanics of carbon tubes: instabilities beyond the linear response. Phys Rev Lett 1996; 76: 2511–2514.

Zhou

Ou-Yang

. Strain energy and Young’s modulus of single-wall carbon nanotubes calculated from electronic energy-band theory. Phys Rev B 2000; 62: 13692–13696.

Ou-Yang

. Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young’s moduli dependent on layer number. Phys Rev B 2002; 65: 233407–233407.

10.

Zhang

Huang

Geubelle

. The elastic modulus of single-wall carbon nanotubes: continuum analysis incorporating interatomic potentials. Int J Solids Struct 2002; 39: 3893–3906.

11.

Odegard

Gates

Nicholson

. Equivalent-continuum modeling of nano-structured materials. Compos Sci Technol 2002; 65: 1869–1880.

12.

Pantano

Parks

Boyce

. Mechanics of deformation of single- and multi-wall carbon nanotubes. J Mech Phys Solids 2004; 52: 789–821.

13.

Odegard

Gates

Wise

. Constitutive modeling of nanotube-reinforced polymer composites. Compos Sci Technol 2003; 63: 1671–1687.

14.

Ding

Chen

S-J

Cheng

. The atomic-scale finite-element modeling of single-walled carbon nanotubes. Proc Inst Mech Eng C J Mech Eng Sci 2007; 221: 613–617.

15.

Katihabwa

Wang

Jiang

. Multi-walled carbon nanotubes/silicone rubber nanocomposites prepared by high shear mechanical mixing. J Reinf Plast Compos 2011; 30: 1007–1014.

16.

Montazeri

Chitsazzadeh

. Effect of sonication parameters on the mechanical properties of multi-walled carbon nanotube/epoxy composites. Mater Des 2014; 56: 500–508.

17.

Shokrieh

Saeedi

Chitsazzadeh

. Evaluating the effects of multi-walled carbon nanotubes on the mechanical properties of chopped strand mat/polyester composites. Mater Des 2014; 56: 274–279.

18.

Haque

Larsen

. Crystallization and mechanical properties of functionalized single-walled carbon nanotubes/polyvinylidene fluoride composites. J Reinf Plast Compos 2012; 31: 1417–1425.

19.

Kashiwagi

Fagan

Douglas

. Relationship between dispersion metric and properties of PMMA/SWNT nanocomposites. Polymer 2007; 48: 4855–4866.

20.

Luo

Koo

. Quantifying the dispersion of mixture microstructures. J Microsc 2007; 225: 118–125.

21.

Basu

Tewari

Fasulo

. Transmission electron microscopy based direct mathematical quantifiers for dispersion in nanocomposites. Appl Phys Lett 2007; 91: 053105–053105.

22.

Hamming

Qiao

Messersmith

. Effects of dispersion and interfacial modification on the macroscale properties of TiO₂ polymer–matrix nanocomposites. Compos Sci Technol 2009; 69: 1880–1886.

23.

Khare

Burris

. A quantitative method for measuring nanocomposite dispersion. Polymer 2010; 51: 719–729.

24.

Thostenson

Chou

. On the elastic properties of carbon nanotube-based composites: modelling and characterization. J Phys D Appl Phys 2003; 36: 573–582.

25.

Iakoubovskii

. Techniques of aligning carbon nanotubes. Cent Eur J Phys 2009; 7: 645–653.

26.

Dai

Wang

. Realign carbon nanotubes in PMMA nano-fibers by improved electrospinning equipment. J Reinf Plast Compos 2011; 30: 1179–1184.

27.

Mori

Tanaka

. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 1973; 21: 571–574.

28.

Selmi

Friebel

Doghri

. Prediction of the elastic properties of single walled carbon nanotube reinforced polymers: a comparative study of several micromechanical models. Compos Sci Technol 2007; 67: 2071–2084.

29.

Sheng

Boyce

Parks

. Multiscale micromechanical modeling of polymer/clay nanocomposites and the effective clay particle. Polymer 2004; 45: 487–506.

30.

Hbaieb

Wang

Chia

YHJ

. Modelling stiffness of polymer/clay nanocomposites. Polymer 2007; 48: 901–909.

31.

Adnan

Sun

Mahfuz

. A molecular dynamics simulation study to investigate the effect of filler size on elastic properties of polymer nanocomposites. Compos Sci Technol 2007; 67: 348–356.

32.

Han

Elliott

. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput Mater Sci 2007; 39: 315–323.

33.

Rafiee R, Rabczuk T, Pourazizi R, et al. Challenges of the modeling methods for investigating the interaction between the CNT and the surrounding polymer. Adv Mater Sci Eng 2013, Article ID 183026, 10 pages.

34.

Wan

Delale

Shen

. Effect of CNT length and CNT-matrix interphase in carbon nanotube (CNT) reinforced composites. Mech Res Commun 2005; 32: 481–489.

35.

Ayatollahi

Shadlou

Shokrieh

. Multiscale modeling for mechanical properties of carbon nanotube reinforced nanocomposites subjected to different types of loading. Compos Struct 2011; 93: 2250–2259.

36.

Chou

. Multiscale modeling of compressive behavior of carbon nanotube/polymer composites. Compos Sci Technol 2006; 66: 2409–2414.

37.

Montazeri

Naghdabadi

. Study the effect of viscoelastic matrix model on the stability of CNT/polymer composites bymultiscale modeling. Polym Compos 2009; 30: 1545–1551.

38.

Chen

Liu

. Square representative volume elements for evaluating the effective material properties of carbon nanotube-based composites. Comput Mater Sci 2004; 29: 1–11.

39.

Luo

Wang

W-X

Takao

. Effects of the distribution and geometry of carbon nanotubes on the macroscopic stiffness and microscopic stresses of nanocomposites. Compos Sci Technol 2007; 67: 2947–2958.

40.

Leclerc

Surville

. Effects of fibre dispersion on the effective elastic properties of 2D overlapping random fibre composites. Comput Materi Sci 2013; 79: 647–683.

41.

Abolfathi

Naik

Karami

. A micromechanical characterization of angular bidirectional fibrous composites. Comput Mater Sci 2008; 43: 1193–1206.

42.

MATLAB help. Natick, Massachusetts. The Mathworks, Inc., 2010.

43.

ABAQUS Documentation, Dassault Systèmes, Providence, RI, USA, 2009.

44.

Jafari

Khatibi

Mashhadi

. Comprehensive investigation on hierarchical multiscale homogenization using representative volume element for piezoelectric nanocomposites. Compos Part B 2011; 42: 553–561.

45.

Andrews

Weisenberger

. Carbon nanotube polymer composites. Curr Opin Solid State Mater Sci 2004; 8: 31–37.

46.

Liu

Xiao

Wang

. Effect of carbon nanotube orientation on mechanical properties and thermal expansion coefficient of carbon nanotube-reinforced aluminum matrix composites. Acta Metall Sin 2014; 27: 901–908.

Effects of carbon nanotubes’ dispersion on effective mechanical properties of nanocomposites: A finite element study

Abstract

Keywords

Introduction

Modeling

Free-space length, L fs

Governing equations

FE simulations

Periodic boundary conditions

Homogenization

Models with uniform CNT dispersion

Numerical results

Conclusion

Footnotes

Conflict of interest

Funding

References

Free-space length, $L_{fs}$