Identification of Variables and Determination of the Mechanism Affecting the Effective Properties of Representative Volume Elements of Unidirectionally Aligned CNT-Based Nanocomposites

Abstract

This study identified the governing variables affecting the effective properties of the representative volume element (RVE) of nanocomposites consisting of unidirectionally aligned carbon nanotubes (CNTs) and determined the mechanism through which they act. For this purpose, multi-walled nanotubes (MWNTs) and polyvinylidene fluoride (PVDF) were selected as the components of the nanocomposites, and Monte Carlo simulations were conducted to examine the variability of the effective properties according to the CNT length. The governing variables affecting the effective properties were identified considering the conditions under which the selected CNTs can be arranged inside the RVE. Using the geometrical relationship between the RVE and CNTs, a simplified two-parameter equation that can calculate the effective properties of the RVE was derived. Using this equation and Monte Carlo simulations, this study confirmed that the characteristics of the effective properties vary with changes in the length of the RVE and the length fraction of the CNTs, and the mechanism of these changes was determined. In addition, the variation in the characteristics of the effective properties according to the coefficient of variation of the CNT length was also determined.

1. Introduction

Due to their excellent mechanical strength, thermal conductivity, and electrical conductivity, carbon nanotubes (CNTs) are widely used in various industries, including the aerospace, defense, automotive, and athletic industries [1]. In CNT-based nanocomposites, CNTs are used as reinforcements to enhance the strength and stiffness of nanocomposites [2]. To apply these nanocomposites to structural designs, their mechanical properties must be well developed [3,4].

Numerical simulation methods such as finite element analysis is used due to the large costs involved in typical experimental methods for determining the mechanical properties of composites [5,6,7,8,9]. Among these methods, the representative volume element (RVE), which represents the geometric distribution of materials that are mixed in a composite, is extensively applied, and many studies have been performed using this approach [10,11,12,13,14,15,16,17,18]. For example, Chen et al. [19] calculated and evaluated the effective properties of a square RVE considering CNT single filaments. Araujo et al. [20] defined RVEs based on square- or hexagonal-packed arrays containing CNT single filaments in the RVEs and reduced the analysis time using parallel arrangements. Further, Moghaddam et al. [21] modeled cylindrical RVEs containing CNT single filaments and analyzed their effective properties according to the variations in the volume fraction, aspect ratio, and orientation of the CNTs. However, the aforementioned studies only investigated RVEs in which extremely simple CNT single filaments were applied.

Moreover, due to their highly complex internal structures resulting from the complex microstructures and dispersion of CNTs, it is difficult to assess the properties of CNT-based nanocomposites. To solve this problem, Alian et al. [22] modeled and analyzed RVEs to investigate the effects of the modulus of elasticity on the CNT shape and dispersion state of the CNT-polyethylene composites. Weid et al. [23] proposed 2D and 3D RVE approaches to identify the effects of nonlinear compressive responses according to the CNT orientation, CNT volume fraction, and CNT aspect ratio of CNT/epoxy nanocomposites. Song et al. [24] proposed a computational homogenization algorithm that can be used to define the RVE for CNT polymer composites. However, no studies have yet developed criteria for defining whether an RVE represents the internal structure of the nanocomposite (e.g., RVE size). To select an appropriate RVE, it is necessary to identify the main variables with the greatest influence on the effective properties of RVEs and to determine the mechanism through which they act.

This study investigated the main variables affecting the effective properties of the RVEs of nanocomposites consisting of unidirectionally aligned CNTs, and analyzed the mechanism through which they act. For this purpose, in Section 2, MWNTs and polyvinylidene fluoride (PVDF) were selected as the components of the nanocomposites, and the main variables were selected considering the variations in the effective properties. Then, in Section 3, using the geometrical relationship between the RVE and CNT lengths considering the main variables, a simplified two-parameter equation that can be used to calculate the effective properties of the RVE was derived. In Section 3, a Monte Carlo simulation was conducted to examine the irregularity of the CNT length, which is one of the main variables, and the suitability of the simplified two-parameter approach for predicting the effective properties of RVE was verified. Finally, Section 4 presents the conclusions of this study.

2. Nanocomposite Modeling Using Monte Carlo Simulations

A CNT-based nanocomposite model was developed to determine the mechanism underlying the varying characteristics of the effective properties of CNT-based nanocomposites. MWNT (K-Nanos 100), having excellent mechanical properties, was selected as a reinforcing agent, and PVDF (Kynar^®761), which is used in various fields such as chemistry, electronics, pharmaceuticals, food, and paper, was selected as a matrix due to its excellent mechanical, thermal, and electrical properties. The mechanical properties are shown in

Table 1. Mechanical properties of MWNTs and PVDF.

	MWNT	PVDF
Density ( $\mathrm{kg}/{\mathrm{cm}}^3$)	2.6	1.78
Young’s modulus (GPa)	940	2
Poisson’s ratio	0.20	0.34

, and data provided by the manufacturer were used. Moreover, 3 wt% was selected as the weight ratio of the CNTs distributed in the CNT-based nanocomposite, having an average CNT length of 2 µm and diameter of 12 nm. To consider a regular orientation of the CNTs, the orientation tensor was positioned only along the x-axis. The homogenization method [25] was used to calculate the effective properties of the modeled CNT-based nanocomposite.

Herein, CNTs of different lengths were randomly arranged inside the CNT-based nanocomposite. In order to apply the variability of the CNT length, Monte Carlo simulation, which is mainly used for generating random variable values from probability distributions using repeated random sampling, was used [26]. Using this, it was performed to assume a normal distribution with a coefficient of variation (COV) of 0.1 and a sample size of 100,000 for the CNT length variable. Figure 1 shows this distribution. In addition, in general, CNTs are arranged in various orientation tensors inside a CNT-based nanocomposite; however, in this case, it is very difficult to understand the geometrical relation between the CNT length and the RVE length. Therefore, in this study, in order to consider the regular orientation of CNTs, only the x-axis was oriented. Using this setup, the main variables that influence the effective properties of the RVE could be analyzed.

3. Results and Discussion

3.1. Analysis of the Variation Mechanism According to the Relationship between the CNT and RVE Lengths

To identify the geometrical relationship between the effective properties of the RVE and the RVE and CNT lengths, we first analyzed the arrangement of CNTs inside the RVE. When selecting the RVE, an arrangement model considering three conditions according to the RVE length could be derived. The first condition considers that the RVE is longer than the CNTs such that some CNTs are located inside the RVE. The second condition considers that the RVE and CNTs have identical lengths such that some CNTs fully meet both opposite faces of the RVE in the x-direction. The third condition considers CNTs that are longer than the RVE such that they are cut. In this study, the third condition was not considered because it does not satisfy the general selection criterion for RVEs in which the RVE should be larger than its internal components. Figure 2 shows the conditions under which the CNTs can be arranged in the RVE.

First, Figure 2a depicts the condition in which the RVE is longer than the CNTs such that some CNTs are located inside the RVE. The cases in which the CNTs can be arranged inside and outside the RVE are denoted as ① to ⑨. Among these, ③ to ⑦ are conditions in which the CNTs can be arranged inside the RVE. Of these, ③ and ⑦ show that the CNTs crossing the RVE edge are cut, and only the CNTs inside the RVE are arranged. In ④ to ⑥, the CNTs are arranged inside the RVE without being cut. The conditions under which the CNTs are not arranged inside the RVE are ①, ②, ⑧, and ⑨, wherein the CNTs are arranged outside the RVE and therefore do not exist inside the RVE.

Second, Figure 2b shows the condition in which the RVE and CNTs have the same length. All cases in which the CNTs can be arranged inside and outside the RVE are denoted as ① to ③ and ⑦ to ⑩, respectively. Further, ①, ②, ⑧, and ⑨ are conditions under which the CNTs are not arranged inside the RVE, whereas ③, ⑦, and ⑩ are conditions under which the CNTs can be arranged inside the RVE. Of these, ③ and ⑦ show that the CNTs crossing the RVE edge are cut, and only the CNTs inside the RVE are arranged. In ⑩, the CNTs fully meet both opposite faces of the RVE.

These arrangements of CNTs according to the RVE length also affect the effective properties of the RVE; as the length of the CNTs inside the RVE increases, the effective properties increase. Therefore, the impact on the effective properties is smaller when the CNTs are cut than when they are not. Accordingly, a detailed analysis of the variation in the effective properties according to the RVE length and CNT length is necessary.

To observe the variations in the effective properties according to the RVE and CNT lengths, cube RVEs were modeled to perform the subsequent analysis. First, as shown in Figure 3, RVEs were modeled in which monofilament CNTs of different lengths were arranged in RVEs of the same length. Elements of C3D8 with eight nodes per element and six degrees of freedom per node were used for this model. There are 27,000 elements and 29,791 nodes in the model. For the boundary condition, one surface in the x-axis direction was completely constrained to derive effective properties $E_1$, and a displacement condition was applied to the opposite surface in the x-axis direction as much as the RVE length. Second, as depicted in Figure 4, RVEs were modeled in which the RVE length was increased while maintaining 3 wt% of CNTs inside the RVEs, and the effective properties were then calculated. Figure 5 shows the effective properties calculated as described above. Figure 5a shows the variations in the effective properties ($E_1$) according to the CNT length ($C_l$) and RVE length ($R_L$), whose values were normalized based on $E_1$ of $C_l/R_L$ = 0.17. As shown in this graph, increasing $C_l/R_L$ resulted in a slight increase in the effective properties, which then sharply increased after this ratio reached 1. This finding suggests that $E_1$ increased because the CNTs fully support both opposite faces of the RVE in the x-direction. Figure 5b depicts the variation in $E_1$ according to the CNT and RVE lengths and normalized based on $E_1$ at RVE = 3 µm. As shown in this graph, the effective properties increased as the RVE length increased. Based on the above analysis results, the main variables affecting the effective properties of the RVE were identified as the CNT and RVE lengths, and further examination of these properties is necessary.

To determine the geometrical relationship between the main variables and effective properties, we first examined the geometrical relationship among the main variables. For this purpose, a simplified two-parameter approach was proposed that can calculate the effective properties using the CNT and RVE lengths considering the arrangement of CNTs inside the RVE, as shown in Figure 6. First, it was assumed that the longitudinal elastic modulus $E_1$ of the RVE is governed by the number of CNT filaments contained in the RVE, and the size of $E_1$ is proportional to the ratio of the CNT length to the RVE length. Accordingly, the effective properties of the RVE are determined by the CNT length fraction and its specific elastic modulus. This process is described in Equations (1) and (2):

$$m_i=\frac{CN{T}_L\left(Length of CNT\right)}{RV{E}_L\left(Size of RVE\right)}\left(0<m\le 1\right); \mathrm{ratio} \mathrm{of} \mathrm{the} i-\mathrm{th} \mathrm{CNT} \mathrm{filament} \mathrm{length} \mathrm{to} \mathrm{the} \mathrm{RVE} \mathrm{length}$$

(1)

$$E_1={\displaystyle \sum }_{i=1}^{CNT number}{E}_{1,i}=E_{1,1}+E_{1,2}+E_{1,3} \cdots$$

(2)

where $E_{1,i}$ is the specific elastic modulus in cases in which only one i-th CNT filament is present inside the RVE. To calculate the effective properties of the RVE using Equation (1), the specific elastic modulus, which is an effective property that exists when a single CNT filament is present in the RVE, is required. Figure 7 shows the variations in the effective properties according to increases in the CNT length when a single filament CNT is arranged inside the RVE. As illustrated in Figure 7a–d, the longitudinal elastic modulus increased as $C_l/R_L$ increased, and the CNT length significantly affected the longitudinal elastic modulus. Using the calculated data, an equation to calculate the longitudinal elastic modulus according to $C_l/R_L$ was derived.

When several CNT filaments are arranged inside an RVE, the longitudinal elastic modulus can be calculated using the specific elastic modulus determined previously and Equation (2). Figure 8 presents the longitudinal elastic modulus (COV = 0.1) calculated using this process.

It was identified that as the RVE length increases,$E_1$ also gradually increases. Accordingly, the effective properties of the RVE are using the relationship between the RVE length and CNT. It is necessary to determine this relationship for specifying the RVE length of the CNT/PVDF nanocomposites.

To identify the geometrical relationship between the effective properties of the RVE and the RVE and CNT lengths, the length fraction of CNTs greater than 2 μm according to the RVE length is shown in Figure 9. In Figure 9a (3 µm RVE), the CNT length fraction sharply increased at a CNT length of 1.8 µm and then decreased. Moreover, the proportion of CNTs greater than 2 µm in length was 10.46%. In Figure 9c (10 µm RVE), the CNT length fraction also sharply increased at a CNT length of 1.9 µm and then decreased, and the proportion of CNTs with lengths greater than 2 µm was 33.37%. This result indicates that even when the RVE length was varied, the largest proportion of CNTs occurred near an average CNT length of 2 µm. Combining the above results, the length fraction of CNTs greater than 2 µm according to the RVE length was analyzed and is shown in Figure 10. As the RVE length increased, the length fraction of CNTs greater than 2 μm increased.

To analyze the variations in the effective properties with increasing RVE length, the rate of change was examined based on the effective properties observed at 30.84 MPa in the 3 µm RVE, as shown in Figure 8. First, the rate at which the effective properties varied with the RVE length is depicted in Figure 11a. When the rate of change in a 3 µm RVE at 30.84 MPa was 100%, the rate of change gradually increased as the RVE length increased and reached 118% at 10 µm RVE. For a more detailed analysis, Figure 11b shows the rate of increase. From 3 to 4 µm RVEs, this rate increased by approximately 5%. However, as the RVE gradually increased, this rate gradually decreased. From 9 to 10 µm RVEs, the change was 0.87%. This finding demonstrates patterns similar to the variation and fraction of CNTs larger than 2 µm, as shown in Figure 10. Because this property is affected by the length of the CNTs inside the RVE, the reason was identified to be the geometrical relationship between the effective properties and CNT length.

3.2. Analysis of the Variation in Effective Properties According to the RVE Length

To identify the changes in the effective properties of the RVE according to the CNT length fraction, the RVE effective properties according to length COVs of 0.1, 0.2, and 0.3 were calculated and are shown in Figure 12. It is evident that even among RVEs of identical length, the effective properties increased as the COV increased.

For a detailed analysis, the length fraction of CNTs according to changes in the COV was analyzed, as shown in Figure 13. Figure 13a–c show the CNT length fraction when the COV is 0.1, 0.2, and 0.3, respectively, based on a 3 µm RVE. It was identified that as the COV increased, the maximum length fraction of the CNTs gradually decreased and the proportion of CNTs with a length greater than 2 μm increased. Figure 13d–f show the CNT length fraction according to changes in the COV based on a 5 µm RVE. A trend similar to that found for the 3 µm RVE was observed, and the maximum CNT length increased as the COV increased. This trend was also observed for the 8 µm RVE shown in Figure 13g–i for COVs of 0.1, 0.2, and 0.3, and 10 µm RVE shown in Figure 13j–l for COVs of 0.1, 0.2, and 0.3, respectively.

Considering the above results, the length fraction of CNTs greater than 2 µm according to changes in the COV was analyzed, as shown in Figure 14. As the COV increased, the length fraction of CNTs greater than 2 μm increased. Moreover, in RVEs of the same length, this parameter increased as the COV increased. Further, as the RVE length increased, the difference in the length fractions of CNTs greater than 2 μm between COVs became smaller. This observation can be explained based on the relationships between the CNT length fraction and COV, and between the arrangement conditions of the CNTs inside the RVE and the RVE length. As shown in Figure 1, the length fraction of CNTs greater than 2 μm increases as the COV increases, and the rate of this increase also increases. Furthermore, as the RVE length increases, the probability that the CNTs are not cut and are present inside the RVE increases, and the difference in the length fraction of CNTs greater than 2 μm decreases even if the COV changes.

To analyze the changes in the effective properties according to increases in the RVE length, the rate of change was examined based on the effective properties of the 3 µm RVE shown in Figure 8. First, the rate of change in the effective properties according to the RVE length is shown in Figure 15a. When the rate of change in the effective properties in a 3 µm RVE was 100%, this rate gradually increased as the RVE length increased and reached 118–120% for a 10 µm RVE. For a more detailed analysis, Figure 15b shows the rate of increase for an RVE that is 1 µm less than the target RVE. From 3 to 4 µm RVEs, this rate increased by approximately 5–6%, but as the RVE gradually increased beyond this point, this rate gradually decreased. From 9 to 10 µm RVEs, the change was approximately 1%. This trend is related to the fraction of the variation in CNTs greater than 2 µm, as shown in Figure 14. As the length fraction of CNTs greater than 2 μm increases, the effective properties increase and the variation range of the effective properties according to increases in the RVE length decreases. These findings demonstrate that the key factor affecting the effective properties of nanocomposites aligned in the x-direction is the length fraction of CNTs inside the RVE. Therefore, the selection of a suitable nanocomposite RVE length may only be achieved by ensuring that the RVE length is greater than the CNT length. In addition, because the effective properties converge to a constant value as the RVE length increases, the nanocomposite RVE must be selected considering the CNT length fraction according to the RVE length.

4. Conclusions

In this study, we identified the main variables affecting the effective properties of the RVEs of nanocomposites consisting of unidirectionally aligned CNTs and determined the mechanisms through which these properties are affected.

• Finite element models of nanocomposite RVEs inside which one CNT filament was arranged inside the RVE were created, and their effective properties were calculated.
• An equation for analyzing the interaction between the CNT length fraction and RVE length was presented, using which the effective properties of unidirectionally aligned nanocomposite RVEs were calculated.
• The interaction between the CNT length fraction and RVE length was analyzed via Monte Carlo simulations, and the CNT length fraction that yielded the greatest influence on the effective properties of the RVE was obtained.
• Using the geometrical relationship between the RVE and CNTs, a simplified two-parameter equation that can calculate the effective properties of the RVE was derived.
• As the RVE increased, it was confirmed that the effective properties converge to a certain value, but only up to 10 μm RVE due to the limitation of H/W. Therefore, there is a need for further studies by expanding the size of the RVE.
• In this study, the orientation tensors of CNTs were placed only along the x-axis. In future studies, analysis of the various orientation tensors of CNTs is also required.