Effect of carbon nanotube type and length on the electrical conductivity of carbon nanotube polymer nanocomposites

Carbon nanotube (CNT) type and length are two key factors that affect the electrical behavior of CNT/polymer nanocomposites. However, numerical studies that consider these two factors simultaneously are limited. This paper presented a stochastic multiscale numerical model to predict the electrical conductivity and percolation threshold of polymer nanocomposites containing single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). The combined effects of CNT type and length on the electrical conductivity and percolation threshold of the polymer nanocomposites were investigated. The model predictions were validated against experimental data of commercially available CNTs. Our results showed that the effect of CNT type varied based on both the length and aspect ratio of the CNTs. Long SWCNTs exhibited the greatest enhancement of the polymer’s electrical conductivity with the lowest percolation threshold among all the CNT types studied.


Introduction
Carbon nanotubes (CNTs) have gained significant attention as a nanofiller in polymer composites due to their remarkable electrical, mechanical, and thermal properties [1,2].They are considered one of the fundamental materials for the 21st century due to their excellent properties and high aspect ratio (length to diameter ratio) [3].CNTs can be categorized into two common types: single-walled carbon nanotubes (SWCNTs), which are composed of a single graphene sheet rolled into a cylindrical shape, and multi-walled carbon nanotubes (MWCNTs), which consist of multiple graphene sheets rolled into a cylindrical shape [4].The incorporation of CNTs into insulative or low-conducting polymer composites turns them into conducting polymer nanocomposites (CPNCs) with a wide range of applications in various fields such as wearable electronics [5], energy storage [6], self-sensing structures [7], aerospace [8,9], automotive [10,11], and electromagnetic shielding [12].
It is well established that the addition of small concentrations of CNTs can enhance the electrical conductivity of polymer composites [13].With the inclusion of CNTs, the composite's conductivity experiences a sudden increase, leading to a transition from insulation to conduction behavior.This typical behavior, known as percolation, occurs when the CNT content reaches a critical limit in the polymer known as the percolation threshold.One of the key aspects of the advancement of CPNCs is decreasing the percolation threshold and increasing their electrical conductivity.Thus, the development of numerical models that can predict the percolation threshold and electrical conductivity is crucial for understanding and enhancing the electrical behavior of CPNCs.
Numerical modeling techniques based on Monte Carlo simulations and resistor networks have been used to analyze the electrical properties of CPNCs [14][15][16][17][18][19][20].In 2008, Hu et al [14] employed a 3D resistor network model to predict the electrical conductivity of MWCNT/polymer nanocomposites for strain sensing applications.They utilized a representative volume element (RVE) with a side length 5 times larger than the average length of CNT to obtain convergent results.However, Bao et al [15] reduced the computational burden associated with large RVE sizes by applying periodic boundary conditions to the RVE.This approach resulted in a smaller RVE size, which was only 1.1 times the average length of CNT.Subsequently, Bao et al [16] investigated the electrical conductivity of SWCNT and MWCNT polymer nanocomposites using the Landauer-Büttiker formula to describe the tunneling resistance, which accounts for the transmission probability and the number of CNT conduction channels.
More recently, Doh et al [17] conducted a study on the effect of CNT chirality on the electrical conductivity of SWCNT/polymer nanocomposites using a 2D RVE with a side length 5 times larger than the average CNT length.They used Simmons' formula to describe the tunneling resistance, which neglects the number of CNT conduction channels.Though, this factor is a key parameter that differentiates between SWCNTs and MWCNTs, which represents the number of energy bands contributing to conductance [16].Later, they extended their numerical model to a 3D RVE with a focus on the uncertainty quantification of the electrical behavior of MWCNT/polymer nanocomposites [18].In another study, Wang et al [19] investigated the effect of non-uniformity of CNT length and waviness on the electrical properties of CPNCs using RVE with a side length 5 times larger than the average CNT length.Gong et al [20] extended their numerical models to consider the agglomeration of CNTs in the polymers.They found that minimizing the size of CNT agglomerates is key to optimizing the electrical conductivity and lowering the percolation threshold of CNT polymer nanocomposites.
Although these numerical models can provide good predictions compared with the experimental results, further development is still necessary to overcome existing limitations.For instance, recent experimental studies suggest that the SWCNT/polymers are advantageous over the MWCNT/polymers in electrical and mechanical properties [21,22].However, most of the numerical studies focused on MWCNT/polymer nanocomposites [14,15,18,19,23,24] with only a few studies [16,17] that focus on SWCNT/polymer nanocomposites.In addition, these numerical studies rarely consider the combined effect of CNT type and length on the electrical behavior.Furthermore, most of the numerical studies utilized large RVE sizes, which is not computationally efficient [14,[17][18][19].Additionally, full matrix inversion methods have been used to solve the matrix equations representing Kirchoff's circuit law [14-19, 23, 24], which increase the computational time significantly [25].Addressing these limitations and gaps is a key objective of our stochastic multiscale numerical model.
In this work, we present a stochastic multiscale numerical model based on Monte Carlo simulations and resistor networks for calculating the electrical conductivity and percolation threshold of CPNCs.We utilize the block-matrix manipulation [25] in a 3D RVE for calculating the electrical conductivity, as opposed to the fullmatrix inversion techniques utilized in most existing numerical models [14-19, 23, 24].To the best of the authors' knowledge, the use of block matrix manipulation for calculating the electrical conductivity in a 3D RVE has not been carried out before.Besides, we are adopting the periodic boundary condition in the RVE generation, which decreases the size of the RVE required to reach convergent results.These adopted approaches should enhance the computational efficiency of the developed numerical model.Moreover, we are using the Landauer-Büttiker formula for calculating the tunneling resistance, which enables the distinct prediction of the electrical properties of SWCNT and MWCNT polymer nanocomposites.
The present research explores, for the first time, the combined effects of CNT types and the geometry of CNTs (length and aspect ratio) on the electrical conductivity and percolation threshold of CPNCs.Additionally, the effect of the CNT intrinsic conductivity, polymer barrier height and the number of CNT conduction channels will be examined systematically to quantify the effect of these parameters on the results.Lastly, the numerical predictions of the electrical conductivity and percolation threshold of CPNCs are compared with experimental studies, providing validation for the proposed model.

Stochastic multiscale modeling 2.1. Representative volume element (RVE) generation
The effective electrical conductivity of CNT/polymer nanocomposites is represented by a randomly generated 3D RVE comprising a network of uniformly distributed CNTs.The RVE is the smallest volume of a material that can statistically represent the bulk material [26], depicted in figure 1(a).To create the RVE, we considered a cube with dimensions L L L , x y z ´´as depicted in figure 1(b).Each CNT in the RVE is represented by a line with a starting point x y z , , )and an ending point x y z , , , ( ) as shown in figure 1(d).The starting point coordinate x y z , ,

(
)is assumed to follow a uniform distribution and is generated by [27]: and the azimuthal and polar angles ( , f q ) that describe the orientation of CNTs are defined by: where, rand is a number generator that uniformly generates random numbers in the range (0,1).Next, the ending point coordinate x y z , , )is generated by [27]: sin cos sin sin cos The length of the CNT l CNT follows a Weibull distribution [28] as follows:  F x e x 1 f o r 0 5 where, F x ( ) is the cumulative distribution function (CDF) for Weibull distribution and a b , ( ) are the scale and shape parameters, respectively.The average length of CNT l avg is then evaluated by: where, G is the gamma function.The shape parameter was set as b 2.4 = based on experimental observations of CNT length distributions [29].Then, the scale parameter a is determined based on the required average CNT length for our simulations, as specified by equation (6).To accurately represent the microstructure of the bulk nanocomposite, the continuity of adjacent RVEs is required.This geometric continuation is achieved by applying periodic boundary conditions (PBC) to the entire RVE [18].Therefore, when generating the ending point of a CNT and if it is found to be generated outside the RVE, that specific CNT will be trimmed by the RVE boundary plane and repositioned to the opposite side of the RVE [16], as shown in figure 2. Finally, the volume fraction V f of CNTs in the polymer nanocomposite is calculated as and 8 where, d CNT is the average diameter of CNT.

Resistances in a percolating network of CNTs
In a typical CNT percolating network, there exist three types of electrical resistances: the CNT intrinsic resistance R , intrinsic ( ) the physical contact resistance between two adjacent CNTs R , physical contact ( ) and the tunneling resistance between two adjacent CNTs R , tunneling ( ) as shown in figure 1(c).The intrinsic resistance along the CNTs is determined by equation (9) [30].
Here, , CNT s l CNT and d CNT are the intrinsic electrical conductivity, the length, and the average diameter of the CNT, respectively.To simplify the numerical calculations, the concept of contact resistance R contact ( ) is introduced [30].The contact resistance is assumed to be the sum of physical contact and tunneling resistances as: R R R 10 For equation (10), it's important to note that this equation serves as a conceptual simplification.It is designed to encapsulate both physical contact and tunneling resistances under a single variable, R , contact for clarity.This representation is not intended to imply a strict physical addition of the resistances but rather serves as a heuristic model to understand their separate contributions.
Physical contact between two neighboring CNTs never occurs due to the Pauli exclusion principle and the closest possible distance between CNTs is the van der Waals distance d 0.34 nm vdW = ( ) [31].Therefore, there is always a gap between adjacent CNTs creating an energy barrier in which electron tunneling occurs if the distance between the CNTs is less than the tunneling cutoff distance d 1.4 nm cutoff = ( ) [16].The contact resistance is approximated by the Landauer-Büttiker formula with a rectangular potential barrier such that [16]: where, h is Planck's constant, e is the electron charge, T is the transmission probability of one electron to tunnel through the energy barrier of the polymer, and M is the number of CNT conduction channels that depends on its type.The differentiation between SWCNTs and MWCNTs is represented by the parameter M. Smaller values of M distinct SWCNTs due to their smaller conduction channels, while for MWCNTs, M is set within the range of 400-500, reflecting the multiple concentric tubes that increase the number of conduction channels.The transmission probability T is estimated by Wentzel-Kramers-Brillouin (WKB) approximation [15,32] as follows: 13 where, d tunnel is the shortest tunneling distance between two adjacent CNTs, d is the minimum distance between two adjacent CNTs and it is calculated mathematically using a computation of the distance between two lines [33], m e is electron mass, E ∆ is the polymer barrier height, which is defined as the work function difference between the polymer and the CNT.The typical values of the physical parameters used in the numerical model are shown in table 1.

Percolating network identification and effective electrical conductivity
The resistor network of CNTs forms an electrical circuit when a voltage source is applied between the left and right surfaces (electrodes) of an RVE, as depicted in figure 3. Thus, we can analyze the percolated network of CNTs using the graph theory and Kirchhoff's circuit laws.Here, the graph that represents the resistor network consists of nodes (junctions between CNTs) and edges (branches between junctions), as shown in figure 4. In graph theory, the incidence matrix is a common representation of a network, with components of 1, −1, or 0 [25].To represent our network, we used an edge-node incidence matrix that has a row for each edge and a column for each node.The incidence matrix A i j , ( ) with i edges and j nodes is constructed as follows: ´-9.909 10 31 ´-1.602 10 19 ´-  A i j i j if edge has end node if edge has start node otherwise To compute the electrical conductivity of the RVE, we need to identify the percolated CNT networks that connect the left and the right electrodes.Before continuing, it is important to distinguish the difference between connected and percolated networks.Here, a connected network is a network of CNTs that are interconnected, but not necessarily connected to the electrodes i.e., not a conductive network.On the other hand, the percolated network is a connected network that is connected to the electrodes i.e., a conductive network.
To identify the connected network, we introduce the Laplacian matrix L , j j , ( ) which is a matrix representation of the graph that is useful in identifying the connected components.The Laplacian matrix is calculated by using the incidence matrix as follows [34]: Next, a Dulmage-Mendelsohn decomposition [35] is applied to the Laplacian matrix to find the connected components of the graph i.e., the connected network.The Dulmage-Mendelsohn decomposition permutes the rows and columns of the Laplacian matrix to form a block upper-triangular matrix.The rows and columns of the Laplacian matrix are rearranged so that the non-zero entries are positioned in smaller blocks along the diagonal.Therefore, each block of the decomposed Laplacian matrix will only contain the connected components of the network.
Then, we analyze each block of the decomposed Laplacian matrix to find the connected network that spans from the left to the right boundary.This step ensures that the connected network is a percolating one, as the CNTs might be interconnected with each other, but not to the electrodes, and therefore will not form a conductive network.After identifying the percolated network, we follow block-matrix manipulation procedures as in ref. [25] and use the reduced equation (16) to calculate the conductance G p of the percolated network.
where L* is the weighted graph Laplacian, and it is computed as follows: where, A is the incidence matrix of the percolated network (i.e., not the whole graph) and R 1 -is the pseudoinverse of the resistance matrix, which is a matrix whose entries are the resistances (intrinsic or contact) of all the components of the percolated network.Depending on the resistor network, there might be more than one percolated network of CNTs that connects both electrodes.Thus, the total conductance of the network is the sum of the individual conductance of the parallel percolated networks.

G G,
where N is the number of percolated networks 18 Finally, we use equation (19) to calculate the effective electrical conductivity s of the CPNC.
where, L is the distance between the two electrodes and A RVE is the cross-sectional area of the RVE, as shown in figure 3. The reported numerical values of electrical conductivity were obtained through at least 100 Monte Carlo simulations for each CNT volume fraction to ensure statistical convergence.It has been established that RVE dimensions significantly impact the numerical results of electrical conductivity at lower CNT concentrations, particularly below the percolation threshold [24].Figure 5(a) shows the electrical conductivity of different RVE sizes with associated error bars representing the variance over 200 simulations.It's clear that as the RVE size increases, the variability (indicated by the size of the error bars) tends to decrease, leading to a more statistically converged result.However, this increase in RVE size is not without its challenges.Larger RVEs, while providing more statistically reliable results, also demand significantly higher computational resources, as shown in figure 5(b).Thus, the optimal RVE size should strike a balance between achieving statistical convergence and ensuring computational efficiency.Therefore, the RVE dimensions were set to 1.1 times the average length of the CNT modeled

Results and discussions
´).It is also worth noting that this RVE size has been successfully adopted in previous studies [15,24].The physical and geometrical parameters of the CNTs used in the simulations are summarized in tables 1 and 2, respectively.SWCNTs tend to form bundles of larger diameters than individual SWCNTs [36].Thus, modeling SWCNTs as bundles with a larger diameter is acceptable.It is possible that the distinct behavior of metallic and semiconducting CNTs may be less pronounced in large SWCNT bundles, and these bundles may have mostly metallic electronic character [40].
Therefore, the number of conduction channels for the SWCNT bundles was chosen to be M 2, = while MWCNTs can have up to M 400 500 = conduction channels [16].The polymer barrier height E ∆ depends on the difference between the CNT and polymer work functions.The work functions of SWCNTs and MWCNTs are found to be 5.05 and 4.95 eV, respectively [41].While the work functions for common polymers such as epoxy, polyurethane, polystyrene, and polycarbonate are 0.5-2.5 eV, 4.0 eV, 4.22 eV, and 4.26 eV, respectively [42,43].Since the work functions for the polymers used are within a narrow range, a value of E 3eV = ∆ c was used for all the simulations.conductivity, and in figure 6(d) for MWCNT volume fractions between 0.01 and 0.03, an overprediction is noted.It is crucial to recognize that these variances arise due to the model's sensitivity to multiple parameters, such as polymer barrier height and intrinsic CNT conductivity.However, it's essential to note that our primary focus is on the behavior of electrical conductivity beyond the percolation threshold, which is well captured using our numerical approach.The second validation study performed was to calculate the percolation probability for the CNT/polymer nanocomposites.A number of simulations N 500 s = ( ) were conducted and the number of percolation occurrences N p ( ) were counted to calculate the percolation probability P p ( ) from: P N N p p s / = [18].The obtained percolation probability results were fitted with a sigmoid curve to determine the percolation threshold, defined as the critical CNT volume fraction at which P 0.5 p = [28].The resulting percolation curves for different CNT types and geometries are shown in figure 7, with the numerical values of the percolation threshold falling within the reported experimental range, as shown in table 3. It is worth noting that the calculated percolation threshold for short MWCNTs is 0.0038, which slightly falls outside the experimental range of 0.004-0.005.However, it's important to consider the close proximity of these values.The slight discrepancy can be seen as lying within acceptable bounds for computational studies, especially when considering the inherent complexities and variables present in experimental setups.Additionally, the rest of our validation studies align closely with experimental benchmarks, reinforcing the overall reliability of our computational approach.
In summary, the comparison of the numerical results with experimental data for electrical conductivity and percolation threshold of CNT/polymer nanocomposites validated the developed numerical model and demonstrated its effectiveness in modeling various types of CNT/polymer nanocomposites.

Effect of CNT type and length
In this section, the combined effect of CNT type and length on the electrical conductivity and percolation threshold of CNT/polymer nanocomposites is studied.The study focuses on two CNT types, SWCNT and MWCNT, each with two different geometries, i.e., short and long CNTs, as shown in table 2. These four CNT types are commercially available [14,21,36,38], and the polymer used in the simulations is assumed to be the same, with a polymer barrier height E eV 3 = ∆ .The simulation parameters, such as the number of Monte Carlo simulations and the RVE size, are consistent with those used in the validation studies.
The first study analyzed the effect of short SWCNTs and MWCNTs on the electrical properties of polymer nanocomposites.The electrical conductivity of the nanocomposites generally increased with the CNTs volume fraction, with MWCNTs performing better than SWCNTs, as shown in figures 8(a) and (c).These results are counterintuitive since SWCNTs have higher aspect ratios than MWCNTs, which may suggest better electrical conductivity enhancement.However, SWCNTs tend to bundle together, forming SWCNT bundles of larger diameters than the individual SWCNTs, which decreases their aspect ratio [36].As a result, MWCNTs have higher aspect ratios than SWCNT bundles, which could explain their better performance.The percolation threshold of MWCNTs was also lower than that of SWCNTs, as depicted in figure 8(e), which was expected due to the larger length and higher aspect ratio of MWCNTs aiding in the percolating network formation.
The second study investigated the effect of long SWCNTs and MWCNTs on the electrical properties of polymer nanocomposites.Figure 8(b) shows the electrical conductivity results while varying volume fractions of long SWCNTs and MWCNTs in polymer nanocomposites.Unlike short CNTs, the results reveal that SWCNTs exhibit superior electrical conductivity performance than MWCNTs for longer CNTs, as depicted in figure 8(d).This outcome may be attributed to the fact that even when long SWCNTs are bundled together having larger diameters than individual SWCNTs, their aspect ratio remains higher than that of long MWCNTs.Furthermore, as shown in figure 8(f), the percolation threshold of SWCNTs is lower than that of MWCNTs, albeit both CNTs have the same length.Once again, this behavior is due to the SWCNT bundles' higher aspect ratio than that of MWCNT, which assists in creating percolating networks at lower volume fractions.
In summary, it was found that the effect of CNT type on the electrical conductivity of polymer nanocomposites is dependent on the CNT length and aspect ratio.For short CNTs, the MWCNTs are performing better in enhancing the electrical conductivity of polymer and having lower percolation thresholds than SWCNTs.Conversely, for long CNTs, the SWCNTs are better than MWCNTs in terms of higher electrical conductivity and lower percolation thresholds for the polymers.Overall, the long SWCNTs with the largest aspect ratio have the best performance in enhancing the electrical properties of polymer nanocomposites compared to the studied CNTs.

Parametric studies
To assess the effect of CNT intrinsic conductivity on long SWCNT/polymer nanocomposites, several CNT conductivities (ranging from 10 3 to 10 5 S m −1 ) were varied while maintaining other numerical parameters constant, as indicated in table 2. The results, presented in figure 9(a), demonstrate a positive effect of CNT intrinsic conductivity on the electrical conductivity.This effect is primarily attributed to the intrinsic resistance, as per equation (9), where higher intrinsic CNT conductivity corresponds to lower intrinsic resistance and thus reduces the overall resistance of the conductive network.
Next, the effect of polymer barrier height on the electrical conductivity of long SWCNT/polymer nanocomposites is studied.To study this effect, numerical modeling was conducted with varying polymer barrier heights (ranging from 1 to 5 eV) while keeping other numerical parameters for long SWCNTs constant, as per table 2. The findings, as demonstrated in figure 9(b), indicate that the polymer barrier height has a negative effect on the electrical conductivity of the nanocomposites.This effect can be attributed to the hindrance of electron transport due to the high energy barrier that the electrons must overcome.As the barrier height increases, the number of electrons able to overcome the barrier decreases, resulting in a decrease in electron tunneling probability, as shown in equations (12) and (13).Therefore, the contact resistance increases as the polymer barrier height increases, subsequently increasing the overall resistance of the nanocomposite.
Finally, the investigation of the effect of the number of conduction channels of CNTs on the electrical conductivity of long MWCNTs/polymer nanocomposites showed that this parameter has a relatively minimal impact on the electrical conductivity of the nanocomposites.The number of conduction channels was varied within the range of 400 to 500 while keeping other long MWCNT parameters constant, as per table 2. The results, depicted in figure 9(c), indicate that there is no significant change in the electrical conductivity of the nanocomposites within the range of 400 to 500 conduction channels.This observation may be attributed to the fact that MWCNTs consist of multiple concentric walls, which provide several pathways for charge transport, thus resulting in a relatively minor effect of the number of conduction channels on the overall electrical conductivity of the nanocomposites.

Conclusions
In this study, a stochastic multiscale numerical model was developed to predict the electrical conductivity and percolation threshold of SWCNT and MWCNT polymer nanocomposites.The model predictions were validated against experimental studies that used different CNT types.The main objective was to examine the combined effect of CNT type and length on the electrical conductivity and percolation threshold of CPNCs.The results indicated that the effect of CNT type on the electrical conductivity of CPNCs was dependent on both the CNT length and aspect ratio.The study showed that long SWCNTs with the largest aspect ratio exhibited the highest enhancement in electrical conductivity with the lowest percolation threshold compared to the other CNT types studied.Interestingly, for shorter CNTs, MWCNTs outperformed SWNCTs in terms of higher conductivity emphasizing the importance of considering both CNT type and geometry in CPNC design.Additionally, other parameters such as CNT intrinsic conductivity, polymer barrier height, and number of CNT conduction channels were investigated, with the findings indicating that CNT intrinsic conductivity had a positive effect on electrical conductivity, while polymer barrier height had a negative effect.However, the number of CNT conduction channels within the range of 400-500 had a negligible effect on electrical conductivity.These results could provide valuable insights for designing CPNCs with tailored electrical properties for a wide range of industrial applications.

Figure 1 .
Figure 1.The different length scales incorporated in the multiscale numerical model.(a) The macroscale nanocomposite shown as a dog-done sample.(b) The microscale representation of the nanocomposite.(c) The nanoscale interactions between CNTs.(d) The CNT representation at the nanoscale.

Figure 2 .
Figure 2. (a) 3D RVE without applying PBC and (b) 3D RVE with applying PBC (The red CNT is highlighted for illustration).

Figure 3 .
Figure 3. 3D RVE with applied bias voltage on the left and right electrodes (shown in blue).

Figure 4 .
Figure 4. (a) Resistor network of CNTs.(b) Equivalent graph representing the resistor network, where the numbers represent the nodes, and the letters represent the edges.

Figure 5 .
Figure 5. (a) Different RVE sizes and their corresponding electrical conductivity.The error bar represents the variance from 200 simulations.(b) The simulation CPU time for each RVE size.

Figure 8 .
Figure 8. Electrical conductivity of (a) short and (b) long SWCNT and MWCNT polymer nanocomposites.Electrical conductivity of (c) short and (d) long SWCNT and MWCNT polymer nanocomposites at certain volume fractions.The percolation probability of (e) short and (f) long SWCNT and MWCNT polymers.

Figure 9 .
Figure 9. (a) Effect of CNT intrinsic conductivity on the electrical conductivity of long SWCNT/polymer nanocomposite.(b) Effect of polymer barrier height on the electrical conductivity of long SWCNT/polymer nanocomposite.(c) Effect of number of conduction channels on the electrical conductivity of long MWCNT/polymer nanocomposite.

Table 1 .
Physical parameters used in the numerical model.

Table 2 .
CNTs geometrical and physical parameters for the simulations.

Table 3 .
Comparison between the numerical predictions and experimental values of the percolation threshold.