Effects of Key Parameters on Thermal Conductivity of Carbon Nanotube–Epoxy Composites by Molecular Dynamics Simulations

Abstract

The application of carbon nanotube (CNT)-reinforced epoxy matrix composites (CRECs) has attracted extensive attention in various industrial sectors due to the significant improvement of material properties imparted by CNTs. The thermal behavior of these nanocomposites is governed by complex heat transfer mechanisms operating at different scales, resulting in a complex relationship between the effective thermal response and the microstructural characteristics of the composite. In order to fundamentally understand the thermal behavior of the CRECs on the nanoscale, in this study, molecular dynamics (MD) simulation methods were used to investigate the thermal conductivity of CRECs, focusing on the effects of key parameters such as the length and volume fraction of CNTs, the degree of cross-linking within the epoxy matrix, and the temperature on the overall thermal properties. First, the thermal behavior of the epoxy matrix was simulated and analyzed. This approach allowed the isolation of the intrinsic thermal response of the epoxy resin as a benchmark for evaluating the enhancement introduced by CNT reinforcement. By systematically varying those key parameters, the study comprehensively evaluates how nanoscale interactions and structural modifications affect the overall thermal conductivity of CRECs, providing valuable insights for optimizing their design for advanced thermal management applications. The simulation results were validated by comparing them with experimental data from literature and analytical predictions. The results show that for the configurations examined, the thermal conductivity of CRECs increases with increasing CNT length and volume fraction, epoxy cross-linking degree, and the system temperature. From a broader perspective, the approach presented here has the potential to be applied to study a wide range of materials and their properties.

1. Introduction

Polymers are widely used in various technological and manufacturing sectors due to their lightweight, low cost, chemical stability, structural reliability, and other properties that are tailored for specific purposes. Epoxy resins, in particular, stand out for their exceptional structural performance and durability against environmental factors, making them ideal for aerospace and automotive applications. However, their thermal conductivities are typically ranging from 0.1 to 0.2 W/m·K, limiting their effectiveness in applications that require efficient heat transfer [1,2,3]. Significant improvement in thermal conductivity often requires increasing filler loadings in the polymer matrices, which can present considerable processing challenges [4]. However, incorporating fibers into the matrix substantially enhances its mechanical properties [5,6]. To address these limitations, recent research has focused on integrating ultra-high thermal conductive nanofillers into epoxy matrices to enhance the thermal conductivity and overall performance of the resulting composites [7,8,9]. Carbon-based nanofillers, such as carbon nanotubes (CNTs), graphene nanoplatelets (GNPs), and biochar, have emerged as important reinforcement materials for polymer nanocomposites due to their exceptional thermal and mechanical properties [10,11,12]. These materials are highly effective in enhancing heat transfer due to their exceptionally high thermal conductivity. In addition to being lightweight, they also possess excellent mechanical properties, which makes them ideal candidates for improving the structural performance of polymer matrix structures [13,14,15].

Epoxy is a widely used thermosetting polymer known for its excellent mechanical properties, thermal stability, and chemical resistance. It is cured from epoxy resin and has a highly cross-linked structure, making it an ideal candidate for coatings, adhesives, and composite matrices. The strength, durability, and adhesion to various substrates, along with resistance to environmental factors of epoxy resins, make them widely used in aerospace, automotive, and electronic applications [16,17].

CNTs are unique cylindrical nanostructures composed of carbon atoms arranged in a hexagonal lattice. They are broadly classified into three categories: single-walled CNTs (SWCNTs), double-walled CNTs (DWCNTs), and multi-walled CNTs (MWCNTs). SWCNTs are formed by a single graphene rolled into a tube, and their diameter and chirality affect their mechanical, thermal, and electrical properties. MWCNTs consist of multiple concentric graphene layers and have enhanced mechanical strength and thermal stability due to their nested structure. DWCNTs are a specific category of MWCNTs with two concentric layers, which combine the flexibility of SWCNTs with the structural robustness of MWCNTs. CNTs have exceptional properties, including high tensile strength, superior electrical and thermal conductivity, and lightweight, which have led to their widespread applications in nanocomposites, electronics, energy storage, and other advanced technologies [18,19].

Based on the properties of CNTs and epoxy resin, the resulting CNT-reinforced epoxy composites (CRECs) have higher thermal conductivity and mechanical strength than pure epoxy resin. This makes them ideal for high-performance applications in industries such as aerospace, automotive, and electronics. Although the addition of CNT increases production costs, the improved material properties justify the investment, especially for applications that require reliable thermal management and high durability, such as electronic devices or lightweight structural components [20,21].

Despite the great potential of carbon nanofillers for enhancing the thermal conductivity of nanocomposites, experimental studies have shown only marginal improvements in thermal conductivity compared to the base matrix materials. The challenge of developing PNCs with thermal conductivities (λ) above a certain amount remains [22]. King et al. and Choi et al. [22,23] reported that MWCNTs randomly dispersed in an organic fluid can enhance thermal conductivity by more than 2.5 times at a nanotube volume fraction of approximately 1%. Biercuk et al. [24] reported a 75% increase in the thermal conductivity of industrial epoxy due to a random dispersion of SWNTs at 1 wt.% nanotube loading and room temperature. Shen et al. [25] reported a 42% increase in thermal conductivity (λ) of polyamide nanocomposites containing 4 wt.% CNTs.

Recent studies have highlighted two major challenges in using CNTs as thermally conductive fillers in polymer composites [26,27]. First, CNTs exhibit a strong tendency to aggregate into bundles due to intrinsic Van der Waals forces and their inert graphite-like surface. This aggregation prevents proper dispersion and limits their full potential in enhancing the thermal conductivity of composites. Second, interfacial thermal resistance arises from the phonon mismatch at the interface between the CNTs and the polymer matrix. This mismatch results in significant phonon scattering at the interface, which leads to a substantial reduction in thermal transport performance [28].

The properties of CNT-reinforced composites are governed by a variety of factors, many of which arise from nanoscale interactions. These complexities make it difficult to comprehensively analyze such materials using traditional experimental methods, continuum-based modeling, or analytical analysis. Traditional trial-and-error experimentation is not only resource- and time-consuming but also impractical in some cases [29]. Therefore, advanced design methods are essential for developing innovative composites with properties tailored for specific applications. Molecular dynamics (MD) and first-principles simulations have emerged as indispensable tools for exploring the behavior of nanoscale materials. MD is a computational technique that simulates the atomic-scale dynamics of materials using interatomic potential to simulate the interactions between atoms. In classical MD, electronic effects are effectively averaged out, and the focus is instead on tracking the changes in atomic positions and velocities over time according to Newton’s equations of motion. This method enables researchers to gain insights into the atomic-scale behavior and dynamic properties of materials.

MD simulations provide valuable atomic-scale insights into material behavior at the atomic scale, offering a controlled approach to studying fundamental mechanisms such as interfacial interactions and nanoscale heat transfer that are difficult to isolate experimentally. However, their inherent limitations in time and length scales raise questions about their direct applicability to real-world experimental systems. MD simulations operate at femtosecond time steps and nanoscale models, making it difficult to capture long-term behavior or large-scale material responses. To bridge this gap, careful parameter selection and validation against experimental data are essential. By using experimentally derived force fields, comparing simulation results with available experimental data, and employing multiscale modeling approaches, MD results can be reliably extrapolated to real-world applications. When combined with continuum models and experimental validation, MD serves as a crucial tool for understanding and predicting large-scale material properties [30,31].

Interatomic potentials, often referred to as force fields, are mathematical models derived from first principles or experimental data to represent atomic interactions. Their accuracy is critical as it directly affects the precision of MD simulations and their ability to be seamlessly integrated with mesoscale modeling approaches [32,33,34]. The COMPASS (Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies) force field is designed to accurately simulate the behavior of condensed-phase systems such as liquids, solids, and polymers. This force field is a well-validated, fully flexible force field designed for polymers, cross-linked networks, and organic–inorganic interactions. It accurately describes bond stretching, angle bending, torsional behavior, and nonbonded interactions within epoxy resins. The force field is derived from ab initio (first principles) quantum mechanical calculations to ensure a precise representation of atomic interactions. It incorporates various energy terms including bond stretching, angle bending, torsional potentials, and non-bonded interactions (van der Waals and electrostatic forces), as well as hydrogen bonding treatments [35]. For modeling polymers, the COMPASS force field can be used as it accurately represents the interactions between polymer molecules. This makes it an ideal tool for simulating the structural and thermodynamic properties of polymers, including their mechanical and thermal behaviors. Using COMPASS, we have gained insights into the molecular dynamics of epoxy resins and their performance in various applications. In addition, the Tersoff potential is a well-known interatomic potential that can effectively describe the bond order based on the local atomic environment (especially the number of neighboring atoms). It is designed for materials with covalent bonds and is widely used in systems where atoms exhibit preferred coordination numbers, such as tetrahedral coordination in diamond or silicon carbide. The potential is able to dynamically adjust the bond strength based on the local atoms [36,37].

There is limited research on how structural and compositional characteristics affect the thermal conductivity of CNT-reinforced epoxy composites. To fill this gap, this study investigated the effects of geometric and chemical parameters, as well as temperature, on the thermal properties of CRECs. The simulation results were validated by comparing them with experimental data, previous computational studies, and analytical predictions [38].

2. Molecular Dynamics Simulation

All simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), an open-source MD simulation platform. The details of the simulation setup are described in the subsequent sections.

2.1. Simulation Details

To investigate the thermal conductivity of pure epoxy, Reverse Non-Equilibrium Molecular Dynamics (RNEMD) based on the Müller-Plathe approach was used. This method reverses the typical cause-and-effect relationship in thermal transport simulations. The simulation box is divided into five bins along the z-axis, with the two end regions designated as cold zones and the middle bin designated as a hot zone (see Figure 1). A heat flux is applied as the primary perturbation. Thermal energy is artificially exchanged between the “cold” and “hot” regions by exchanging the velocity of the hottest atom (v_hot) in the cold region with the velocity of the coldest atoms (v_cold) in the hot region. This creates a temperature gradient, which is then measured during the simulation. RNEMD is used because it converges faster than NEMD. While NEMD measures heat flux via temperature gradients, resulting in fluctuations and slower convergence, RNEMD artificially imposes heat flux via velocity exchanges between particles to stabilize the system, which improves simulation efficiency and accuracy, particularly for small systems [37,39]. By periodically repeating the transfer, an artificial heat flux j_z is generated and calculated as follows [40]:

$$j_z={\displaystyle \frac{1}{2tA}} \sum {\displaystyle \frac{m}{2}}(v_{hot}^2-v_{cold}^2)$$

(1)

where A is the cross-sectional area of the simulation box, t is the simulation time, and m represents the atomic mass. v_hot is the velocity of the selected atoms in the hot region, while v_cold is the velocity of the selected atoms in the cold region. Once a steady-state condition is achieved, the heat flux (j_z) can be calculated, which represents the energy flow per unit time and area from the hot region to the cold region due to heat conduction. This heat flux induces a temperature gradient ⟨∂T/∂z⟩ throughout the system. From this gradient, the thermal conductivity (λ) is derived using Fourier’s law [41], which establishes the relationship between the heat flux and the temperature gradient in the system.

$$\lambda =\underset{\partial T/\partial z\to 0}{\lim }\underset{t\to \infty }{\lim }-{\displaystyle \frac{{(j}_z\left(t\right))}{(\partial T/\partial z)}}$$

(2)

After the simulation was complete, the temperature values, z-direction coordinates, and heat flux within the simulation box were systematically recorded for each time step. These recorded data points provide the information necessary to calculate the thermal conductivity of the epoxy. By applying the collected data to Equation (2), the thermal conductivity can be computed using data analysis tools such as Excel 365 or MATLAB R2024a, allowing for precise evaluation of the thermal properties of the materials.

In this study, the thermal conductivity of the composites was evaluated using the Non-Equilibrium Molecular Dynamics (NEMD) method. In heterogeneous systems such as nanocomposites, atoms from different phases (e.g., CNTs and epoxy) may exchange atomic velocities between hot and cold regions, each with unique vibrational modes. This interaction can produce non-uniform heat fluxes in the simulation region, which can introduce biases into the results [42]. To achieve a uniform heat flux, the NEMD method was used in a micro-canonical ensemble, with Langevin thermostats coupled to the epoxy and CNT atoms in the hot and cold regions, respectively. The system was first equilibrated in an isothermal–isobaric ensemble, i.e., maintaining a constant number of particles, pressure, and temperature (NPT), at 300 K and 1 bar to establish initial stability. The simulation box was divided into 5 bins, as shown in Figure 1. The central bin was heated to a high temperature (400 K), while the end regions were cooled to a lower temperature (200 K). The bins between were run in a microcanonical ensemble. The microcanonical ensemble is an MD simulation method that conserves the number of particles, volume, and total energy (NVE) and is suitable for studying isolated systems without external energy or heat exchange. The Langevin thermostats were used to regulate the temperature of the isothermal regions and facilitate accurate heat exchange analysis within the computational domain.

2.2. Computational Details

In this study, the epoxy resin and curing agent are Diglycidyl Ether of Bisphenol A (DGEBA) and Diethylenetriamine (DETA), respectively. The nanofillers considered are SWCNTs of varying lengths (Figure 2), which, therefore, affect the volume fraction of the nanocomposite. DGEBA is an epoxy resin containing two reactive epoxide groups, which enables it to undergo cross-linking reactions during the curing process. DETA is a curing agent with five reactive sites consisting of primary and secondary amine groups. These reactive sites interact with the epoxide groups in DGEBA to form a highly cross-linked three-dimensional network. Each DETA molecule has five reactive sites that can react with up to five DGEBA molecules, while each DGEBA molecule contains two epoxide groups that can be attached to two separate DETA molecules. This reactive interaction typically results in a DGEBA to DETA composition ratio of approximately 5:2 in the mixture, which has been taken into account in the simulations [43,44]. The degree of cross-linking in epoxy resin refers to the extent to which the chemical bonds are formed between the curing agent and reactive functional groups in the epoxy resin, thereby forming a three-dimensional network structure. It quantifies how many potential cross-linking sites in the polymer matrix are occupied, indicating the level of curing or polymerization that has been achieved. The degree of curing directly affects the mechanical strength, thermal stability, and chemical resistance of the polymer [45].

The partially cross-linked epoxy system is first relaxed by energy minimization and then further equilibrated by a brief MD simulation in the canonical ensemble. The process is repeated until the target degree of curing or maximum reaction cutoff is achieved. Unless otherwise stated, the simulations performed in this study used a 70% cross-linking degree. The physical properties of epoxy resins, such as mechanical strength and thermal conductivity, improve with increasing cross-linking degree. However, simulations above 80% cross-linking become computationally expensive and more difficult to equilibrate due to excessive material strain. Thus, a 70% cross-linking degree was chosen for most simulations to balance the accuracy of the material properties with computational efficiency [46,47].

The CNTs were modeled using the Tersoff force field, and the epoxy resin was modeled using the COMPASS force field. The Lennard–Jones interactions between the epoxy resin and the CNTs were determined using the Lorentz–Berthelot mixing rule (Equation (3)). Each system was initially energy minimized and then equilibrated at a constant temperature of 300 K, with a time step of 1 fs. Thermal properties were then evaluated using the RNEMD method [48,49].

$${\epsilon }_{ij}=\sqrt{({\epsilon }_i\ast {\epsilon }_j)}\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}{\sigma }_{ij}={\displaystyle \frac{{\sigma }_i+{\sigma }_j}{2}}$$

(3)

where ε is the energy parameter representing the interaction strength, and σ is the distance at which the interparticle potential is zero, defining the effective particle size.

3. Results and Discussion

The thermal properties of the pure epoxy resin and SWCNT-reinforced epoxy composites were investigated using MD simulations. The results of these simulations are presented and analyzed in detail in the subsequent sub-sections.

3.1. Thermal Properties of Epoxy Matrix

The thermal properties of pure epoxy resin were evaluated using a cubic simulation box containing 920 DGEBA and 368 DETA molecules with a degree of curing, γ, ranging from 65% to 85%, as shown in Figure 2.

The degree of curing in the simulations can be controlled by adjusting the polymerization parameters such as temperature, duration, pressure, and catalyst presence. To analyze the effect of γ on thermal properties, a 1 ns-cured epoxy resin was simulated under ambient conditions (300 K and 1 bar) in the chosen ensemble. The density of the simulated box, ρ_epoxy = 1.1 g/cm³, is very close to the experimental value for DGEBA-DETA epoxy, which is reported to be 1.16 g/cm³ [50]. The RNEMD method was used to calculate the thermal conductivity (λ) of the epoxy at various curing degrees. As shown in the schematic diagram in Figure 1, heat flux was applied between the cold and hot zones of the simulated epoxy resin using the Müller-Plathe algorithm. The calculated thermal conductivity (λ) shows an increasing trend with increasing cross-linking degree, as shown in Figure 3 and

Table 1. Thermal conductivity of pure epoxy at varying cross-linking degrees (300 K, 1 bar) compared with the results from the literature.

γ (%)	65	70	75	80	85	Note
λ from MD Simulation	0.157 ±0.0031	0.159 ±0.0032	0.181 ±0.0031	0.183 ±0.0036	0.201 ±0.0040	This Study
γ (%)	65	68	75	80	88	Note
λ from Literature	0.16	0.17	0.185	0.19	0.215	[1,51,52]

, which can be attributed to the density enhancement of the epoxy matrix. As shown in Figure 3, the thermal conductivity (λ) exhibits an exponential trend with a coefficient of determination R² greater than 0.93:

$$\lambda =A e^{(B.\gamma )}$$

(4)

The results show that the regression constants for the thermal conductivity of pure epoxy at 1 bar pressure and 300 K are A = 0.0677 and B = 0.0127.

The red-squared dots in Figure 3 represent the results from the literature, which are in good agreement with our findings. Wan et al. [53] reported that the thermal conductivity of epoxy resin at 300 K increased significantly with increasing cross-linking degrees, reaching 0.18 W/m·K and 0.21 W/m·K at the cross-linking degrees of 75% and 87%, respectively. Similarly, Choi et al. [54] also demonstrated that the thermal conductivity increased with increasing cross-linking degree of epoxy resin, which further supports the consistency of our results with previous studies.

3.2. Thermal Properties of CNT-Reinforced Epoxy Composites (CRECs)

To investigate the thermal conductivity of CRECs, a cubic simulation box filled with DGEBA-DETA resin and reinforcement of five (6,6) SWNTs was initially considered, as shown in Figure 4. The thermal conductivity of pure SWCNTs was taken from our previous study [33], where the thermal conductivity of SWCNTs was determined using MD simulations where the SWCNTs were modeled in a free-standing state without direct interaction with the epoxy matrix. In fact, in a composite system, the actual thermal conductivity of SWCNTs is affected by the interfacial phonon scattering caused by the interactions with the atoms of the surrounding epoxy molecules. This results in a lower effective thermal conductivity compared to the free-standing case. To minimize the effect, in this study, the free-standing value was used as a common approximation in the models due to the lack of direct measurements of the thermal conductivity of the embedded SWCNTs, and the thermal interaction between the interfaces of the SWCNTs and the epoxy molecules was created. To evaluate the thermal conductivity of CRECs, the simulation domain consisted of 49,160 atoms with periodic boundary conditions applied on each Cartesian axis. All simulations were performed using a time step of 1 fs. Five (6,6) SWCNTs were introduced into the simulation domain and then filled with epoxy molecules. The system was energy minimized and relaxed for 1 ns in the isothermal–isobaric (NPT) ensemble at 300 K and 1 bar using the Nosé–Hoover thermostat. Once the steady-state condition was achieved, the cross-linking procedure was initiated. After reaching the target density of the epoxy, the system was further relaxed for 100 ps in the micro-canonical (NVE) ensemble. Finally, NEMD simulations were performed on this fully relaxed configuration to determine the effective thermal conductivity of the nanocomposite.

To gain a deeper understanding of the key factors affecting the thermal conductivity (λ) of SWNT-reinforced epoxy composites, the structural, compositional, and material properties of the polymer nanocomposite were systematically varied, and their effects on λ were analyzed.

Studying the effect of the cross-linking degree of epoxy on thermal conductivity is essential as it directly affects the network structure, thermal conductivity, and overall performance of the polymer, thereby optimizing effective thermal management in various applications. The effect of varying cross-linking degrees in the epoxy matrix on the thermal conductivity of the CRECs with a cube length of 8.1 nm was investigated by evaluating γ values of 65%, 70%, 75%, and 80% on the CNT major axis. The results in Figure 5 and

Table 2. Thermal conductivity of CRECs at varying cross-linking degrees (300 K, 1 bar) compared with the results from the literature.

γ (%)	65	70	75	80	Note
λ from MD Simulation	1.67 ±0.0167	1.74 ±0.0174	1.78 ±0.0178	1.79 ±0.0179	This Study
γ (%)	68	75	80	85	Note
λ from Literature	0.16	0.17	0.185	0.19	[55,56,57]

show that the thermal conductivity (λ) follows an exponential trend and the coefficient of determination (R² > 0.89) follows the relationship described by Equation (4).

The simulation results show that the regression constants of the thermal conductivity of the CRECs at a pressure of 1 bar and T= 300 K are A = 1.248 and B = 0.0046.

The positive correlation observed between the cross-linking degree and the thermal conductivity is attributed to two main factors: The thermal conductivity of the polymer matrix is enhanced and the thermal resistance at the SWNT–epoxy interface is reduced with increasing curing degree. This results in a more efficient heat transfer in the composite materials. Furthermore, the thermal conductivity values of the CRECs are higher than that of pristine epoxy at all degrees of cure, highlighting the dominant role of CNT fillers in enhancing the thermal conductivity of the composites. The CNTs facilitate the transfer of thermal energy, making them a critical component in the heat transfer mechanism of the nanocomposite relative to the pure epoxy resin.

The effect of CNT length on thermal conductivity (λ) was simulated and analyzed for a cross-linking degree (γ) of 70%. Simulations were performed by varying the CNT length (L) from 2 nm to 8.1 nm to evaluate how the length of the SWCNTs affects the thermal conductivity of the nanocomposite. The results in Figure 6 and

Table 3. Effect of SWCNT length on thermal conductivity of CRECs at 300 K, 1 bar, and 70% cross-linking degree (γ), compared with the results from the literature.

L (nm)	0	2	4	6	8.1	10	20	Note
λ from MD Simulation	0.159 ±0.004	1.20 ±0.036	1.69 ±0.050	1.74 ±0.052	1.75 ±0.052	1.75 ±0.050	1.75 ±0.052	This Study
L (nm)	0	2	4.2	6	8	16	24	Note
λ from Literature	0.17	1.38	1.73	1.79	1.8	1.8	1.8	[58,59,60]

show that the thermal conductivity (λ) follows a logistic growth function:

$$\lambda ={\displaystyle \frac{A}{1+e^{-B(L-L_0)}}}-C$$

(5)

As shown in the figure, the thermal conductivity (λ) of the reinforced systems is improved compared to pure epoxy. For CNT lengths ranging from 2 to 8.1 nm, λ exceeds the thermal conductivity (λ) of pristine epoxy, demonstrating the positive impact of CNT reinforcement on the heat transfer properties. However, for CNT lengths exceeding 6 nm, the thermal conductivity remains nearly constant. The simulation results show that the regression constants for the thermal conductivity of the composite at 1 bar pressure, 70% cross-linking degree, and T = 300 K are A = 1.907, B = 1.308, C = 0.163, and L₀ = 1.3

Figure 7 and

Table 4. Thermal conductivity of CRECs at varying temperatures under 1 bar pressure, 70% cross-linking degree (γ), and CNT length of 8.1 nm, compared with the results from the literature.

T (k)	250	300	350	400	Note
λ from MD Simulation	1.49 ±0.030	1.74 ±0.035	1.88 ±0.038	1.93 ±0.038	This Study
λ from Literature	1.57	1.85	2	2.1	[61,62]

illustrates the relationship between thermal conductivity (λ) and temperature for the studied nanocomposite systems. The simulations were performed at four different temperatures: 250 K, 300 K, 350 K, and 400 K, with a curing degree of γ = 70% and a nanotube length of 8.1 nm. The figure highlights how λ varies with increasing temperature, providing insights into the thermal behavior of the material under different thermal conditions.

The simulation result in Figure 7 shows that the thermal conductivity (λ) exhibits an exponential trend with a coefficient of determination R² > 0.88:

$$\lambda =A T^{\left(B\right)}$$

(6)

The simulation results show that the regression constants for the thermal conductivity of the composites at 1 bar pressure, γ = 70%, and L = 8.1 nm are A = 0.0705 and B = 0.5584.

Jakubinek et al. [61] and Gardea et al. [63] reported that a similar trend was observed. This behavior stems from the fact that the thermal conductivity of CNTs and epoxy resin are proportional to temperature, at least in the range considered in this study [64,65].

The study concluded by analyzing the effect of CNT volume fraction (f) on the thermal conductivity (λ) of the composites using a simulation box with dimensions of 8.1 nm × 8.1 nm × 8.1 nm. The domain consisted of epoxy resin with a curing degree of γ = 70% and reinforced with five SWCNTs of length L = 8.1 nm. The simulations were performed for CNTs with varying chiral indices: (6,6), (8,8), (10,10), and (12,12), corresponding to volume fractions of f = 1.5%, 3%, 5.2%, and 6.4%, respectively. It is worth noting that the SWCNTs were considered hollow and not filled with epoxy resin inside.

Figure 8 and

Table 5. Thermal conductivity of CRECs at a varying volume fraction of SWCNTs, under 1 bar pressure, 70% cross-linking degree (γ), and CNT length of 8.1 nm, compared with the results from the literature.

f (%)	0	1.5	3	5.2	6.4	Note
λ from MD Simulation	0.159 ±0.004	1.44 ±0.043	1.73 ±0.051	1.80 ±0.055	1.80 ±0.057	This Study
f (%)	0	1.2	4	5.5	6.2	Note
λ from Literature	0.17	1.51	1.85	1.9	1.9	[64,65,66,68,69,70]

shows a logistic growth relationship between the thermal conductivity (λ) of the composites and the CNT volume fraction (f). This observation is consistent with the results from previous studies, indicating that increasing the volume fraction of SWCNTs increases the thermal conductivity of the composite [66,67]. The increase in the thermal conductivity of the composite is attributed to the high intrinsic thermal conductivity of CNTs, which significantly contributes to the overall heat transport within the system as the volume fraction of CNTs increases. However, when the CNT volume fraction exceeds 5%, the thermal conductivity remains almost constant, and the regression follows the following equation:

$$\lambda ={\displaystyle \frac{A}{1+e^{-B\left(f\right)}}}-C$$

(7)

The simulation results show that at γ = 70%, L = 8.1 nm, and at 1 bar pressure and T = 300 K, the regression constants for the thermal conductivity of the composite are A = 3.32, B = 1.35, and C = 1.5.

There are several analytical analyses that provide theoretical models for estimating the thermal conductivity of composites based on the properties of the matrices and fillers. When the fibral filler is parallel aligned, the effective thermal conductivity of the composite can be estimated using the following equation based on the rule of mixture [38]:

$${\lambda }_{eff,z}=\left(1-f\right){\lambda }_m+f{\lambda }_f$$

(8)

For the transverse direction (perpendicular to the reinforcement, such as fibers or CNTs), the following model based on the inverse rule of mixture is used to estimate the effective thermal conductivity of composite materials [71]:

$${\lambda }_{eff,x}=({{\displaystyle \frac{f}{{\lambda }_f}}+{\displaystyle \frac{(1-f)}{{\lambda }_m}})}^{-1}$$

(9)

where λ_m and λ_f represent the thermal conductivity of the polymer matrix and the filler material, respectively, and f denotes the volume fraction of the filler. The thermal conductivity of the SWCNT used in this study was obtained from previous work [33], in which its intrinsic thermal conductivity was determined using molecular dynamics simulations for a free-standing state.

The effective thermal conductivity of the composite (λ_c) was assessed in both the axial/longitudinal direction of the CNTs and the transverse direction. The thermal conductivity of the composite was evaluated in the axial and transverse directions relative to the CNT axis at a cross-linking degree of γ = 70%. The results are shown in Figure 9a,b, showing that the thermal conductivity of the composite along the CNT axis and perpendicular to the CNT axis are λ_c,z = 1.744 W/m·K and λ_c,x = 0.393 W/m·K, respectively. The thermal conductivity of the composite in the transverse direction is improved by a factor of 2.5 compared to that of the pure epoxy matrix. In contrast, the axial/longitudinal thermal conductivity of the composite is significantly improved by a factor of 11 relative to that of the pure epoxy, highlighting the enhanced heat transfer achieved by the oriented CNT network. The reason is that polymer nanocomposites reinforced with continuously aligned fibers exhibit pronounced anisotropic thermal conduction properties. In the direction in which the fibers are aligned, CNTs play a dominant role in enhancing the thermal conductivity of the composites, thereby significantly contributing positively to heat transfer. However, in the transverse direction, the heat conduction is severely hampered by the thermal resistance at the interfaces between the CNTs and the polymer matrix, creating a substantial thermal barrier that limits the effective heat transfer [72].

Although this study investigated the effects of many key factors (e.g., degree of cross-linking of the epoxy matrix, CNT length and volume fraction, and temperature) on the thermal conductivity of CRECs, there are many other factors (e.g., orientation of CNTs, their distribution quality and heat transfer resistance at the interface, the aspect ratio and chirality of the CNTs, the type of CNTs (such as SWCNTs, DWCNTs, and MWCNTs)) that play significant roles in determining the overall thermal conductivity of CNT-based composites. While these other factors are undoubtedly important, they were outside the intended scope of this study to avoid unnecessary complexity and allow for a targeted analysis. Future studies will be aimed at addressing these aspects.

4. Conclusions

In this study, an MD computational framework was developed to investigate the effects of key factors, such as cross-linking degree, CNT length and volume fraction, and temperature, on the thermal conductivity of CRECs. The effective thermal conductivity of the composites was simulated and analyzed using MD simulations. This approach allows for a detailed understanding of how structural, physical, and compositional parameters affect heat transfer in CRECs, providing valuable insights for optimizing their thermal performance in various applications. The simulation results show that the effective thermal conductivity of the nanocomposite is significantly affected by multiple factors, including the length and volume fraction of the CNTs, the degree of cross-linking within the epoxy matrix, and the temperature of the composite. As the length of the nanotubes increases, the enhanced thermal paths provided by the extended CNT structure contribute to improved heat transfer efficiency. Similarly, higher CNT volume fractions lead to denser conductive filler networks in the composites, further enhancing thermal conductivity. Overall, these results highlight the synergistic effects of structural, compositional, and thermal parameters in optimizing the thermal properties of CRECs, providing valuable insights for the design and development of advanced thermal management materials. The simulation results of the thermal conductivity are in good agreement with the experiments and simulations from the literature and are consistent with the predictions by the analytical method. As observed, the heat transfer of the composites reinforced by the aligned CNTs in the axial/longitudinal direction is significantly higher than that in the transverse direction, highlighting the anisotropic thermal behavior of the CRECs. Future studies will aim to address the effects of other key factors (e.g., the orientation of CNTs, their distribution quality and heat transfer resistance at the interface, the aspect ratio and chirality of the CNTs, and the type of CNTs) on the thermal behavior of the CNT-based polymeric matrix composites.