Multi-scale Friction Simulation and Experimental Verification of Carbon Nanotube-Reinforced PTFE Composites

The synergistic analysis of friction properties of carbon nanotube (CNT)-reinforced polymers at the nanoscale and macroscale can help to obtain the intrinsic mechanism of carbon nanotubes to reduce the friction coefficient of polymers, which is important to guide the modification of polymer friction properties. However, the huge gap in spatial scales makes it difficult for molecular dynamics (MD) simulations at the nanoscale to predict the friction coefficient of virtual contact interfaces, and conducting a large number of macroscopic experiments to obtain natural frictional laws could be more efficient. This study proposes a multi-scale model to investigate the frictional behavior of copper (Cu)-CNT/polytetrafluoroethylene (PTFE). By using the micromechanics Mori–Tanaka homogenization method as a bridge, the nanoscale simulations of the CNT/PTFE elasticity and frictional behavior and the macroscopic finite element simulation of the Cu ring-CNT/PTFE block contact are coupled, thus integrating the nanoscale frictional laws of Cu-CNT/PTFE obtained from molecular dynamics simulations into the actual contact interface. The results of multi-scale friction simulations show that the filling of CNTs can effectively improve the elastic and frictional properties of the PTFE matrix, and the degree of improvement is related to the orientation and mass fraction of the CNTs. Under a mean contact pressure of 0.5 MPa and a rotating speed of 30 rpm, the friction coefficient continuously decreases (from 0.198 to 0.156) with increasing CNTs mass fraction (0%, 1.25%, 2.5%, 5%). The simulation results were verified by copper ring-CNT/PTFE block friction experiments.


Introduction
Polymer materials have great potential in the field of mechanical structures due to their excellent comprehensive performance and processing properties [1][2][3][4]. However, the relatively weak mechanical properties, self-lubricating properties, and wear resistance of polymers limit their further development as materials for mechanical friction pairs. The addition of force-enhancing fillers or lubricating fillers such as molybdenum disulfide, graphite, carbon fibers, and silicon carbide is a common practice to improve the mechanical and frictional properties of polymers [5][6][7][8]. Among them, CNTs with their unique structure and excellent mechanical properties, self-lubrication, and wear resistance, have shown great potential as additives for polymer lubrication and wear resistance [9][10][11][12]. Many factors influence the frictional properties of CNT-modified polymers, including CNT size, content, and dispersion. For different polymer matrices, it is worth conducting in-depth research on how to design CNT composites with optimal frictional performance.
The conventional and most direct approach to investigate the frictional properties of carbon nanotube-reinforced polymer composites is through the preparation of various composite materials and conducting macroscopic friction experiments. Ring-block friction experiments under dry conditions showed that the friction coefficient of the composites decreased with increasing CNTs content when CNTs of mass fraction 2.5-30% were introduced as fillers into PTFE, and PTFE composites blended with 15-20% CNTs exhibited excellent anti-wear properties [13]. Similarly, Vail et al. [14] tested the tensile properties, frictional behavior, and surface electrical properties of single-walled carbon nanotube (SWCNT)-filled PTFE composites with CNT content ranging from 0 to 15%. The results showed that the natural tensile stress increased by 200%, and the frictional properties were significantly improved when 2% CNTs were filled. Friction comparison experiments between carbon nanotube (CNT)-reinforced PEEK and pure PEEK, prepared using a deformation-driven process, demonstrated that the friction coefficient and wear rate of 3.0 wt% CNTs/PEEK were lower by 7.32% and 6.71% compared to pure PEEK [15]. In addition to the mass fraction, the length and diameter of carbon nanotubes also influence the frictional properties of the composite material. Li et al. [16] prepared carbon nanotubefilled lithium-based lubricants and investigated the effect of nanotube length and diameter on their friction performance. The results showed that increasing the nanotube length led to a reduction in the friction coefficient and wear rate, while the effect of nanotube diameter on the friction coefficient was relatively low. Apart from pristine carbon nanotubes, various functionalized carbon nanotubes, such as carboxylated, silanized, carbonylated, and aminated, have been found to improve the mechanical and frictional properties of composite materials [17,18]. Further, the PTFE composites doped with carbon nanotubes and other force-reinforced fillers also exhibited excellent mechanical properties. When a 20% mass fraction of oxybenzene ester was added, even a 1% mass fraction of CNTs could significantly enhance the tensile strength, modulus, and creep resistance of PTFE composites [19]. Internal friction and compressional deformation tests using low-frequency independent amplitude (AIIF) and independent amplitude (ADIF) showed that the frictional properties of the new PTFE-based nanocomposites doped with 5 wt% and 10 wt% Co clusters were enhanced in both elastic and inelastic deformation domains [20]. Zhilin et al. [21] successfully prepared one-dimensional carbon nanotube/nanorod (CNT/CNR) hybrid nanocarbon materials and blended them with PTFE as fillers. The experimental results showed that the volume wear rate of PTFE nanocomposites filled with 0.3% CNT/CNR was only 1/700 of pure PTFE under a load of 200 N and a rotation speed of 200 r/min.
Although traditional friction experimental studies described above can provide some insight into the frictional properties of carbon nanotube-reinforced polymers, this process requires the preparation of a large number of comparative specimens, and the repetitive nature of the experimental procedure is inefficient. To efficiently prepare composite materials with excellent frictional properties and accurately predict their friction behavior, reliable friction simulation methods need to be developed.
In recent years, with the rapid development of computer hardware performance, the use of MD methods to simulate the mechanical and frictional properties of polymers from the nanoscale has become a hot research topic. The polymer model, constructed from the molecular chain, is the basic physical structure model. The simulation of micromechanical phenomena such as interatomic force transfer, molecular structure extrusion, bond formation, and fracture is an effective tool for analyzing the deformation, stress, and strain patterns of polymers and its composites [22][23][24][25]. The relative motion generated by the frictional pairing of PTFE molecular structures can be used to investigate the frictional properties and intrinsic wear mechanism of PTFE [26][27][28][29]. Barry et al. [30][31][32] designed a series of MD simulations under different working conditions to analyze the effects of temperature, sliding velocity, and molecular chain length on the frictional behavior of PTFE. They explained the degree to which these factors influence the frictional properties of PTFE at the molecular level. Regarding the engineering application of PTFE friction, MD simulations of metal-PTFE pairings have also progressed. Song et al. [33] conducted MD simulations of Cu-PTFE friction pairs under dry friction and water lubrication conditions. They explained the molecular interactions between PTFE molecules, water molecules, and Cu atoms. Based on the MD simulation of Cu-PTFE molecular friction, they also conducted groundbreaking research on the frictional performance enhancement of PTFE by carbon nanotubes (CNTs). The simulation results showed that the average coefficient of friction of PTFE sliding against a Cu layer in the stable phase decreased from 0.169 to 0.127 after CNTs reinforcement [34]. Similarly, simulations of Fe-PTFE and Fe-CNT/PTFE friction pairs at the molecular level have been investigated by others, revealing the intrinsic mechanism of improving the frictional properties of PTFE by CNTs from the nanoscale [35][36][37]. It should be noted that considering the significant influence of carbon nanotube orientation on its mechanical properties, it is necessary to conduct separate studies of different orientations of carbon nanotube fillers when performing MD simulations of carbon nanotube-polymer composites [38,39].
Although MD studies have provided valuable theoretical insights into the frictional behavior of CNT-reinforced composites, validating the MD simulations through experimental investigation at the nanoscale is challenging. The spatial scale of molecular dynamics simulations is also minimal due to the high computational cost. More unfavorably, it is even more challenging to integrate the mechanical and frictional properties obtained from molecular dynamics simulations with macroscopic contact friction models due to the significant disparities in spatial and temporal scales of measurement [40,41]. Therefore, the combination of nanoscale simulations and macroscopic simulations through a multi-scale modeling approach is precious for guiding the optimization design of composites [42].
As a high-performance polymer material, PTFE possesses excellent characteristics such as a low friction coefficient, wide operating temperature range, and low reactivity [43]. Compared with pure PTFE, CNT-reinforced PTFE has been proven to possess better mechanical properties, friction performance, and wear resistance, making it widely used in engineering technologies such as coatings [44,45], thin films [46,47], and textiles [48]. In this paper, a joint multi-scale simulation of mechanical and frictional properties of CNT/PTFE composites is performed, and the multiscale simulation consists of two parts:One part of this study involves the multi-scale simulation of the mechanical properties of CNT/PTFE composite materials. The Mori-Tanaka model [49], a numerical homogenization method commonly used to investigate composite materials' mechanical and elastic properties at the microscopic scale, has been effectively applied to composite material damage [50][51][52]. In this study, the elastic properties of mesoscale CNT/PTFE composites were obtained by homogenizing the nanoscale isotropic pure PTFE and transverse isotropic CNT/PTFE representative volume element (RVE) elastic mechanical parameters obtained from molecular dynamics simulations using the Mori-Tanaka method as a bridge. The composite elastic parameters were then imported into a macroscopic finite element friction model to predict the contact stress distribution of the macroscopic friction model. The other part of this study involves the multi-scale simulation of the frictional characteristics of the Cu-CNT/PTFE friction pair. The molecular dynamics simulation was used to obtain the friction coefficients of the copper-PTFE friction pair and the copper-CNT/PTFE friction pair at the microscopic scale. Considering the mass fraction and directional distribution of CNTs in the composite material, the frictional characteristics obtained from the MD simulation were used as input to the friction coefficient equation at the contact interface of the macroscopic finite element friction model. The two parts of this work are closely coupled, with the multi-scale simulation of mechanical properties forming the basis for the multi-scale friction modeling. The two simulation tasks were carried out simultaneously and jointly in a macroscopic finite element friction model, establishing a complete multiscale friction model for CNT/PTFE composite materials. Finally, the simulation results are compared and analyzed with the experimental data of copper ring-CNT/PTFE block friction. Establishing this multi-scale friction model using a novel approach provides new insights into the study of the frictional properties of polymers.

Methods
As this study is a novel attempt, only the PTFE matrix's elastic deformation stage and its composites were considered. Both friction simulations and experiments were limited to dry sliding conditions. MD simulations were performed as the first step in the multi-scale modeling, including the materials' elastic properties and dry friction characteristic simulations. The elastic properties of the materials to be simulated include the isotropic tensile modulus and Poisson's ratio of pure PTFE, as well as the elastic parameters of the transversely isotropic CNT/PTFE RVE. The friction coefficients of Cu-PTFE and Cu-CNT/PTFE RVE friction pairs need to be calculated for simulation. Then, the Mori-Tanaka method in micromechanics is used to homogenize the elastic mechanical parameters of the pure PTFE matrix and CNT/ PTFE RVE. The obtained CNT/PTFE material parameters were used as input for the contact mechanics parameters of the continuum macroscopic finite element friction model. The frictional characteristics of the molecular simulations were used to determine the friction coefficient function of each element on the contact interface, realizing the multiscale friction simulation study. Under a mean contact pressure of 0.5 MPa and a rotational speed of 30 rpm, multiscale friction simulations were conducted on four types of CNT/PTFE polymer composites with CNT mass fractions of 0 (pure PTFE), 1.25%, 2.5%, and 5%. Finally, a copper ring-PTFE block friction experiment was conducted to validate the effectiveness of the multi-scale friction model.

MD Simulations
In this study, we combined the strengths of various molecular simulation software. We utilized the commercial software Materials Studio (MS version 2019) for molecular structure modeling and the large-scale atom/molecule parallel simulator (LAMMPS) [53] for system optimization and MD simulation. The polymer consistent force field (PCFF) was used in this molecular dynamics simulation [27]. The non-bonded interactions between atoms were represented by Coulomb and van der Waals (vdW) interactions, and the cutoff distance for non-bonded interactions was set to 14.0 Å [54].

Modeling and Mechanical Simulations of Pure PTFE and CNT/PTFE RVE
The modeling process of pure PTFE was performed first. PTFE blocks are formed by many amorphous and randomly arranged chain structures. C-C bonds connect PTFE chains and consist of multiple repeating units (C2F4). In the MS software, we constructed a box with dimensions of 50 Å × 30 Å × 30 Å. Based on the actual density of pure PTFE, the box density of 2.1 g/cm 3 was preset. Several PTFE chains with several repeating units of 20 were randomly filled into the periodic box using the Monte Carlo method [55,56] until the preset density was reached. The generated pure PTFE unit cell is shown in Fig. 1a.
Then the modeling of CNT/PTFE RVE was performed. In this work, the armchair (6, 6) single-walled carbon nanotube (SWNT) was selected as the object of investigation. The single-walled carbon nanotube has a diameter of 8.14 Å and a length of 29.51 Å. Firstly, a CNT (6, 6) structure was placed at the center of a 50 Å × 30 Å × 30 Å box, with its axis aligned parallel to the X-direction of the box. Then, the Monte Carlo method was employed to randomly fill a periodic box with a certain number of PTFE chains, each containing 20 repeating units until the mass fraction of the carbon nanotube reached 10%. The CNT/PTFE RVE with a CNT mass fraction of 10% was constructed here in preparation for the homogenization process and multi-scale friction simulation of CNT/PTFE materials with different mass fractions (5%, 2.5%, 1.25%) later. For example, by homogenizing a volume fraction of 0.5 of CNT/PTFE RVE with a volume fraction of 0.5 of PTFE matrix, the mechanical properties of a macro-scale (micrometer-level) CNT/PTFE composite material with a mass fraction of 5% can be obtained. This enables the incorporation of the 5% CNT composite material into a finite element model for macro-scale friction simulations. The resulting CNT/PTFE representative volume element (RVE) had a density of 1.90 g/ cm 3 , as shown in Fig. 1b.
Once the model structure was established, the periodic models were imported into LAMMPS using the msi2lmp function module in MS. Since the initial amorphous unit cell's high total energy was extremely unstable, it required optimization. The conjugate gradient method was employed to minimize the energy of the entire molecular system, with energy convergence set to 10 -5 kcal/mol and force convergence set to 10 -4 kcal/mol/Å [26]. Furthermore, to eliminate singular connections and forces between molecular chains and molecules within the unit cell, 1 ns of MD equilibrium was performed in the NPT (constant-pressure, constant-temperature) ensemble, with a temperature set to 300 K and a pressure of 0.1 MPa atmospheric pressure. Subsequently, the equilibrated PTFE unit cell was subjected to the annealing treatment, with the temperature range set to 250 K-500 K and the temperature step size set to 50 K for 10 cycles. Through a cycle of annealing processes from high to low temperatures, the molecular structure within the system was better relaxed, and the molecular chain structure was better optimized. Finally, in the NVT (canonical ensemble), at a temperature of 300 K and a pressure of 0.1 MPa, the entire system was subjected to 1 ns of MD equilibrium. Such system optimization results in the lowest energy at the set temperature of 300 K and a pressure of 0 in all directions.
Before performing the simulation of elastic mechanical properties, it is necessary to examine the independent elastic parameters of the material. Based on the principles of elastic mechanics, the stresses and strains in the elastic deformation stage of the material conform to the following instantaneous equations: where and are the flexibility tensor and the stiffness tensor of the material, respectively.
Since pure PTFE is an isotropic polymer, its flexibility tensor takes the form of: The Young's modulus E and Poisson's ratio are two independent parameters, and the shear modulus G satisfies the following relationship: The CNT/PTFE RVE is a transversely isotropic material with a flexibility tensor of the form: where E 11 is the axial (x-direction) Young's modulus, E 22 is the transverse elastic modulus ( 12 is the axial Poisson's ratio ( 12 = 13 ), G 12 is the axial shear modulus, and 23 is the transverse Poisson's ratio. Due to the transverse isotropy in the 2-3 plane, the transverse shear modulus G 23 satisfies the following relationship: A tensile simulation was conducted on the optimized PTFE unit cell to obtain Young's modulus and Poisson's ratio. A continuous strain was applied to the unit cell x (11) direction with a straining step of 0.0025, and the maximum strain was set to 0.05. In this study, the elastic mechanical properties of the material were obtained by applying the constant strain method in both the tensile and shear simulations, which has been proven to be reliable in molecular simulations of carbon nanotube-reinforced composites [57].
To investigate the effect of the direction of transverse isotropy of carbon nanotubes on the mechanical properties of the composite material, three mechanical simulation processes (axial tensile, axial shear, and transverse tensile) were designed to obtain the five elastic parameters ( E 11 , E 22 , 12 , G 12 , 23 ) of the CNT/PTFE RVE. Continuous tensile and shear strains were applied to the tensile and shear motion simulations, respectively, and the strain loading steps were set to 0.005 for both tensile and shear, and the maximum strain was set to 0.05.

Friction Simulations of Cu-PTFE and Cu-CNT/PTFE RVE
A metal Cu crystal box with a thickness of 10 Å was constructed and placed on top of the PTFE box. The bottom region of the original PTFE box with a thickness of 5 Å was defined as the fixed layer, while the rest was defined as the mobile layer. The normal load was applied on top of the copper layer, and a sliding speed was imposed in the tangential direction to simulate sliding friction. Several load pressures of 0.05-0.80 MPa (corresponding to the contact interface load distribution of the finite element simulation) were designed. Dry sliding friction tests were performed at a sliding velocity of 0.055 m/s. The friction coefficient was obtained by calculating the ratio of the total reactive force on the copper layer in the direction of atom motion to the applied load. Each friction test was conducted for 300 ps, and the mean value of the friction coefficient over a stable period of 150 ps was taken as the effective friction coefficient. The time step of the sliding friction is 1 fs.
Similar to the Cu-PTFE friction simulation, dry sliding friction simulations were performed on Cu-CNT/PTFE RVE. However, considering the influence of CNT orientation on the frictional properties of CNT/PTFE, separate simulations were conducted for different directions of CNT/ PTFE. Since CNT/PTFE exhibits isotropic characteristics in the Y and Z directions, the frictional properties in these two directions are basically the same [38]. Therefore, this friction process simulation was carried out in X and Y directions, as shown in Fig. 2. The load and sliding velocity settings for the Cu-CNT/PTFE RVE friction simulation were identical to those for Cu-PTFE.

Mori-Tanaka Homogenization Process
In actual CNT/PTFE composite materials, the CNTs are randomly dispersed in the matrix and the macroscopic elastic properties of the CNT/PTFE composite exhibit isotropic elastic properties. Therefore, a method is needed to combine the elastic mechanical parameters of the isotropic PTFE matrix obtained from the previous section with those of the transversely isotropic CNT/PTFE RVE. Homogenization methods in composite materials mechanics can combine the elastic mechanical properties of two-phase materials (filler fibers and matrix) in three-dimensional space to predict the mechanical properties of composite materials. In this section, the Mori-Tanaka method is employed to calculate the elastic properties of CNT/PTFE. Figure 3 shows the homogenization process of CNT/PTFE RVE and pure PTFE matrix.
For composites with a defined matrix (pure PTFE) and filled fibers (CNT/PTFE RVE), the mean internal stresses of the matrix and fibers are related at their common interface as: The bridging tensor [ ] is defined here. The effective compliance tensor [ ] of the homogenized composite material can be expressed as: and are the stiffness tensor of the PTFE matrix and CNT/PTFE RVE, respectively, and [ ] denotes the Eshelby tensor [58].
It should be noted that the composite flexibility tensor [ ] is obtained here in the ordered arrangement of CNT/ PTFE RVEs in the matrix. Considering the random distribution of CNT/PTFE RVE directions, it is necessary to transform [ ] from the local coordinate system(x y z ) to the global coordinate system ( x ′ y ′ z ′ ) and take the average value. The bridging tensor ′ after coordinate transformation can be expressed as: here [ ] is the rotation operator matrix: where , , are Euler angles and the coordinate transformation relations are shown in Fig. 4. The averaging of Q with a random distribution of directions results in the following expression [59]: By substituting the bridging tensor obtained from the averaging of the randomly distributed directions into Eq. (8), the effective stiffness tensor of the homogeneous and isotropic CNT/PTFE composite material can be calculated, and the elastic parameters of CNT/PTFE composites are determined.
In this section, Mori-Tanaka homogenization was performed on three types of CNT/PTFE composites with CNT mass fractions of 1.25%, 2.5%, and 5%. In the previous section of MD modeling, the initial CNT mass fraction in the CNT/PTFE RVE was set to 10%. Therefore, when the CNT mass fractions are 1.25%, 2.5%, and 5%, the corresponding V R values are 0.125, 0.25, and 0.5, and the corresponding V M values are 0.875, 0.75, and 0.5.

Finite Element Simulations
The finite element method is a classical technique for computing the mechanics of macroscopic continua. It is also a practical approach for solving nonlinear problems such as friction and contact. In finite element models, the material is divided into numerous small elements, and the mechanical response of the system is obtained by transferring stress and strain between the nodes. Specifically, for the finite element problem of rolling contact, the arbitrary Lagrangian-Eulerian (ALE) method [60] can be used to solve it. To facilitate subsequent experimental validation, the structure and size of the copper ring-CNT/PTFE block model constructed in this section remain consistent with the values used in subsequent experiments, with specific parameters shown in Table 1. The frictional pair of the finite element ring block is shown in Fig. 5. The finite element modeling and meshing were performed using ANSYS software (version 2022R1), and the ALE method was employed to solve the problem.

Effect of Contact Interface Meshing on Pressure Distribution
The homogenized elastic properties of the CNT/PTFE composite material from Sect. 2.2 were incorporated into the finite element friction model to compute the mechanical transfer between individual elements and nodes and determine the pressure distribution on the contact interface. In finite element simulations, mesh generation is a critical factor that affects the simulation results. In this particular finite element simulation, the frictional contact interface is the direct region where frictional forces are transmitted, thus it is necessary to investigate the mesh generation for the contact interface. Considering the overall area of the contact interface and the hardware capabilities of the computer, an initial local mesh refinement of 0.5 mm and an overall mesh refinement of 1 mm were used for the load distribution calculation, resulting in a load of 0.5 MPa. Subsequently, based on the preliminary calculation results, the local mesh refinement for the contact interface was improved to 0.3 mm, and finally refined to 0.2 mm. The load distribution at the contact interface of pure PTFE under three different local mesh refinements is shown in Fig. 6. Comparing the three load distribution patterns in Fig. 6, it can be observed that the local maximum loads in Figs. 6b and 6c both occur at the curved edges of the specimen, whereas this phenomenon is not evident in Fig. 6d. Since the thickness of the copper ring is greater than that of the specimen, stress concentration occurs at the edges of the specimen during normal loading, thus the load distribution patterns in Figs. 6b and 6c are reasonable. Additionally, after refining from 0.3 mm to  0.2 mm, the load distribution becomes more balanced. Therefore, the final choice for the local mesh refinement of the contact interface is set at 0.2 mm. Table 2 provides a statistical summary of the parameters for the three different local mesh refinements.

Calculation of Friction Coefficient
With the model structure and material elastic parameters known, the friction coefficient i for each element on the contact interface is a function of the local contact pressure and velocity of the element [61]: here i is the index of all contacting elements on the contact interface, p i is the local contact pressure of the element, and v is the relative displacement velocity. The friction coefficient function for each element is determined based on the MD friction simulation results from Sects. 2.1.2. Therefore, the overall friction coefficient for the copper ring-CNT/ PTFE block interface can be expressed as: here p is the overall external pressure, A c is the contact area, and A i is the area of each individual contacting element. FE friction simulations were performed for CNT/PTFE composites with CNTs mass fractions of 0, 1.25%, 2.5%, and 5%, respectively. The copper ring speed was set to 30 rpm (0.055 m/s), and the pressure load was set to 0.5 MPa. The simulation duration was 5 s, and the mean value of the friction coefficient during the stable period of 2.5 s was taken as the practical value. Three replicate simulations were performed for each group of materials.

Experimental Setup
Considering the potential frictional mechanical applications of carbon nanotube-reinforced PTFE polymer in oil-free rotating mechanical structures such as bearings, it is necessary to validate the proposed multiscale friction model on a ring-block friction testing machine. The above-established multi-scale friction model of copper ring-CNT/PTFE block was verified on a UMT friction tester manufactured by Bruker Company. The friction tester configuration is shown in Fig. 7, and the dimensional parameters of the copper ring and CNT/PTFE sample blocks are detailed in Table 1.
Due to the exploratory nature of this multiscale simulation, a mild operating condition was chosen for simulation  and experimental validation on the friction testing machine. The main motor speed was set at 30 rpm (0.055 m/s) considering the potential instability at low rotational speeds. Similarly, the external pressure was set to a mild value of 0.5 MPa. The purchased single-walled carbon nanotubes (SWC-NTs) have a diameter ranging from 3 to 5 nm and lengths ranging from 2 to 5 μm. The PTFE powder used has a particle size ranging from 20 to 200 μm. The physical blending of carbon nanotubes and PTFE powder, as well as the preparation of CNT/PTFE composite materials, were independently conducted by our research institution. Four specimens with CNT mass fractions of 0%, 1.25%, 2.5%, and 5% were sequentially tested for friction properties. Each sample was tested for 1 min, and the friction coefficient during the sound stage of the 30 s was taken as the practical value. Friction experiments were conducted with friction coefficients recorded every 1 s, and each set of specimens underwent three repetitions to ensure reproducibility.

Mechanical Performance Simulation Results
Firstly, the tensile behavior of the pure PTFE matrix was analyzed. Figure 8 shows the stress-strain relationship of the PTFE tensile simulation for a constant strain step of 0.025. It can be seen from the figure that the stress-strain exhibits an excellent linear elastic relationship when the axial strain is 0-0.5. The linear fitting equation of pure PTFE in Fig. 7 gives a simulated value of Young's modulus of 1.51 GPa at 300 K room temperature, consistent with the experimental range value (1.5-1.6 GPa) reported in the literature [62]. These results confirm the effectiveness of the system optimization and simulation methods in this molecular simulation study. Based on the relationship between the axial and transverse deformation of the PTFE box during the tensile process, Poisson's ratio is calculated to be 0.37.
To facilitate comparison, the stress-strain relationships for both axial and transverse tensile simulations of the CNT/ PTFE RVE were presented in Fig. 8. The results showed that the addition of carbon nanotubes significantly enhanced the elastic properties of the CNT/PTFE composite material. The axial and transverse tensile moduli of the CNT/PTFE RVE were increased by 368% and 225%, respectively, compared to pure PTFE. This enhancement can be attributed to the solid non-bonding interactions between PTFE molecules and CNTs. Furthermore, it can be observed from Fig. 8 that the elastic properties of the CNT/PTFE RVE are closely related to the direction of carbon nanotubes, with the axial tensile modulus being more significantly improved than the transverse tensile modulus. Based on the relationship between the axial and transverse deformations of the CNT/PTFE box during stretching in two directions, Poisson's ratio can be obtained ( 12 = 0.38 , 23 = 0.39).
Similarly, the results of the axial and transverse shear simulations of CNT/PTFE RVE and the shear modulus of pure PTFE are compared in Fig. 9 for easy comparison. The transverse shear modulus of CNT/PTFE RVE and the shear modulus of PTFE are calculated according to Eq. (6) and Eq. (4). As with the tensile modulus simulation, the comparison results in Fig. 9 show that the axial shear modulus of CNT/PTFE RVE is higher than the transverse shear modulus. This is due to the unique crystal structure of the CNTs and the direction of their physical binding with PTFE. The

Shear tress(GPa)
Strain results of MD simulations indicate that the addition of CNTs significantly enhances the strength of PTFE, and the degree of enhancement is related to the orientation of CNTs.

Friction Performance Simulation Results
As described in the Methods section, the results of MD friction simulations will be used to determine the friction coefficient function of each contact element at the contact interface in Eq. (13). Therefore, the determination of the load pressure in MD simulations is dependent on the pressure distribution at the Cu-CNT/PTFE contact interface. In this study, multi-scale simulations of the material mechanical properties were performed before the MD friction simulations. Thus, the pressure distribution on the contact interface (ranging from 0.0599 to 0.7635 MPa) has already been obtained. The simulation of the pressure distribution will be described in detail in Sect. 3.3. Figure 10 shows the variation of friction coefficients for Cu-PTFE and Cu-CNT/PTFE RVE friction pairs at a sliding velocity of 0.055 m/s and a load pressure of 0.5 MPa.
As shown in Fig. 10, under dry friction conditions, the practical mean value of the Cu-PTFE friction coefficient is 0.198, which is the most considerable value among the three sets of simulation results. The addition of CNTs effectively improves the friction performance of PTFE. In the frictional process between pure PTFE and Cu counterpart, the PTFE molecular chains tend to adsorb onto the surface of Cu. However, with the incorporation of carbon nanotubes (CNTs), the strong interaction between CNTs and PTFE molecular chains reduces the adsorption of PTFE chains on the Cu surface. As a result, the adhesive interaction at the contact interface decreases, leading to a reduction in friction. In addition, it can be found that the friction coefficients of CNT/PTFE in the X and Y directions decreased by 44.4% (0.110) and 40.4% (0.118), respectively, compared to pure PTFE. This is because the structural feature of the CNT with a large aspect ratio makes the number of carbon atoms in the X-direction (along the length of the CNT) significantly more than that in the Y-direction. Therefore, the carbon nanotubes have a stronger adsorption force on the PTFE molecular chains in the X-direction, which makes the PTFE molecular chains adsorbed on the upper copper atomic layer decrease. At the same time, the friction coefficient decreases with the decrease of adsorption at the contact interface.
Based on t he pressure distr ibution range (0.0599-0.7635 MPa) of each element on the contact interface of the FE model, several sets of friction simulations with sliding velocity 0.055 m/s and different pressure loads were designed, and the friction coefficient results are statistically presented in Fig. 11.

Mori-Tanaka Homogenization Results
Based on the mechanical properties obtained from MD simulations of the PTFE matrix and the CNT/PTFE RVE, the Mori-Tanaka model was used to homogenize the elastic properties of the CNT/PTFE composite materials. Table 3 summarizes the elastic parameters of three homogeneous CNT/PTFE composites with carbon nanotube mass fractions of 1.25%, 2.5%, and 5% after homogenization. Similar to the results of MD simulations, the mesoscale mechanical results after homogenization also show that the filling of carbon nanotubes improves the elastic properties of the PTFE matrix significantly. Specifically, the elastic modulus was enhanced by 322% with 5% mass fraction of CNTs. Meanwhile, the tensile modulus of the CNT/PTFE composite material also increases with the increase in CNT mass fraction.

Finite Element Simulation Results
Prior to conducting finite element friction simulations, normal loading is first applied. The homogenized material parameters of the CNT/PTFE block using the Mori-Tanaka model are used as the material parameters. Figure 12 illustrates the pressure distribution at the contact interface of four CNT/PTFE materials with CNT mass fractions of 0%, 1.25%, 2.5%, and 5% after applying a mean contact pressure of 0.5 MPa.
After the loading and boundary conditions have been set, the finite element friction simulation can be performed. The friction coefficients for each element on the contact interface, as mentioned in Eq. (13), have been simulated in the MD friction section (see Fig. 11). If the pressure values of the cells are not among the nine pressure values shown in Fig. 10, the nearest of these values is considered. When considering the friction pair of each contact unit, the corresponding coefficient V M or V R should be chosen as the probability coefficient for Cu-PTFE or Cu-CNT/PTFE RVE, respectively. After the friction coefficients of each contact element are calculated, the overall friction coefficient can be obtained using Eq. (14). Figure 13 shows the dry friction characteristics of the four materials under a mean contact pressure of 0.5 MPa and a constant speed of 30 rpm(0.055 m/s).
It is evident from Fig. 13 that the filling of CNTs leads to optimizing the frictional properties of PTFE at a steady state. As the mass fraction of CNTs increases, the friction coefficient continuously decreases. Most prominently, a 17.8% reduction in dry friction coefficient can be achieved with a CNT mass fraction of 5%. This is because with an increase in the mass fraction of CNTs, the adsorption of CNTs onto PTFE

Comparison of Experimental Results and Simulations
The corresponding copper ring-CNT/PTFE block friction experiments were performed on a Bruker UMT friction tester with the same friction condition settings as the finite element friction simulation. The experimental results were compared with the results of the multi-scale friction simulation in Fig. 14. It can be seen that the trend of the observed friction coefficient changes with the increase in CNT mass fraction is consistent with that of the multi-scale model, validating the effectiveness of the multi-scale friction model. Although the trend of the friction coefficient obtained from the simulation was validated, there still exist specific errors in the friction coefficient values (the maximum error is 5.26%). An analysis of the sources of error revealed several potential factors: Firstly, due to the limitation of computer hardware performance, the length size of carbon nanotubes (nano-scale) in molecular dynamics simulation is difficult to reach the experimental carbon nanotube size (micron scale). To minimize the simulation error caused by the inconsistency between the simulated and experimental carbon nanotube dimensions, we simulated the mechanical and frictional properties of CNT/PTF in both axial and radial directions by considering the structural characteristics of actual carbon nanotubes with large aspect ratios in the molecular simulation process. In addition, we try to keep the mass fraction of the simulated and actual carbon nanotubes as consistent as possible to minimize the error caused by the inconsistent size of carbon nanotubes. Secondly, the fabrication process of CNT/PTFE composites cannot guarantee uniform dispersion of carbon nanotubes within the matrix. Additionally, experimental variables such as temperature control can also contribute to discrepancies between experimental and simulation results.

Conclusion
This letter introduces a new multi-scale polymer friction model developed to couple the mechanical and frictional properties at three spatial scales (from the nanoscale to mesoscale to macroscale). The multi-scale model is applied to investigate the dry frictional behavior of Cu-CNT/PTFE pairs. By combining the advantages of simulations at different scales, this model considers the effects of CNT orientation and mass fraction on the mechanical and frictional performance of PTFE. Friction experiments between a copper ring and a CNT/PTFE block validate the conclusions of the multi-scale friction simulation. This multi-scale simulation study also yields some valuable insights: The MD simulations have shown that adding carbon nanotubes (CNTs) can significantly enhance the elastic modulus of PTFE. Due to the directional nature of CNTs, the axial distribution of the CNT/PTFE composite exhibits better improvement in elastic properties than the transverse distribution. The friction results from MD show that the filling of carbon nanotubes also effectively reduces the friction coefficient of Cu-PTFE dry friction. The carbon nanotubes' orientation also affects the friction properties' improvement. The Mori-Tanaka homogenization method can effectively combine the elastic properties of the nanoscale PTFE matrix and CNT/PTFE RVE to obtain macroscopically available elastic properties parameters of CNT/PTFE composites. This macroscopic finite element copper ring-CNT/PTFE block friction model utilizes the friction results from MD simulations and the fine-scale composite mechanical properties. The frictional and mechanical properties of carbon nanotube (CNT)-reinforced composite materials are closely interrelated. Therefore, when investigating the frictional performance of these materials, it is essential to conduct a synergistic analysis with their mechanical properties. The multiscale friction results show a decreasing friction coefficient (from 0.198 to 0.156) with increasing carbon nanotube mass fraction (0, 1.25%, 2.5%, 5%). This trend remains consistent with the experimental results.
Although experimental results have validated the trend of the multi-scale friction model presented in this study, errors in the simulated friction coefficients still exist. Further analysis is required to identify the sources of these errors and optimize the multi-scale model. Nevertheless, establishing the multi-scale friction model provides a novel approach to investigating the frictional properties of CNT-reinforced polymers.