Orientation and Dispersion Evolution of Carbon Nanotubes in Ultra High Molecular Weight Polyethylene Composites under Extensional-Shear Coupled Flow: A Dissipative Particle Dynamics Study

The property of carbon nanotubes (CNTs)-based composites are significantly dependent on the orientation and dispersion evolution of CNTs in the polymer matrix. In this work, the dissipative particle dynamics (DPD) simulations are employed to discover the orientation and dispersion evolution of CNTs in ultra–high molecular weight polyethylene (UHMWPE) under extensional–shear coupled flow conditions for the first time. In this paper, we investigate the roles of the increasing extensional-shear coupled rate in morphology of CNTs/UHMWPE composites by varying CNTs concentration and observe that the system under consideration lies in the same evolution morphologies. When comparing our results for various morphologies, we notice that the orientation is affected more significantly by changing the extensional-shear coupled rates. A good alignment appears with an increase of extensional-shear coupled rates, which transform it into ordered morphology. In addition, a higher extensional-shear coupled rate does not necessarily contribute to better dispersion even though CNTs concentration varies, as shown by the mean square displacement (MSD) and the relative concentration distribution functions of CNTs in CNTs/UHMWPE composites.


Introduction
Ultra-high molecular weight polyethylene (UHMWPE), with molecular weight numbering in the millions, is composed of extremely long polymeric chains of ethylene monomers, which exhibit superior toughness, high abrasion resistance, a low friction coefficient, low moisture uptake, and excellent chemical stability. This demonstrates a variety of technological applications ranging from bearing components and super tough engineering plastics to medical materials in total joint replacement. In spite of its excellent properties, the usage of UHMWPE in some demanding applications has been limited due to various constraints such as its low load bearing capacity, thermal Every UHMWPE chain consists of 13,914 beads. The box contains 24,000 beads and the number density is kept as 3. The SWCNTs particle is stationary and treated as a rigid body. To incorporate rigidity in the bending of the nanotube, the bond length is taken as 0.75 DPD unit and the bond constant as 20 k B T for CNTs. The angle and angle constant for CNTs are taken as 180 • and 40 k B T, respectively. The volume of the CNTs particle is about 480Å 3 . The solubility parameter of UHMWPE and SWCNTs is 16.50 and 18.69 (J/cm 3 ) 0.5 , which is obtained from References [27,28], respectively. All the simulations were carried out for 1 × 10 5 steps with a time step of 1 fs in reduced DPD units. The coarse-grained model of single-walled carbon nanotubes (SWCNTs) (6,6) with a diameter of 8.14 Å and length of 103 Å is displayed in Figure 1, where 24 carbons are grouped as one particle. Every UHMWPE chain consists of 13,914 beads. The box contains 24,000 beads and the number density is kept as 3. The SWCNTs particle is stationary and treated as a rigid body. To incorporate rigidity in the bending of the nanotube, the bond length is taken as 0.75 DPD unit and the bond constant as 20 kBT for CNTs. The angle and angle constant for CNTs are taken as 180° and 40 kBT, respectively. The volume of the CNTs particle is about 480Å 3 . The solubility parameter of UHMWPE and SWCNTs is 16.50 and 18.69 (J/cm 3 ) 0.5 , which is obtained from Reference [27]and Reference [28], respectively. All the simulations were carried out for 1 × 10 5 steps with a time step of 1 fs in reduced DPD units. Figure.1 The coarse-grained model of SWCNTs (6,6).
For a combination of shear flow and extensional flow, the velocity gradient is given by Equation (1) [29] below.
where γ̇ is the shear rate and ε̇ is the extensional rate.
As proposed in our previous study [23], in DPD simulation, pressure can be imposed on each bead to drive the flow because the normal pressure is in proportion to the force and simple extensional flow is generated using the Souza-Martins method. Consequently, in this study, referring to the velocity gradient u , the extensional-shear coupled flow can be described as Equation (2) in which the equivalent hydrostatic pressure is zero. The quantitative relationship between pressure P (GPa) and the shear rate γ̇ or extensional rate ε̇ is still unkown.

Results and Discussion
In this simulation, the Souza-Martins method is performed for driving the extensional-shear coupled flow field, by combining the simple extensional flow field and the simple shear flow field. For a combination of shear flow and extensional flow, the velocity gradient ∇u is given by Equation (1) [29] below.
where . γ is the shear rate and . ε is the extensional rate. As proposed in our previous study [23], in DPD simulation, pressure can be imposed on each bead to drive the flow because the normal pressure is in proportion to the force and simple extensional flow is generated using the Souza-Martins method. Consequently, in this study, referring to the velocity gradient ∇u, the extensional-shear coupled flow can be described as Equation (2) in which the equivalent hydrostatic pressure is zero. The quantitative relationship between pressure P (GPa) and the shear rate . γ or extensional rate . ε is still unkown.
P xx P xy P xz P yx P yy P yz P zx P zy P zz

Results and Discussion
In this simulation, the Souza-Martins method is performed for driving the extensional-shear not displayed in this study and beads of CNTs are displayed in CPK style with CPK scale of 2.5 (CPK style indicates that atoms are rendered as spheres with radii that are related to the radii of the beads). These results confirm that the final states of the orientation is highly dependent on the extensional-shear coupled rate. When at a low coupled rate (i.e., 0.1), CNTs tend to form disordered morphology. By increasing the coupled rate, CNTs tend to align along the elongated direction. In the case of a high coupled rate (1.0), the ordering alignment is noticeable. However, an increment of extensional-shear coupled rate yields less effect on the dispersion of CNTs in UHMWPE. This means that the higher extensional-shear coupled rate does not necessarily contribute to much better dispersion. To describe the orientation and dispersion evolution of CNTs (10%) in UHMWPE under extensionalshear coupled flow, morphological structure evolution under a low, moderate, and high extensionalshear coupled rates (0.1, 0.5, 1.0) are calculated, where the extensional-shear coupled rate is composed of shear rate γ̇ and extensional rate ε̇, ε̇=γ̇ in the current paper. The typical simulation results are illustrated in Figure 2. To observe the CNTs morphology clearly, beads of PE are not displayed in this study and beads of CNTs are displayed in CPK style with CPK scale of 2.5 (CPK style indicates that atoms are rendered as spheres with radii that are related to the radii of the beads). These results confirm that the final states of the orientation is highly dependent on the extensional-shear coupled rate. When at a low coupled rate (i.e., 0.1), CNTs tend to form disordered morphology. By increasing the coupled rate, CNTs tend to align along the elongated direction. In the case of a high coupled rate (1.0), the ordering alignment is noticeable. However, an increment of extensional-shear coupled rate yields less effect on the dispersion of CNTs in UHMWPE. This means that the higher extensionalshear coupled rate does not necessarily contribute to much better dispersion. Compliant to proven flow facts on orientation and dispersion of CNTs in UHMWPE, mass fraction of CNTs in UHMWPE is taken into consideration. Figure 3 displays the morphologic structures of system of CNTs/UHMWPE composites at 3% and 7% CNTs under a low, moderate and high extensional-shear coupled rates (0.1, 0.5 and 1.0). A common feature between different flow is that the alignment and dispersion of CNTs in UHMWPE behaves rather similarly under different rates since CNTs concentration varies. The orientation arrangement of CNTs is improved with increasing extensional-shear coupled rates from 0.1 to 1.0, which suggests that the increase of coupled rates promotes the CNTs to transform into ordered morphology. Furthermore, the quality of the dispersion is not significantly improved by imposing a higher extensional-shear coupled rate, even though CNTs concentration varies because CNTs dispersed evenly in the UHMWPE matrix within the considered CNTs content, in accordance with experimental results by Yin et al. Compliant to proven flow facts on orientation and dispersion of CNTs in UHMWPE, mass fraction of CNTs in UHMWPE is taken into consideration. Figure 3 displays the morphologic structures of system of CNTs/UHMWPE composites at 3% and 7% CNTs under a low, moderate and high extensional-shear coupled rates (0.1, 0.5, and 1.0). A common feature between different flow is that the alignment and dispersion of CNTs in UHMWPE behaves rather similarly under different rates since CNTs concentration varies. The orientation arrangement of CNTs is improved with increasing extensional-shear coupled rates from 0.1 to 1.0, which suggests that the increase of coupled rates promotes the CNTs to transform into ordered morphology. Furthermore, the quality of the dispersion is not significantly improved by imposing a higher extensional-shear coupled rate, even though CNTs concentration varies because CNTs dispersed evenly in the UHMWPE matrix within the considered CNTs content, in accordance with experimental results by Yin et al. In Figure 4, the dynamic process of CNTs alignment in CNTs/UHMWPE composites at 10% CNTs under extensional-shear coupled flow field, obtained with a coupled rate of 1.0, is presented. The simulation snapshots are taken every 2 × 10 4 timesteps. Frame 1 is the initial structure and some CNTs clusters can be found. Frame 21 shows that the CNTs begin to align along the flow direction and are not disordered absolutely. After 1 × 10 5 timesteps, the CNTs are highly oriented and the initial disordered morphology disappears, instead of the orientation arrangement. The findings confirm that the extensional-shear coupled flow has a significant influence on the CNTs orientation. Since extensional-shear coupled flow is imposed, the system is elongated and then the CNTs are far apart and gradually align along the flow direction. Furthermore, the orientation arrangement still exists In Figure 4, the dynamic process of CNTs alignment in CNTs/UHMWPE composites at 10% CNTs under extensional-shear coupled flow field, obtained with a coupled rate of 1.0, is presented. The simulation snapshots are taken every 2 × 10 4 timesteps. Frame 1 is the initial structure and some CNTs clusters can be found. Frame 21 shows that the CNTs begin to align along the flow direction and are not disordered absolutely. After 1 × 10 5 timesteps, the CNTs are highly oriented and the initial disordered morphology disappears, instead of the orientation arrangement. The findings confirm that the extensional-shear coupled flow has a significant influence on the CNTs orientation. Since extensional-shear coupled flow is imposed, the system is elongated and then the CNTs are far apart and gradually align along the flow direction. Furthermore, the orientation arrangement still exists over another 2 × 10 4 time steps (see Frame 81 and 101). Therefore, the orientation arrangement under extensional-shear coupled flow field is stable, based on Souza-Martins method. In addition to the morphologic structures and dynamic process of CNTs in CNTs/UHMWPE composites, the mean square displacement (MSD) and the relative concentration distribution functions of CNTs in CNTs/UHMWPE composites are evaluated to further understand the structural and dynamical properties under extensional-shear coupled flow field. Figure 5 illustrates the MSD plots of CNTs in response to different extensional-shear coupled rates and CNTs concentrations. For a given coupled rates range, it shows that the MSD of CNTs in a 10% CNTs system under coupled rate of 1.0 is much higher than that under a coupled rate of 0.1 and 0.5. This can be attributed to the CNTs alignment as extensional-shear coupled rates increase. It is also found that the change of MSD with time depends on the CNTs concentration in a nonlinear fashion, for a given time. An increase of the CNTs concentration restricts the CNTs dynamics and, therefore, reduces its diffusion. In addition to the morphologic structures and dynamic process of CNTs in CNTs/UHMWPE composites, the mean square displacement (MSD) and the relative concentration distribution functions of CNTs in CNTs/UHMWPE composites are evaluated to further understand the structural and dynamical properties under extensional-shear coupled flow field. Figure 5 illustrates the MSD plots of CNTs in response to different extensional-shear coupled rates and CNTs concentrations. For a given coupled rates range, it shows that the MSD of CNTs in a 10% CNTs system under coupled rate of 1.0 is much higher than that under a coupled rate of 0.1 and 0.5. This can be attributed to the CNTs alignment as extensional-shear coupled rates increase. It is also found that the change of MSD with time depends on the CNTs concentration in a nonlinear fashion, for a given time. An increase of the CNTs concentration restricts the CNTs dynamics and, therefore, reduces its diffusion. Figure 6 demonstrates the relative concentration distribution functions of CNTs in CNTs/UHMWPE composites corresponding to the morphologies in Figures 2 and 3. The relative concentration is given by the ratio of concentration of a type of bead in the slab to its average concentration across the entire system. By comparing the number and magnitude of the peak values existing in the relative concentration distribution functions, the dispersion state can be indirectly evaluated. The fewer peaks and the smaller the peaks are, more uniform the dispersion presents in CNTs/UHMWPE composites. As seen in Figure 6a, it is clear that the relative concentration evolution is similar even though extensional-shear coupled rate varies, which suggests that the higher extensional-shear coupled rate does not necessarily contribute to much better dispersion, consistent with the above results from morphologies in Figures 2 and 3. Judging from Figure 6b, it can be concluded that the dispersion of CNTs in UHMWPE is slightly impacted by CNTs concentration because they are uniformly dispersed for the systems below 10% CNTs.   Figures 2 and 3. The relative concentration is given by the ratio of concentration of a type of bead in the slab to its average concentration across the entire system. By comparing the number and magnitude of the peak values existing in the relative concentration distribution functions, the dispersion state can be indirectly evaluated. The fewer peaks and the smaller the peaks are, more uniform the dispersion presents in CNTs/UHMWPE composites. As seen in Figure 6a, it is clear that the relative concentration evolution is similar even though extensional-shear coupled rate varies, which suggests that the higher extensional-shear coupled rate does not necessarily contribute to much better dispersion, consistent with the above results from morphologies in Figures 2 and 3. Judging from Figure 6b, it can be concluded that the dispersion of CNTs in UHMWPE is slightly impacted by CNTs concentration because they are uniformly dispersed for the systems below 10% CNTs.

Conclusions
In this contribution, we focus on DPD simulation studies performed to provide a molecularlevel understanding of flow-induced orientation and dispersion evolutions of CNTs in UHMWPE, subjected to extensional-shear coupled flow, for the first time. Through the morphological transition at different extensional-shear coupled rate levels, it is found that the final states of the orientation are

Conclusions
In this contribution, we focus on DPD simulation studies performed to provide a molecular-level understanding of flow-induced orientation and dispersion evolutions of CNTs in UHMWPE, subjected to extensional-shear coupled flow, for the first time. Through the morphological transition at different extensional-shear coupled rate levels, it is found that the final states of the orientation are highly dependent on the extensional-shear coupled rate and the CNTs all undergo a shift from disordered morphology to well-ordered alignment by increasing the extensional-shear coupled rate. Additionally, the dispersion of CNTs in UHMWPE is not significantly enhanced by imposing a higher extensional-shear coupled rate and is slightly impacted by the CNTs concentration. These findings agree well with those obtained from the MSD and the relative concentration distribution functions of CNTs in CNTs/UHMWPE composites. The results of this work lead to a better understanding regarding the orientation and dispersion induced by the external flow field.