Before performing the DPD simulation, the repulsive interaction parameters should be obtained first and are listed in Tables , , and for 10/10/1, 6/14/1, and 2/18/1 volume fractions, respectively. In the DPD simulation, all the repulsive interaction parameters between the same materials are 38.403. When the repulsive interaction parameter between different materials is larger than that between the same material, it means that these two materials have stronger repulsive interaction.
The repulsive interaction parameter at 10/10/1 volume fraction
The repulsive interaction parameter at 6/14/1 volume fraction
The repulsive interaction parameter at 2/18/1 volume fraction
From Table , , and , we can observe that the repulsive interaction parameter between PE polymers and CNTs decreases with an increase in PE polymer volume fraction. It indicates that the CNTs are easily dispersed into the polymer matrix at a lower CNT fraction. At a lower polymer fraction (a higher CNT fraction), the much higher repulsive parameters between PE and CNT beads lead to the aggregation of CNTs surrounded by the polymer matrix. The characteristic of repulsive parameters at different fractions corresponds to the related experimental observation. Chen et al. demonstrated that CNTs with smaller weight fraction in the polymer matrix will be easily dispersed [11
]. In addition, we found that the repulsive interaction parameter between PLLA and PE polymers increases from 6/14/1 to 2/18/1 volume fractions. The reason for this is that the calculation of cohesive energy density includes the weight function for a pure component, which is shown in Equation 2.
After the DPD simulation was performed, all equilibrated structures were obtained at different volume fractions with blend and di-block copolymer methods, which can be seen in Table . All equilibrated structures of different volume fractions with these two methods are shown in Figure . The red, green, and blue beads represent the PLLA, PE polymers, and CNTs, respectively. Figure shows the lamellae structures, which are found in the 10/10/1 volume fraction in the blend method. In many DPD studies, most of the equilibrated structure is lamellae structure. However, for the corresponding di-block copolymer system, the polymer beads will form the perforated lamellae structure in the polymer/CNTs bead matrix, as shown in Figure . In Figure , the PE polymers and CNTs aggregated and formed one layer, and PLLA polymers formed another layer by themselves because of the relationship of repulsive interaction parameters. The CNTs did not aggregate and form the cylindrical shape because of the similar repulsive interaction parameter between the CNT and PE polymers. In addition, the value of that between PE polymer and CNT is obviously smaller than both that between PLLA polymers and CNTs and between PLLA and PE polymers at 10/10/1 volume fraction. This means that the PLLA polymer has a very strong repulsive interaction to PE and CNTs. Therefore, PLLA polymers form one layer by themselves, excluding other materials. Because CNTs with similar repulsive interaction parameters were not forced to connect to PE or PLLA polymers, CNTs also disperse inside the PE polymer matrix. From Figure , we found that the layer in Figure is thinner than that in Figure . In the di-block copolymer method, one PE polymer chain was forced to connect to a PLLA polymer chain, and the movement of these two polymers is restrained in the polymer/CNTs matrix. For example, the PE polymer only can adsorb on the PE side of other di-block copolymer chain and arrange parallel to form the perforated lamellae structure. However, in the blend method, every material can aggregate together easily because they do not have any movement limitations. Therefore, the thickness of the layer in the blend method was larger than that of the di-block copolymer method.
The equilibrated structure at three volume fractions with blend and di-block copolymer methods
The equilibrated structure at (a-b) 10/10/1, (c-d) 6/14/1, and (e-f) 2/18/1 fractions.
Figure shows the equilibrated structures, which are perforated lamellae and tube-like structures at 6/14/1 volume for blend and di-block copolymer methods, respectively. Figure shows the CNT structure which forms three cylindrical structures. Compared to Figure , the CNTs do not disperse at this volume fraction. The reason for this is that the repulsive interaction parameter between CNTs and PE polymers is larger than that between the same materials. As can be seen from Table , the repulsive interaction parameter between PLLA polymer and CNTs is the largest, and that between PLLA and PE polymer is just smaller than that between PLLA and CNTs. Therefore, there are two possible structural types for the CNTs in the polymer/CNT matrix. First, they form the cylindrical structure and are covered by PLLA polymers. Second, they are surrounded by PE polymers, and these PE polymers are surrounded by PLLA polymers. Figures and show the two structural types in the polymer/CNTs matrix. In Figure , almost all of the CNTs are surrounded by PE polymers. This is due to the restrained movement and the relationship of repulsive interaction parameters. It is impossible for CNTs to exist in the middle of PE and PLLA polymers because of the connection between PE and PLLA polymers. In addition, the repulsive interaction parameter between PE and CNT is significantly smaller than that between PLLA and CNT. Therefore, CNTs can only be inside the PE polymers which are covered by the PLLA polymers.
The equilibrated structure at 6/14/1 volume fraction with blend method.
Figure illustrates the equilibrated structures at 2/18/1 volume fraction with blend and di-block copolymer methods. In the blend method, the PE polymers aggregate themselves to form the cluster because of the unrestrained structure and the lower volume fraction. Similarly, the CNTs form cylindrical structures were similar to the 6/14/1 volume fraction. In addition, there are the fewest PE polymers at the 2/18/1 volume fraction such that PE polymers do not cover all CNTs. Figure shows an equilibrated structure similar to that in Figure . The reason for forming the same equilibrated structure is almost the same. Because the number of PE polymers is the lowest, they cannot cover all of the CNTs. Hence, some CNTs are in contact with the PLLA polymers. In addition, the CNTs form more cylindrical structures and the PE polymer of the di-block copolymer can easily cover the CNTs.
In order to analyze the relationship between the micro-structures of PE and PLLA polymers and equilibrated structures, the square radius of gyration Rg2
is examined to provide information on the mass distribution of the chain in the system, which also plays a central role in interpreting light scattering and viscosity measurements. If all beads have the same mass:
denotes the coordinate of the particle, rc
denotes the coordinate of center of mass of the polymer chain, and n
is the bead number in a chain. Additionally, it can be represented as the tensor in different directions as follows:
where rix and riy denote the position vector of the particle i, whereas rcx and rcy denote the position vector of the center of mass of polymer chain. The three eigenvalues of G are denoted by Rg12 (major axial, which is the largest eigenvalue) Rg22, and Rg32, which can be used to determine roughly the structural arrangement of a chain in the system. If the values of Rg22 and Rg32 are almost the same, it means that the micro-structure of this material is spherical structure. The summation of Rg12, Rg22, and Rg32 is Rg2 which can be used to determine roughly the structural arrangement of a chain in the system. The larger Rg2 means that the structure is extended, whereas the lower dimension phase has the more collapsed structure in the polymer chain. When the Rg12 is larger than Rg22 and Rg32, the micro-structure of material is ellipsoid structure. All of the values of PE and PLLA polymers with two methods at different volume fractions are listed in Table . All micro-structures of PE and PLLA polymers are spherical at three volume fractions because all polymers are not restricted, and it is easy for polymers of the same material to aggregate by themselves because they have the same repulsive interaction parameter. Hence, they have the similar micro-structure at three volume fractions.
The radius of gyration at three volume fractions with blend and di-block copolymer methods
From the value of Table , all micro-structures of entirely di-block copolymers are ellipsoid in structure. The ellipsoid structures elongate with the increase of PE volume fraction. We use the ratio of Rg12/Rg22 and Rg12/Rg32 to compare their micro-structures. When the equilibrated structure is a perforated lamellae structure, the ratios are 0.38 and 0.32 at 10/10/1 volume fraction. When the equilibrated structure is tube-like, the ratios are about 0.8 at 2/18/1 and 6/14/1 volume fractions. This can be attributed to the different equilibrated structure at three volume fractions. The equilibrated structure at 10/10/1 volume fraction is perforated lamellae. The micro-structure of the entirely di-block copolymer is the longest and thinnest. In particular, this shape can arrange parallel to form the perforated lamellae structure. When the volume fraction of PE polymer increases, the equilibrated structure changes from perforated lamellae to the tube-like structure. If the micro-structure is thin and elongated, it is difficult for PLLA polymers to fill the spaces which are not occupied by the PE polymer and CNTs. Hence, it is easy for the shorter and wider ellipsoid structure to form the tube-like structure. Unlike the blend system, where no relationship exists between the micro-structure and the equilibrated structure, in the di-block copolymer system, the micro-structure and equilibrated structure have specific relationships such as the long and thin ellipsoid forming the perforated lamellae. It is difficult in the blend system to find the relationship between micro-structures and equilibrated structures due to the unrestricted conditions between PE and PLLA polymers.