# A Coarse-Grained Molecular Model for Simulating Self-Healing of Bitumen
## 3 Results and Discussion
### Relative Concentration and Crack Ratio  
The relative concentration represents the fraction of molecules in a given direction. For a 3D box with XYZ coordinates, the analysis of relative concentration is more like a slice of the box. The larger the relative concentration at a point in the \(Z\) direction, the more molecules in the XY slice at this point. The relative concentration of molecules will be close to 0 at the position where voids completely appear in the box. The relative concentration can be calculated as:  
\[RC=\frac{n_{s}/V_{s}}{n/V} \tag{6}\]  
Among them, \(RC\) is the relative concentration of molecules, \(n_{s}\) is the number of particles in a cross-section perpendicular to the coordinate axis in the box, \(V_{s}\) is the volume of the cross-section, \(n\) is the number of particles in the calculated box, and \(V\) is the volume of the calculated box. Figure 8: MSD of each component. Figure 9: Diffusion coefficient of each component before and after healing. Taking 333 K as an example, the relative concentration of molecules along the length of the box in the simulation process of the model is shown in Figure 10. Among them, 40 ps roughly corresponds to the moment the model achieves initial self-balance, 80 ps roughly corresponds to the moment the bitumen's overall shape changes, 110 ps roughly corresponds to the moment the model's healing rate starts to accelerate, and 300 ps roughly corresponds to the moment the model achieves healing. The change in relative concentration is consistent with the molecular trajectory. As the simulation progresses, the crack in the center of the model becomes narrower, and the interval with the value of 0 in the center of the relative concentration diagram becomes shorter. The box's length steadily reduces as a result of the NPT ensemble simulation, and the relative concentration broken line diagram gradually gets shorter. As the bitumen at both ends of the model gradually moves towards the central crack, the relative concentration of bitumen molecules at both ends gradually decreases. For the relative concentration along the length direction at a specific time, the maximum value of the abscissa represents the box length at that time, and the length of the interval with the value of 0 in the center represents the crack width at that time. Due to the NPT ensemble simulation, the box shrinks with the simulation, and the crack width is the result of the reduction in the box length, which cannot directly represent the degree of crack healing. Therefore, the parameter crack rate is defined in this paper to characterize the degree of crack healing. \[c=\frac{w}{l} \tag{7}\]  
Among them, \(c\) is the crack rate at a certain time, \(w\) is the crack width at that time, and \(l\) is the box length at that time. Taking 333 K as an example, the crack width, box length, and corresponding crack rate of the model before healing are shown in Figure 11.  
Figure 10: Relative concentration at each time during the self-healing process of the model. The model enters a state of self-balancing during the first 40 ps, during which time, the box length, crack width, and crack rate all increase. The box length essentially maintains a constant rate of reduction after 40 ps, which is in line with the features of NPT ensemble simulation. The crack width basically maintained a constant rate of decline within the 40-80 ps, accelerated from the 80 ps, slowed down at about 290 ps, and decreased to 0 at about 300 ps. The crack rate remained basically unchanged in the 40 ps to 80 ps, accelerated from the 80 ps, slowed down at about 290 ps, and decreased to 0 at about 300 ps. According to the molecular trajectory, 80 ps is the time when the model completes the internal molecular reorganization, and 290 ps is the time when the bitumen molecules at both ends of the model begin to contact.