# Effect of Aggregate Crystalline Surface Anisotropy on
## 3 Results and Discussion
### Diffusion of Asphalt Compositions on Aggregate Surface  
The diffusion characteristics of asphalt molecules on the aggregate surface significantly influence the properties of the asphalt-aggregate interface. Over time, diffusion occurs within the asphalt-aggregate system. In general, the diffusion behavior represents the mutual penetration process between two compatible systems. When the asphalt is in contact with the aggregate, it can be expressed as the invasion of the asphalt into the aggregate surface. This diffusion property reflects the level of compatibility and is closely related to the adhesion properties. The diffusion coefficient, calculated by mean square displacement (MSD), is commonly used to represent the diffusion performance. In this study, the aggregate was fixed during the simulation process to better replicate the actual conditions. Consequently, the diffusion behavior of asphalt on the aggregate surface was interpreted as the free movement of the asphalt away from the aggregate surface. Faster movement away from the surface indicates more unstable interfacial properties and lower adhesion performance. The diffusion coefficient of the asphalt-aggregate system was calculated using MD by simulating and analyzing the MSD curve of the asphalt system, so as to study the microscopic interface dynamics of asphalt-aggregate diffusion behavior. In MD calculations, atoms are constantly moving from their initial positions, resulting in different positions at each instant. The \(MSD\) is recorded by summing the squares of the differences between the moving positions \(r(t)\) and the initial position \(r(0)\) of all particles in the system after a time \(t\). The \(MSD\) is used to determine whether the system is in equilibrium. Its value is increased with the simulation time and is stabilized once the system has reached equilibrium. The calculation formula for the \(MSD\) is shown in Equation (2):  
\[MSD(t)=\frac{1}{N}\underset{i=1}{\overset{N}{\underset{i=1}{\sum}}}[ri(t)-ri(0 )]^{2} \tag{2}\]  
where \(t\) is the movement time; \(r_{i}(t)\) is the position of particle \(i\) at time \(t\); \(r_{i}(0)\) is the position of particle \(i\) at time \(0\); and \(N\) is the number of molecules in the system. According to the [PERSON] diffusion model, the molecular diffusion coefficient can be calculated as shown in Equation (3):  
\[D=\frac{1}{6}\underset{i\rightarrow\infty}{\lim}\frac{d}{dt}\underset{i=1}{ \overset{N}{\underset{i=1}{\sum}}}[ri(t)-ri(0)]^{2} \tag{3}\]  
The optimized equilibrium model was used to analyze the \(MSD\) of each asphalt component on the aggregate surface using the _Forcite_ module. The \(MSD\) was observed to grow rapidly after 150 ps. To better observe the \(MSD\) curves, only the data from 0 to 150 ps were plotted. The resulting \(MSD\) curves of asphalt and its components on the surfaces of two aggregates with different crystalline surfaces with respect to the simulation time are shown in Figure 9.  
As shown in Figure 9, the MSD curves of all components gradually increased with the simulation time. However, the growth rate varied among different components and crystalline surfaces. Since the number of particles (N) was kept constant in the NPT system, the slope of the MSD curve with respect to time (\(t\)) was directly proportional to the diffusion coefficient. A larger slope indicated a higher diffusion coefficient and stronger molecular diffusion ability. The diffusion coefficient was calculated from the MSD curves to provide a better understanding of the diffusion properties of the molecules. Data from 40 ps to 150 ps of each MSD curve were selected for linear regression to obtain the slope, and the diffusion coefficient was calculated as shown in Figure 10.  
The average diffusion coefficient of bitumen on the three surfaces of calcite minerals is 30.12% higher than on the three surfaces of quartz. As shown in Figure 10, the diffusion coefficient of asphalt in the asphalt-SiO\({}_{2}\) system was significantly larger than that in the asphalt-CaCO\({}_{3}\) system.