## 5 Rheological Properties of the Control and xGNP Modified Asphalt Models
### MD Viscosity Aimulation Methods and Results  
In the MD experimental simulation, there are four common methods to calculate the dynamic shear viscosity in the MD systems [16]: (1) a non-equilibrium MD (NEMD): the unit cell is sheared by \"fix deform\" and the temperature is controlled; (2) a NEMD: the viscosity is computed through the velocity and pressure in the systems; (3) a reverse non-equilibrium MD (rNEMD): the momentum flux is transferred between different layers in the unit cell through the Muller-Plathe algorithm; (4) equilibrium MD (EMD): the Green-Kubo (GK) formula is used to compute the viscosity, and continuous momentum flows are applied in the unit cell.  
In this MD study, the Muller-Plathe method was used to calculate the viscosities of the control and xGNP modified asphalt systems. The unit cell of asphalt models was split into 20 layers. The viscosity calculation is shown in Equation (6). During the calculation of viscosity in the MD simulation, unit conversion is needed. The viscosity unit in the MD simulation is gram/mol/angstrom/femtosecond, but the unit in the laboratory test is kilogram/meter/second. Avogadro's constant is used to convert from microscopic to macroscopic states. In addition, the momentums transferred in the control and xGNP modified systems at the temperature of 443 K are shown in Figure 6a. The viscosities of the control and xGNP modified asphalt systems at a temperature of 423 K are shown in Figure 6b,c. The MD and laboratory viscosity results of the control and xGNP modified asphalt systems at different temperatures are shown in Section 5.4.  
\[j_{z}(p_{x})=-\eta\frac{\partial v_{x}}{\partial z}\text{ and }j(p_{x})=\frac{P_{x}}{2 IA} \tag{6}\]  
where \(\eta\) is the dynamic shear viscosity; \(\frac{\partial v_{x}}{\partial z}\) is the shear rate; \(j(p_{x})\) is the input momentum flux; \(P_{x}\) is the input momentum; \(t\) is the simulation time; \(A=L_{x}L_{y}\), \(L_{x}\) is the length of unit cell in the \(x\) direction; and \(L_{y}\) is the length of unit cell in the \(y\) direction.  
Figure 6: \(Cont.\)  
Figure 6a displays the momentum transferred in the control and xGNP modified asphalt binder systems at a temperature of 443 K, as well as some of the results from different temperatures for repeatability. The momentum transferred in the xGNP modified asphalt binder model is more than that of the control model. The temperatures in the control asphalt binder model fluctuate more than those of the xGNP modified asphalt binder model due to there being fewer molecules. The temperatures are also around 443 K, and do not have large variations. It indicates that more molecules in the MD system lead to less data variation and a stable structure of materials. It coincides with the conclusion that the vibrating range of MD data is within a \(1/\sqrt{N}\) variation [40] (\(N\) is the number of molecules in the system). Figure 6b,c demonstrates the viscosities of the control and xGNP modified asphalt binder systems at a temperature of 423 K, and it is also part of data analysis under different temperatures. The temperature fluctuation of the control model is larger than that of xGNP modified asphalt binder model due to fewer molecules compared to xGNP modified asphalt binder system. The temperatures  
Figure 6: Viscosity test and MD calculation for the control and xGNP modified asphalt systems. (**a**) Momentum transferred in the control and xGNP modified asphalt binder systems at the temperature of 443 K; (**b**) Viscosities of the control and xGNP modified asphalt binder systems at the temperature of 423 K; (**c**) Viscosities (from 0 to 500 cp) of the control and xGNP modified asphalt binder systems at the temperature of 423 K.  
are also centered at 423 K with a 20 K variation. It coincides with the temperature setting of simulations. Furthermore, the statistical analysis of viscosities of the control and xGNP modified asphalt models was performed to analyze the distributions of viscosity data in MD simulation.