## 6 Thermal Property of the Control and xGNP Modified Asphalt Models
### MD Simulation Methods and Results  
In the MD simulations, there are four methods to compute the thermal conductivity for MD systems: (1) NEMD: energy is applied to the hot region, and an equal amount of energy is subtracted from the cold area in the simulation cell. The heat flux is monitored between different temperature layers; (2) NEMD: Energy is added or subtracted in two regions, and the temperature difference of the intermediate region is monitored; (3) rNEMD: The kinetic energies of two atoms in different layers are swapped, and the temperature gradient is monitored; (4) EMD: the heat flux can be computed from the fluctuations of per-atom potential and kinetic energies, as well as the stress tensor. It is common in NEMD (non-equilibrium MD) for calculating the thermal conductivity of systems to impose the temperature gradient and the responded heat flux is measured. However, a reverse non-equilibrium MD (rNEMD) algorithm is used in the Muller-Plathe method [47]. The heat flux is applied in the system and the temperature gradient is measured. When the heat flux is imposed in the simulation cell, which is divided into N slabs (N is an even number, 20 was used in this study) with identical thickness. Energy transfer (Figure 10a) is produced from hot to cold slabs through the \(z\)-direction and it causes the temperature difference (Figure 10b) between these two slabs. Velocity exchange occurs in two particles, and the energy conservation is satisfied. The formulas for thermal conductivity and heat flux are shown in Equation (9). In addition, the Avogadro constant was used to complete the  
Figure 9: Thermal conductivity test apparatus and results. (**a**) KD2 Pro thermal conductivity tester in the laboratory; (**b**) the chamber for temperature control in the thermal conductivity test; (**c**) Thermal conductivity results of the control and xGNP modified asphalt binders. unit conversion due to different scales from microscopic to macroscopic states based on Equation (9). The mass/volume effect in the MD simulation was also considered in the unit conversion. The results of thermal conductivity of the xGNP modified and control asphalt binder models are presented in Figure 10c. \[J=-\lambda\
abla T\ \text{and}\ \lambda=-\frac{\sum_{trans\ \text{\small{f}}res}\frac{m}{2}(v_{hot}^{2}-v_{cold}^{2})}{2 tL_{x}L_{y}( \partial T/\partial z)} \tag{9}\]  
where \(\
abla T\) is the temperature gradient (scalar) in the simulation cell; \(J\) is the energy transferred (scalar) through the surface of layers; \(\lambda\) is the thermal conductivity; \(t\) is the simulation time; \(v_{hot}\) is the velocity of the hot particle; \(v_{cold}\) is the velocity of the cold particle; \(m\) is the identical mass of particles; \(L_{x}\) is the length of the simulation box in the \(x\)-direction; and \(L_{y}\) is the length of the simulation box in the \(y\)-direction; and \(\partial T/\partial z\) is the temperature gradient in the \(z\)-direction. Figure 10: Thermal conductivity calculations in the MD simulations. (**a**) Cumulative delta energies of the control and xGNP modified asphalt binder systems at a temperature of 298.15 K; (**b**) Temperature difference of the control and xGNP modified asphalt binder systems at a temperature of 298.15 K.  
Figure 10a shows the cumulative energy input in different molecular binder systems at a temperature of 298.15 K. The control asphalt binder system has a relatively low energy input in contrast to the energy of the xGNP modified asphalt binder system due to the small number of molecules and low volume in the control binder system. The energy of the xGNP modified asphalt binder system is four times greater than that of the control asphalt binder system, and it is the same as the mass ratio of the xGNP modified asphalt binder system to the control binder system. The temperature variation of the xGNP modified asphalt binder model is also lower than that of the control binder system due to the large number of molecules.