## 4 Physical Properties of the Control and xGNP Modified Asphalt Models
### Glass Transition Temperature  
The glass transition temperatures (Tg) of materials are influenced by their components, and the addition of a new component in the material results in a difference in the glass transition temperature. The formula [42] of the glass transition temperatures of composite materials is shown in Equation (4). The asphalt transfers from the viscoelastic state to a brittle one. The internal stress increases and thermal energy are insufficient below the glass transition temperature in the materials. Therefore, the glass transition temperature is an important parameter or property for amorphous materials, and it should be low for a good low-temperature performance [42]. The glass transition temperature is the temperature where two asymptotes intersect on the specific volume-temperature curve. In the laboratory, the glass transition temperature of materials can be tested by differential scanning calorimetry (DSC). In this MD simulation study, the control and xGNP modified asphalt systems were relaxed under the NPT ensemble with a temperature range of 233.15 K-443.15 K. The specific volumes of these systems were calculated and the glass transition temperatures of the models were determined. The MD simulation results of the control and xGNP modified asphalt models are shown in Figure 4.  
\[\frac{1}{T_{g}}=\sum\frac{w_{i}}{T_{gi}} \tag{4}\]  
where \(T_{g}\) and \(T_{gi}\) are the glass transition temperature of the composite material and its component, respectively; and \(w_{i}\) is the mass fraction of the component. Figure 4a or Figure 4b shows the specific volumes of the control system or xGNP modified asphalt model, respectively. Figure 4c shows the specific volumes of both control and xGNP modified asphalt models. The specific volumes increase with the increasing temperatures of the models. The glass transition temperature of the control asphalt model is around 300 K [24]. As shown in the Figure, it is deduced that the Tg of the xGNP modified asphalt system is around 250 K. Based on the laboratory data in the reference [43], the Tg of asphalt ranges from 223 K to 303 K. The Tg of the modified asphalt model is within the laboratory data, and it is also better than the reference data [18] (298 K-358 K) of the asphalt model. In order to get better thermal relaxation in the asphalt, a low Tg of asphalt is expected. Figure 4d shows the comparisons of glass transition temperatures of the references and MD simulations. The glass transition temperatures of the Strategic Highway Research Program (SHRP) asphalt binders are all around 250 K [42], including SHRP asphalt AAA-1, AAB-1, AAC-1, AAD-1, AAF-1, AAG-1, AAK-1 and AAM-1. The Tg of xGNP modified asphalt model is close to the glass  
transition temperatures of SHRP asphalt binders. Therefore, the Tg of xGNP modified asphalt is reasonable and better than the results of the control [24] and reference asphalt models [18]. Figure 4: _Cont._  
Figure 4: Specific volumes and temperatures of the control and xGNP modified asphalt systems. (**a**) specific volumes of the control asphalt system: amplified section to show the Tg temperature with asymptotes intersect for control asphalt model; (**b**) specific volumes of the xGNP modified asphalt system: amplified section to show the Tg temperature with asymptotes intersect for modified asphalt system; (**c**) specific volumes of the xGNP modified and control asphalt system; (**d**) Glass transition temperatures of different asphalt types and models: the data of the glass transition temperature of different binders is from the reference [18,24,42]: 248.55 K for AAA-1 binder, 252.25 K for AAB-1, 253.75 K for AAC-1, 249.15 K for AAD-1, 253.95 K for AAF-1, 265.35 K for AAG-1, 252.55 K for AAK-1, 252.35 K for AAM-1, which were tested by [PERSON] (1997) [42]; 300 K for control asphalt model which is from the MD simulation by [PERSON] at al.