# Dynamic Characteristic Master Curve and Parameters of Different Asphalt Mixtures
## 3 Characteristic Parameters of Master Curves
### Phase Angle Master Curve Parameters
#### 3.2.2 Phase Angle Peak Position Parameter-\(\omega_{\rm c}\)  
\(\omega_{\rm c}\) is a parameter that varies with temperature. Figure 8 shows that the \(\omega_{\rm c}\) value of the phase angle master curve moves to the high-frequency position with the increase of temperature, which shows that the viscoelasticity of the asphalt mixture changes with temperature, which has nothing to do with aggregate but is only caused by the temperature sensitivity of asphalt. Figure 9 shows the relationship between \(\omega_{\rm c}\) and the phase angle of the asphalt binder at the same temperature, with R\({}^{2}\) greater than 95%, which is sufficient to illustrate this temperature sensitivity issue. Figure 6: Correlation between master curve parameters of phase angle and asphalt indexes. Figure 7: Phase angle relationship between asphalt and mixture. In the theory of polymer physics, the \(\omega_{\rm c}\) position is related to the relaxation characteristics of polymer materials, which is defined as the relaxation peak [10]. Its significance is to characterize the response ability of the viscoelasticity of materials under external action. Only when the load time (t) is close to the material relaxation time (\(\tau\)), that is, the action frequency (\(\omega_{\rm r}\)) is the same as the reciprocal of the relaxation time (\(\tau^{-1}\)) (\(\omega_{\rm r}=\tau^{-1}\)), is the energy consumption of the material the highest, which reflects the peak value of the phase  
Figure 8: Relationship between \(\omega_{\rm c}\) and temperature. Figure 9: Relationship between \(\omega_{\rm c}\) and asphalt phase angle. angle or loss angle tangent; when the frequency is high, the action time of external force is short, and the viscous pot effect of the material is too late to respond, reflecting the elastic characteristics and glass state. When the frequency is low, the action time of the external force is long, the spring effect of the material is completely restored, which shows the full sticky pot effect, and the material shows more viscous characteristics and a rubber state. Because the asphalt mixture will not appear with absolute flow behavior, there will not be multiple relaxation characteristic peaks like polymer materials, and there is only one phase angle peak or loss angle tangent peak. Therefore, at a certain temperature, the peak position of the phase angle of the asphalt mixture can be used as a reflection of its relaxation ability, which is also important information given by the phase angle master curves, which is often ignored by researchers. According to the peak position of the phase angle master curve, the relaxation time and ability of the mixture at a certain temperature can be calculated and investigated according to the relationship \(\omega=\tau^{-1}\). The relaxation times of three kinds of asphalt mixtures at different temperatures are shown in Table 3.  
Figure 10 shows the relationship between relaxation time and temperature. It shows that in the range of 20 to 50 \({}^{\circ}\)C, the higher the temperature is, the smaller the relaxation time of the asphalt mixture is, and it shows an exponential attenuation relationship. For penetration grading 50 asphalt, the relaxation time of AC20 asphalt mixture is 121.3 s at 20 \({}^{\circ}\)C, but it can be reduced to 0.08 s at 50 \({}^{\circ}\)C. This explains the reason for the cracking disease caused by the stress concentration of the asphalt mixture under the condition of low temperatures. In addition, for the same kind of asphalt mixture, the high-grade asphalt has less relaxation time; for example, at 20 \({}^{\circ}\)C, the relaxation times of grades 50, 70, and 90 asphalt mixtures are 121.3, 40.5, and 16.4 s, respectively. This shows that high-grade asphalt is helpful for improving the cracking resistance of the mixture. At the same time, the relaxation time is of great significance to the characterization of crack resistance of the asphalt mixture.