## 5 Results and Discussions
### Temperature Susceptibility of THFS Modified Asphalt Binder
#### 5.5.5 Frequency susceptibility based on CAM model  
The Christensen-Andersen-Marasteanu (CAM) model is a kind of rheological model adding a parameter, \(m_{e}\), on the basis of basic shape parameters of the CA model [30; 31; 32; 33]. Most rheological models are mainly focused on researching the rheological properties of asphalt, however, the CAM model can not only describe the rheological properties of asphalt but also analyze the rheological properties of mixtures. The CAM model includes four equations: the complex modulus master curve equation, the storage modulus master curve equation, the phase angle equation and temperature shift factor equation. These equations characterize the proportion of viscosity and elasticity of asphalt, and  
Figure 8: \(G^{*}\) and \(\delta\) Master Curves of Unaged Asphalt Binders over Frequencies. also exhibit the temperature sensitivity of viscoelastic properties of asphalt. The equation of the CAM model is shown in Equation (6)  
\[G^{*}=G_{e}^{*}+\frac{G_{g}^{*}-G_{e}^{*}}{\left[1+\left(f_{c}/f^{\prime}\right)^ {k}\right]^{m_{e}/k}} \tag{6}\]  
where: \(G_{e}{}^{*}\) is the equilibrium complex modulus of asphalt (kPa). It is \(G^{*}\) when frequency reaches zero or the temperature is very high;  
\(G_{g}{}^{*}\)is the glassy complex modulus (kPa). It is \(G^{*}\) when frequency reaches infinity or the temperature is very low;  
\(f_{c}\) is the elastic limit threshold (Hz);  
\(f^{\prime}\) is the reduced frequency, related to temperature and strain (Hz);  
\(k\) and \(m_{e}\) are shape parameters, without sizes. The zones beyond the threshold frequency are the low-frequency stable zone and high-frequency stable zone. Correspondingly, it is called equilibrium complex modulus \(G_{e}{}^{*}\) and glassy complex modulus \(G_{g}{}^{*}\). The breakpoint is called the low-frequency break point when the low-frequency stable zone changes into the rheological zone. The break point symbolizes the change in physical form of asphalt. The intercept between \(G_{g}{}^{*}\left<f_{c}\right>\) and \(G_{g}{}^{*}\) is recorded as \(R\), which is related to shape parameters \(k\) and \(m_{e}\). R refers to the width of the relaxation spectrum. A larger value of \(R\) indicates that it is easy to change from elastic to viscous behaviors. It also indicates less sensitivity to frequency changes. The parameter R is shown in Equation (7). \[R=lg\frac{2^{m_{e}/k}}{1+(2^{m_{e}/k}-1)G_{e}^{*}/G_{g}^{*}} \tag{7}\]  
Since \(G_{e}{}^{*}=0\) for asphalt binder, the equation of \(R\) is as follows (8):  
\[R=\frac{m_{e}}{k}lg2 \tag{8}\]  
Table 9 lists the fitting parameters of asphalt binders based on the CAM model. The test results in Table 9 presented that the correlation coefficients \(R^{2}\) were higher than 0.9990, very close to 1, which indicated that the CAM model was very reliable in predicting the rheological behavior and frequency susceptibility of asphalt binder. In the CAM model, the \(G_{g}{}^{*}\) values of asphalt binders were higher than that of SK-90, and 10% THFS modified asphalt binder had the highest \(G_{g}{}^{*}\) value. It showed that THFS modified asphalt binder had more resistance to permanent deformation at frequencies ranging from 0.1 Hz to 50 Hz. Subsequently, asphalt binder with higher \(f_{c}\) value indicates it had a good resistance to thermal cracking at low temperatures. Compared with SK-90, the \(f_{c}\) value of THFS modified asphalt binder decreased and 10% THFS modified asphalt binder had the lowest \(f_{c}\) value. It could be predicted that THFS modified asphalt binder was more prone to thermal cracking at low temperatures. This remark was similar to the BBR test results.