# Mechanism of HCB-Modified Asphalt and Dynamic Properties of Mixtures
## 6 Dynamic Properties of HAM
### The Master Curve Analysis of Dynamic Response Parameters  
The master curve of a dynamic modulus is explained by the sigmoidal model [18], i.e., the horizontal translation of the shift factor is realized by adopting nonlinear least-squares fitting, as exhibited in Equation (1). \[\text{log}|E^{\star}|=\delta+\frac{\alpha-\delta}{1+\text{e}^{\beta+\gamma( \text{log}\log f_{\gamma})}} \tag{1}\]  
where \(E^{\star}\) is dynamic modulus; \(f_{\gamma}\) is the reduced frequency; \(\delta\) is the minimum value of the dynamic modulus; \(\alpha\) is the maximum value of the dynamic modulus; and \(\beta\) and \(\gamma\) are regression parameters, which are determined by the properties of the binder. For a given reference temperature, the horizontal movement number value of frequency is a shift factor. The relationship between the reduced frequency \(f_{\gamma}\) and shift factor \(\alpha\)(T) is exhibited in Equation (2). \[f_{\tau}=f\cdot a(T) \tag{2}\]  
where \(f_{\gamma}\) is the reduced frequency in Hz; \(f\) is the loading frequency in Hz; and \(\alpha\)(T) is the shift factor. When the master curve is determined, the shift factor of the dynamic modulus can be acquired via Equation (3). \[\text{log}[a(T)]=\frac{\Delta E_{a}}{19.14714}\Big{(}\frac{1}{T+273.15}-\frac{ 1}{T_{r}+273.15}\Big{)} \tag{3}\]  
where \(\Delta E_{a}\) is activation energy in J/mol; \(T_{r}\) and \(T\) are reference temperature and test temperature in \({}^{\circ}\)C, and the shift factor \(\alpha\)(T) = 1 a the reference temperature. The reduction frequency can be acquired via Equation (4), adopting the Arrhenius equation. \[\text{log}f_{\tau}=\text{log}f-\frac{\Delta E_{a}}{19.14714}\Big{(}\frac{1}{T+ 273.15}-\frac{1}{T_{r}+273.15}\Big{)} \tag{4}\]  
where, \(f_{\gamma}\) is the reduced frequency in Hz and \(f\) is the test frequency in Hz. The master curve is exhibited in Equation (5). \[\text{log}\text{log}|E^{\star}|=\delta+\frac{\alpha-\delta}{1+\text{e}^{\beta +\gamma\ast[\text{log}f-\frac{\Delta E_{a}}{19.14714}(\frac{1}{T+273.15}-\frac {1}{T_{r}+273.15})]}} \tag{5}\]  
Dynamic modulus test results were ascertained via sigmoidal function, using the numerical analysis software Origin and adopting 21.1 \({}^{\circ}\)C as the reference temperature. The initial values of the parameters were: \(\beta\) = \(-1.0\), \(\gamma\) = \(-0.5\), \(\delta\) = \(0.5\), \(\Delta E_{a}\) = \(200\),\(000\). The activation energy \(\Delta E_{a}\) and shift factor \(\alpha\)(T) of mixtures are exhibited in Table 6, and the fitting parameter of the master curve for the dynamic modulus is exhibited in Table 7. \begin{table}
\begin{tabular}{c c c c} \hline
**Mixtures** & **Transition Temperature** (\({}^{\circ}\)C) & **Activation Energy** \(\Delta\)**Ea (T)** & **Shift Factor log[\(\alpha\)(T)]** \\ \hline \multirow{4}{*}{BAM} & 4.4 & 216631.4 & 2.314 \\  & 21.1 & 0 & 0 \\  & 37.8 & 230313.7 & \(-\)2.194 \\  & 54.4 & 221054.2 & \(-\)3.989 \\  & 4.4 & 208657.3 & 2.228 \\ \multirow{2}{*}{HAM modified asphalt} & 21.1 & 0 & 0 \\  & 37.8 & 231061.2 & \(-\)2.203 \\ \multirow{2}{*}{mixture} & 54.4 & 225021.7 & \(-\)4.060 \\ \hline \end{tabular}
\end{table} 
Table 6: Shift factors of mixtures. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Mixtures** & **Transition Temperature** (\({}^{\circ}\)C) & **Activation Energy** \(\Delta\)**Ea (T)** & **Shift Factor log[\(\alpha\)(T)]** \\ \hline \multirow{4}{*}{BAM} & 4.4 & 216631.4 & 2.314 \\  & 21.1 & 0 & 0 & 0 \\  & 37.8 & 230313.7 & \(-\)2.194 \\  & 54.4 & 221054.2 & \(-\)3.989 \\ \multirow{2}{*}{HAM modified asphalt} & 4.4 & 208657.3 & 2.228 \\  & 21.1 & 0 & 0 & 0 \\  mixture & 37.8 & 231061.2 & \(-\)2.203 \\ \multirow{2}{*}{mixture} & 54.4 & 225021.7 & \(-\)4.060 \\ \hline \end{tabular}
\end{table} 
Table 7: Fitting parameters of dynamic modulus master curve. The master curve of the dynamic modulus was obtained by moving the data horizontally along the logarithmic frequency axis. As a function of the loading frequency and temperature, the dynamic modulus was reduced to a function of frequency.