## 3 Results and Discussion
### Master Curves
The crossover modulus decreases and DI increases due to the chemical oxidation that favours the aggregations between molecules. Analogously, [PERSON] et al. [34] correlate the inverse of \(\log\mid G_{\infty}\mid\) with the total oxygen content. Of course, the trend is the same observed for \(R\) and thus we can read these data as an indication of a lower DI and lower oxygen uptake while dealing with LS-modified binders. This point will be further discussed in Section 3.3. Another possible way to quantitatively correlate the master curves and aging is through the variation of the shift factors used in the TTSP. Although there are many possibilities in the analytical description of the horizontal shift factors [36], in bitumen practice the most widely used ones are the William-Landel-Ferry (WLF) equation,  
\[\log a_{T}(T,T_{r})=\frac{-C_{1}(T-T_{r})}{C_{2}+(T-T_{r})}, \tag{16}\]  
and Arrhenius equation,  
\[\log a_{T}=\frac{0.4347E_{a}}{R}\bigg{(}\frac{1}{T}-\frac{1}{T_{r}}\bigg{)}, \tag{17}\]  
where \(C_{1}\) and \(C_{2}\) are the WLF parameters, \(T_{r}\) is the reference temperature (K), \(R\) is the ideal gas constant, and \(E_{a}\) is the activation energy for flow. The variation of the shift factors, or alternatively of the parameters of Equations (16) and (17), can be monitored as a function of aging. As an example, [PERSON] et al., showed that variations of parameters \(C_{1}\) and \(C_{2}\) with aging correlates with the growth of carbonyl groups [4]. Analogously, the same parameters showed a relationship with the asphaltenes content [37] and thus with aging, \(C_{1}\) being the most reliable indicator [38]. In our case, the shift factors showed a better correlation with Equation (17) and the obtained values of the activation energies are reported in Table 2 together with the corresponding aging index, defined as the ratio between the activation energies in the aged (E\({}_{a,a}\)) and unaged (E\({}_{a,u}\)) samples. The trend in the aging indexes confirms the previous findings, but the numerical values seems to be slightly sensitive to aging and no significant variation in the activation energies is observed for the longest aging. With regard to the vertical shift, all the samples showed a similar behaviour that does not correlate with aging. The variation in the vertical shift factors with temperature is reported in Figure 5, where all the data corresponding to the B and LS-B at 0, 25, 40, 65, and 90 h of PAV are represented with the same symbol, in order to visualize the common trend. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & & \multicolumn{3}{c}{**B**} & & & & **LS-B** & \\ \hline _PAV_ & \(R\) & \(\omega_{c}\) **(rad/s)** & \(AI_{R}\) & \(AI_{w}\) & **Log (\(\mid G_{c}\mid\))** & \(R\) & \(\omega_{c}\) **(rad/s)** & \(AI_{R}\) & \(AI_{w}\) & **Log (\(\mid G_{c}\mid\))** \\ \hline
0 & 1.80 & \(5.36\cdot 10^{-1}\) & 0 & 0 & 7.2 & 1.86 & \(2.57\cdot 10^{-1}\) & 0 & 0 & 7.14 \\
25 & 2.42 & \(4.94\cdot 10^{-3}\) & 0.35 & 7.53 & 6.58 & 2.16 & \(1.19\cdot 10^{-2}\) & 0.16 & 2.26 & 6.84 \\
40 & 2.56 & \(2.19\cdot 10^{-3}\) & 0.42 & 8.83 & 6.44 & 2.44 & \(5.51\cdot 10^{-3}\) & 0.31 & 2.83 & 6.56 \\
65 & 2.97 & \(5.74\cdot 10^{-5}\) & 0.66 & 14.7 & 6.03 & 2.85 & \(4.28\cdot 10^{-4}\) & 0.54 & 4.71 & 6.15 \\
90 & 3.20 & \(9.87\cdot 10^{-6}\) & 0.78 & 17.5 & 5.8 & 2.72 & \(3.02\cdot 10^{-4}\) & 0.47 & 4.97 & 6.28 \\ \hline \end{tabular}
\end{table} 
Table 1: Values of \(R\), \(\omega_{c}\), and \(\log(\mid G_{\infty}\mid)\) from the experimental master curves and the corresponding aging indexes defined in Equation (15). In both cases, there is an evident shift of the right part of the distributions toward higher molecular weights; this being an immediate confirmation that the dispersion index increases with aging, as suggested by the \(\mid\)G\({}_{\rm c}\mid\) values. Moreover, the shape of the AMWD shows two other important changes related to aging.