## 3 Results and Discussion
### Master Curves
This model describes the complex modulus of bituminous binders and asphalt mixtures by the following expression:  
\[|G^{*}(\omega)|=|G_{\infty}|\left[1+\left(\frac{\omega_{c}}{\omega}\right)^{ \frac{log2}{R}}\right]^{\frac{-R}{log2}} \tag{14}\]  
which contains only three parameters: \(|G_{\infty}|\), the crossover frequency \(\omega_{c}\), and the so-called rheological index \(R\). The latter is a \"shape parameter\" and represents the difference between the logarithmic values of \(|G_{\infty}|\) and \(|G_{c}|\) (crossover modulus = complex modulus at the crossover frequency). Generally, the master curve of \(|G^{*}|\) becomes flatter as a consequence of aging and thus the increase in \(R\) can be used to estimate the level of aging. [PERSON] et al., suggested a power law for the dependence of \(R\) from the aging time [33]. Moreover, the above-described shift to the left of the curve can be identified through the crossover frequency and thus two out of the three parameters of the CA model are directly connected with aging. The third parameter is the glassy modulus and often is not directly available since it may require recording data at low temperatures by means of torsion bars. In some cases, this value is simply treated as a fitting parameter and thus extrapolated from the intermediate temperatures. Alternatively, a constant value (1 GPa) is arbitrarily assumed as valid irrespective of the aging degree and in this case the CA model has only two parameters, with \(R=R(\,|G_{c}|\,)\)[34]. Both solutions are questionable, but probably the assumption of a fixed glassy modulus has a better chance to fit with reality. It is important to underline that thanks to their physical meaning, the parameters can be directly extrapolated from the \(|G^{*}|\) master curves without applying the CA model to fit the curve and this is what we did. In other words, here Equation (14) is reported only to introduce the physical meaning of these parameters and not used to fit the master curves, and the values of \(R\) and \(\omega_{c}\) were taken directly from the experimental data reported in Figure 4. The numerical values for these parameters are reported in Table 1, together with the corresponding aging indexes \(AI_{R}\) and \(AI_{\omega}\) defined as follows:  
\[AI_{R}=\frac{(R_{a}-R_{u})}{R_{u}}\;\;AI_{\omega}=\frac{(\log\omega_{c,a}-\log \omega_{c,u})}{\log\omega_{c,u}} \tag{15}\]  
where the subscript \(a\) and \(u\) indicate aged and unaged, respectively, and \(\omega_{c}\) is expressed in rad/s. For the calculation of \(R\) in Table 1, \(|G_{\infty}|\) was assumed equal to 1 GPa. The variation in \(R\) indicates that the crossover moduli varies 1.4 orders of magnitude without clay and less than one order of magnitude with the clay. Even more pronounced is the shift of the crossover frequency, which varies of about five orders of magnitude for B and only three with clay (LS-B binder). The aging indexes clearly reflect these differences, \(AI_{\omega}\) being the one that has the higher sensitivity in this case. Moreover, it is interesting to observe that between 65 and 90 h of PAV the properties of LS-B remain almost constant, while those of B are still subjected to significant changes. As already underlined, even if useful and widely accepted, the disadvantage of \(R\) is its dependence on \(\mid G_{\infty}\mid\), which can be uncertain. Therefore, as an alternative with a higher chance of being included in the experimental master curve, the evolution of the crossover modulus \(\mid G_{\infty}\mid\) can be considered. Indeed, due to the assumption of \(\mid G_{\infty}\mid\) being constant and equal to \(10^{9}\) Pa, the logarithm of the crossover moduli is given by 9-R, also reported in Table 1. [PERSON] et al. [35] underline that \(\log\mid G_{\infty}\mid\) can be linked to the dispersion index (DI, ratio between the weight and number average molecular weights), which, in turn, has a direct link with the material composition and thus with aging.