## 3 Results and Discussion
### Relaxation Spectrum  
The relaxation spectrum, usually indicated as \(H(\widetilde{\tau})\), is one of the most important rheological functions, since once it is known, it allows calculating any other viscoelastic function and vice versa [11]. For this purpose, a number of algorithms has been proposed to obtain \(H(\widetilde{\tau})\) from the master curve of a viscoelastic function [25; 38; 43; 44; 45]. Moreover, since it describes a distribution of relaxation times, \(H(\widetilde{\tau})\) can be viewed as an indirect measurement of the MWD [46] and specific relationships have been proposed for polymers [38; 47; 48] and bituminous binders, thus providing an alternative to the \(\delta\)-method [38]. For all these reasons, \(H(\widetilde{\tau})\) can be used to characterize the aging of the binders [38; 49]. [PERSON] et al., observed a horizontal shift of the relaxation spectra towards higher relaxation times and suggest the use of the mean value, variance, and skewness of the spectra to characterize its evolution with time [50]. [PERSON] et al., derived an indicator to evaluate the effect of rejuvenators directly from \(H(\widetilde{\tau})\)[51]. As described in the materials and method section, the relaxation spectra were derived from the 2S2P1D model and in what follows, they are reported on a logarithmic scale, where the range of the \(x\)-axis corresponds to experimental data. In other words, the 2S2P1D model in this case is used only in the frequency range covered by the experimental master curve and there is no extrapolation of the data outside that range. Therefore, the reported spectra represent only a portion of the usual asymmetric bell shape corresponding to a full range of relaxation times. This portion is located after the bell peak, and thus the curves decrease monotonically with the relaxation time. For this reason, the variance and skewness are not very useful parameters. Nevertheless, the shift of the spectra toward higher relaxation times is clearly visible as well as how the clay reduces this shift. Figures 12 and 13 show the relaxation spectra for the B binder at different levels of aging and the usual comparison with the LS-B binder, respectively. For both binders, the relaxation spectra converge to similar values at low relaxation times and show the main differences in the right-hand side of the spectra. This is consistent with the already observed variations in MWD. The low MW molecules (lower relaxation times) are less affected by the oxidation, while the other molecules change their composition, favouring a higher degree of aggregation. This determines the shift to the right of both the AMWD and \(H(\widetilde{\tau})\) curves. Figure 11: Comparison of the relative amounts (% by weight) of the four populations obtained by deconvolution and HPTLC. The red dotted lines represent the linear regressions for P2-aromatics and P3-resins. Since in a logarithmic plot all the relaxation spectra appear parallel at high relaxation times, an aging index could be defined in order to take into account this horizontal shift of the right-hand side of the curves:  
\[AI_{H}=\frac{\left(\log\widetilde{\tau}\right)_{a}}{\left(\log\widetilde{\tau} \right)_{u}} \tag{23}\]  
where \(\widetilde{\tau}\) indicates the numerical value of time expressed in seconds and the index is evaluated at any \(H(\widetilde{\tau})\) in the range were the behaviour is linear in the logarithmic plot, which means below approximately 100 Pa. As an example, the vertical arrows in Figure 12  
Figure 12: Relaxation spectra for binder B at all investigated levels of aging. Figure 13: Comparison of the unaged, 25, and 90 h PAV aged B and LS-B binders. indicate the values of \(\tau\) for B and B25 corresponding to \(H(\widetilde{\tau})\) = 1 Pa. The obtained values are reported in Table 7.  
Eventually, it is interesting to correlate the data obtained from the master curves, AMWD, and relaxation spectra.