## 2 Materials and Methods
### Apparent Molecular Weight Distributions  
For several years rheologists have been trying to link the molecular weight distributions (MWD) with the rheological data. The first papers were dedicated to polymers [12; 13] and then the same approach was extended to bituminous binders [14; 15]. One of the suggested procedures is the so-called \(\delta\)-method, recently developed by [PERSON] and co-workers [5]. The method is described in detail in the original paper and is based on the idea that relaxation frequencies of the molecules are directly correlated with their molecular weight. In inhomogeneous systems, such as polymers or bituminous binders, this corresponds to a distribution of the relaxation frequencies (\(\omega\)) that is directly correlated with the phase-angle (\(\delta\)) master curve. At a given frequency, the higher the number of molecules with a lower relaxation frequency, the higher the phase angle. This leads to the assumption that the cumulative molecular weight distribution (_CMWD_) is proportional to \(\delta(\omega)\). In other words, the master curve of the phase angle is directly converted into _CMWD_ by applying the simple normalization:  
\[CMWD=\frac{1}{90}\delta(MW) \tag{1}\]  
where \(\delta\) (_MW_) is the phase angle expressed as a function of the molecular weight. Therefore, all that is needed to convert the phase-angle master curve into an _MWD_ is a correlation between \(\omega\) and the molecular weight (_MW_). The chosen correlation is the one developed by [PERSON] et al. [14] through vapour pressure osmometry. \[\log\left(MW\right)=2.88-0.06768\log\left(\omega_{\mathrm{c}}\right) \tag{2}\]  
where \(\omega_{\mathrm{c}}\) is the crossover frequency (at which the phase angle is equal to 45\({}^{\circ}\)) at 0 \({}^{\circ}\)C. The last step to convert the cumulative distribution into the more commonly used weight fraction (_w_) is the differentiation of the _CMWD_ with respect to \(\log(MW)\):  
\[w(MW)=\frac{d(CMWD)}{d(\log(MW))} \tag{3}\]  
It is worth underlining that a complete _MWD_ may be obtained only if a master curve covering the whole range of phase angles (from 0 to 90\({}^{\circ}\)) is available. Moreover, the smoothness of the curve is important for the derivation of the _MWD_ by differentiating the _CMWD_[16]. For these two reasons, it may be useful to fit the experimental data with a rheological model. This allows both extrapolating the curve where experimental data are missing (phase-angle values close to 0\({}^{\circ}\)) and an easier data manipulation. Among the long list of available models (see the review by [PERSON] et al. [17]), we chose 2S2P1D [18; 19; 20; 21], consisting of a combination of two springs, two parabolic elements, and one dashpot (Equation (4), Figure 1). \[G^{*}=G_{o}+\frac{\left(\left|G_{\infty}\right|-\left|G_{o}\right|\right)}{1+ \alpha\left(i\omega\tau\right)^{-k}+\left(i\omega\tau\right)^{-h}+\left(i \omega\beta\tau\right)^{-1}} \tag{4}\]  
where \(G^{*}\) is the complex modulus; \(\left|\,G_{\infty}\,\right|\) and \(\left|\,G_{0}\,\right|\) are the magnitudes of the complex modulus at angular frequency \(\omega\) approaching infinite and zero, respectively; \(\alpha\), \(\beta\), \(\tau\), \(k\), and \(h\) are the model parameters; and \(i\) is the imaginary unit. It is worth nothing that in the literature, \(\left|\,G_{\infty}\,\right|\) is often indicated as \(\left|\,G_{\mathrm{g}}\,\right|\) and referred as the \"Glassy modulus\". The expression for the phase angle can be derived from Equation (4) by applying the properties of complex numbers as follows. \(G^{*}\) can be expressed in terms of real and imaginary parts:  
\[G^{*}=G^{\prime}+iG^{\prime\prime} \tag{5}\]  
If we introduce the complex number \(A^{*}\) defined as  
\[A^{*}=1+\alpha(i\omega\tau)^{-k}+(i\omega\tau)^{-h}+(i\omega\beta\tau)^{-1}=A ^{\prime}-iA^{\prime\prime}\]  
where \(\left|\,A^{*}\right|\) is the magnitude of \(A^{*}\).