## 3 Results and Discussion
### Oscillatory Shear Experiments  
The viscoelastic properties of bitumens were analysed by oscillatory experiments (frequency-sweeps and temperature sweeps) at 25 \({}^{\circ}\)C. Preliminary stress-sweep tests were also performed by applying small strain amplitudes in order to define linear viscoelastic conditions. The frequency dependence of the experimental complex modulus 1G*1 measured at 25 \({}^{\circ}\)C for both unmodified and modified bitumens is illustrated in Figure 2 in correspondence to different thermal cooling gradients of, respectively, 1 \({}^{\circ}\)C/min (A), 5 \({}^{\circ}\)C/min (B) and 10 \({}^{\circ}\)C/min (C). Clear evidence can first be observed at low rates (1\({}^{\circ}\)/min) and fast (10\({}^{\circ}\)/min), showing substantial differences of 1G*1 among the various samples, attributable to the presence of the additives (see Figure 2A,C). An exception can be found for P2 KA and PPA modified bitumens, characterised by 1G*1 data, which are almost overlapped for 1 \({}^{\circ}\)C/min (Figure 2A). However, the effect of the additives seems to be cancelled out at the intermediate cooling rate of 5 \({}^{\circ}\)C/min, where the respective rheological behaviours are hardly distinguishable from those of neat bitumen (see Figure 2B). As observed from AFM images, the development of amorphous/crystalline polydomains in the bitumens is controlled by nucleation and growth rates of disperse catana/peri-phase induced by different cooling rates, which in turn are affected also by the presence of additives. It should be expected that upon rapid cooling small aggregates would engender the formation of a network stabilised by physical interactions with a consequent increase of complex modulus 1G*1. To understand whether the added compounds are somewhat engaged in the formation of noncovalent interactions between asphaltene aggregates able to stabilise or even destabilise supramolecular networks upon cooling, the rheological data have been analysed in the framework of the colloidal gel model. Figure 2: Complex modulus 1G*1 vs frequency determined at 25 \({}^{\circ}\)C for neat bitumen (diamonds), and bitumens modified by addition of 2% P2 KA (circles), LCS (squares) and PPA (triangles), respectively, after the specimens were subjected to different thermal treatment: (**A**) 1 \({}^{\circ}\)C/min, (**B**) 5 \({}^{\circ}\)C/min and (**C**) 10 \({}^{\circ}\)C/min. Solid lines represent the best non-linear fits according to the power law of Equation (3), whose fitting parameters \(z\) and \(A\) have been listed in Tables 2 and 3, respectively. According to the theory of [PERSON] [33] and [PERSON] [34], which has been applied to several colloidal complex systems [35; 36; 37] and widely reported in literature as the \"weak-gel model\" [38], a weak-gel material is defined as a complex system characterised by a cooperative arrangement of flow units connected by weak physical interactions that cooperatively ensure the stability of the structure. Thus, the weak-gel model provides a direct link between the microstructure of the material and its rheological properties. The most important parameter is the \"coordination number\", \(z\), which is the number of flow units interacting with each other to give the observed flow response. It was shown in reference [33] that, above the Newtonian region, there exists a regime characterised by the following flow equation:  
\[|G^{*}(\omega)|=\sqrt{G^{\prime}(\omega)^{2}+G^{\prime\prime}(\omega)^{2}}=A \omega^{\frac{1}{z}} \tag{3}\]  
where \(A\) is a proper constant related to the overall stiffness or resistance to deformation of the material within the linear viscoelastic region at an angular frequency of 1 rad/s. Clearly, a log-log plots of \(\mid\)G\({}^{*}\)\(\mid\) vs. frequency should yield a straight line with slope \(1/z\) and intercept \(A\). In Tables 2 and 3, the parameters \(z\) and \(A\), calculated from non-linear fitting of viscoelastic data to Equation (1), are listed for all investigated samples subjected to different cooling rates.