# Polysaccharides-Reinforced Bitumens: Specificities and Universality of Rheological Behavior
## 3 Results and Discussion
### Viscosity (\(\eta\)) as a Function of Temperature (T)--Arrhenius Model  
The temperature dependence of both \(G^{\prime}\) and \(G^{\prime\prime}\) allows further analysis. To do so, a more synthetic indicator has been derived: the complex viscosity (\(\eta^{\star}\)). It is the ratio of \(G^{\star}\) to angular frequency \(\omega\), i.e., [25]:  
\[\eta^{\star}=\frac{G*}{\omega}=\frac{\sqrt{G^{\prime 2}+G^{\prime\prime 2}}}{\omega} \tag{2}\]  
Here, \(\eta^{\star}\) has been considered because it is a synthetic indicator. In fact, differently from \(G^{\prime}\) and \(G^{\prime\prime}\), which give separate information on the energy absorbed by the system in elastic (\(G^{\prime}\)) and dissipating (\(G^{\prime\prime}\)) form, \(\eta^{\star}\) instead synthetically yields information on the _total_ amount of energy that the system absorbs. In addition, besides being more synthetic, \(\eta^{\star}\) has the physical meaning of representing the resistance to flow under oscillatory shear conditions. The temperature dependence of \(\eta^{\star}\) is reported in the inset of Figure 1 as ln \(\eta^{\star}\) vs. 1000/T (Arrhenius plot). The choice of the Arrhenius plot, far from following usual habits to show the temperature dependence of viscosity, is instead the result of our approach based on the molecular interpretation of the macroscopic properties (an approach we are keen on) which lead to the following observations and considerations. From the microscopic point of view, in fact, the complex viscosity modulus is determined ultimately by the force field between the molecules, i.e., the strength of the interactions involved in the intermolecular network in bitumen. The theoretical description is quite complex [26], and several approaches have been developed, both theoretical [27; 28; 29] and numerical as based on molecular dynamic simulations [30]. On the other hand, for a simple analytical description of the temperature dependence of viscosity, several forms have been derived. The most common is the Arrhenius model, but other models are also present, like the Vogel-Tamman-Fulcher (VTF) equation, the polynomial expansion [31] and other formulas taking into account for eventual tunneling [32]. These forms are nicely commented in [30] and [31]. Basically, as some authors shown [31], the main distinction lies on the linearity of the ln \(\eta^{\star}\) vs. 1/T trend (Arrhenian behavior). The trend is usually linear for simple liquids, which follows the basic two-wells potential model. If the trend is not linear then other equations must be used according to the reason of non-linearity: if cooperative effects are present, as occurring for example for strongly interacting fluids like ionic liquids, heavy oils and fuels, polymers, melting salts, and glasses, then, the VTF can be used. It is usually used, indeed, for glass, with the phenomenological description of the glass forming behavior close to Tg. If tunneling is present, then the Nakamura-Takayanagi-Sato (NTS) or Aquilanti-Sanches-Coutinho-Carvalho (ASCC) formulas can be considered. Other approaches, like a polynomial expansion, with interpolation of experimental data, can be used. Further details of these aspects can be found in many academic and specialized textbooks so the eager reader is redirected to those readings. Here, this preliminary distinction will be used to justify the choice of Arrhenius model. As can be seen by the Arrhenius plot (inset in Figure 1), the linear trend of the data is pretty well observed in an adequate interval around the temperature chosen as representative of the working condition (50 \({}^{\circ}\)C).