# Polysaccharides-Reinforced Bitumens: Specificities and Universality of Rheological Behavior
## 3 Results and Discussion
### Correlation between Activation Energy (Ea) and Pre-Exponential Factor (As)  
Before rationalizing the specific effect of the various additives on the activation energy and the pre-exponential factor, it must be unveiled another interesting observation: the activation energies are correlated with the pre-exponential factors (Figure 4). The goodness of this result is supported by the observed R-squared value (0.93) and the mean squared errors (MES) value of less than 0.2. Although the R-squared value may seem to be not strikingly high, it must be considered that the pre-exponential factor, as derived by the Arrhenius model, is an extrapolation to the \(Y\)-axis of the data in the Arrhenius plot, i.e., at infinite temperature, which is an un-physical situation. Moreover, it must be considered that the determination of viscosity in bitumens is problematic for several reasons. As an example, it can be considered that measurements from different laboratories on carefully collected samples of the same bitumen have given values differing even of a factor of two ([PERSON], 2019). After these considerations, and given the statistically significant number of the samples (fifteen samples, including the reference) of the present work, in our opinion the correlation seems safe and deserves attention. Figure 3: Plot of Ea vs. \(G^{\prime}\)@50 \({}^{\circ}\)C showing lack of correlation. The correlation between \(\mathrm{Ea}\) and \(\mathrm{In}\) As has been already observed for simple liquids [31]. This is an important aspect also for applicative purposes. In fact, if this correlation holds, then the temperature dependence of the viscosity of liquids can be expressed through one parameter only (the only one known, \(\mathrm{Ea}\) or \(\mathrm{As}\)). This would have the advantage to allow for the viscosity description even if only one parameter is known, since the other can be derived from it. Although there is not a theory explaining such correlation [31] this can be helpful for practical uses. The data presented in that paper are reported in the inset of Figure 4 (open circles) together with the data of the present work. It must be noted that the viscosity calculated for our data refers to a region quite far from the Tg (generally occurring around \(0\,^{\circ}\mathrm{C}\)) and where the contribution of the elastic response is negligible (\(G^{\prime}\) is lower than \(G^{\prime\prime}\) by two orders of magnitude) so it can be approximated to the zero-shear viscosity. For this reason, the comparison with the literature data shown in the inset of Figure 4 can be considered at first approximation as safe. Interestingly, we found a nice consistency between the two sets of data, despite they greatly differ in:  
1. viscosity (the viscosity of our samples is several orders of magnitudes higher);
2. activation energies (\(\mathrm{Ea}\) of our samples is from 2 to 20 times higher);
3. pre-exponential factor (our samples have pre-exponential factors several orders of magnitude lower);
4. structure (bitumens are highly complex materials--see introduction - contrasting the structure of simple liquids). The data in the present work, despite these big differences, extend the number and types of systems for which this linear correlation holds, suggesting a quite universal behavior although some difference in the slope is still (reasonably) present. The data of the present paper open a door for a more general view of fluids, based on a deeper understanding of the key ingredients dictating their behavior. For this reason, we hope that our data can be precious for further theoretical investigation in order to model such a universal behavior. This correlation lets us to schematically depict the additive effect on the microstructure as a more or less pronounced \"shrinking\" or \"stretching\" of the structure and, consequently, of the microscopic energetic landscape that the system has to explore in the flowing process. This principle is represented in Scheme 1.