# Non-Newtonian Behaviors of Crumb Rubber-Modified Bituminous Binders
## 3 Results and Discussion
### Zero Shear Viscosity  
Since bitumen viscosity at relatively low temperatures is highly shear rate dependent, directly comparing the viscosity value at a certain shear rate seems quite tedious. A parameter describing  
Figure 4: Schematic representation of shear-thinning mechanisms (**a**) polymer solutions/melts; (**b**) suspensions; (**c**) dispersions, adapted from [6]. the viscosity that independent of shear rate is plausible. At very low shear rates, Brownian motion dominates and causes all molecules or particles to be oriented randomly despite the initial effects of any applied shear stress. Under this circumstance, structured liquids behave similarly to Newtonian liquids, with a defined viscosity independent of shear rate or angular frequency. This viscosity, measured in the shear deformation when the shear rate approaches zero, is called zero shear viscosity (ZSV). ZSV is reported to be a good indicator of rutting resistance of bituminous binders [32]. Various methods were proposed to estimate the theoretical ZSV value [13]. In this study, data obtained from the frequency sweep tests were applied to the Cross model to determine the ZSV. The Cross model defined in the form of a four-parameter equation (Equation (1)) was able to fit the flow behavior of binders over a wide range of shear rates. \[|\eta^{*}|=\frac{\eta_{0}-\eta_{\infty}}{1+\left(K\omega\right)^{m}}+\eta_{\infty} \tag{1}\]  
in which \(\eta^{*}\) is the complex viscosity; \(\eta_{0}\) is ZSV or the first Newtonian region viscosity; \(\eta_{\infty}\) is infinite shear viscosity in the second Newtonian region; \(\omega\) is angular frequency (rad/s); \(K\) is a material parameter with the dimension of time; and \(m\) is a dimensionless material parameter. In the frequency range tested in this study, it is reasonable to assume that \(\eta^{*}\gg\eta_{\infty}\). Therefore, the Cross model reduces to a three-parameter equation as shown in Equation (2). \[|\eta^{*}|=\frac{\eta_{0}}{1+\left(K\omega\right)^{m}} \tag{2}\]  
Non-linear regression analysis was used to estimate ZSV at 50 \({}^{\circ}\)C with the available data. The measured viscosity and the model predicted results are plotted in Figure 5. The complex viscosities of two binders containing wax additives did not approach a plateau when the shear rate approaches zero. This was observed through the model fitting results that the solver to the solution of binder 70/100-W and CRMB-W cannot converge with defined constraints and convergence. Therefore, there are no practical ZSV values for these two binders. ZSV values and corresponding model parameters for the other four binders are summarized in Table 4. ZSV of CRMB is much higher than the base bitumen 70/100, which indicates superior high-temperature rutting resistance. Besides, ZSV also reveals the molecular weight of binders [6]. CRMB with higher ZSV possesses higher molecular weight than base bitumen due to the rubber-bitumen interaction. Both natural rubber and synthetic rubber in CRMs are macromolecules with much higher molecular weight than bitumen [33]. The formed polymer network within the bitumen matrix after rubber-bitumen interaction increases the viscosity and molecular weight of binder [34]. The addition of chemical additives increases the ZSV of both binders. From the definition of the Cross model, it can be noted that parameter \(m\) is related to shear susceptibility. Based on the data in Table 4, it is inferred that the rheological behavior of a binder with smaller \(m\) is closer to Newtonian behavior. Furthermore, the level of non-Newtonian behavior is reflected in parameter \(K\). The higher value \(K\) is, the higher level of non-Newtonian behavior the binder has. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Binder Type** & **70/100** & **70/100-C** & **CRMB** & **CRMB-C** \\ \hline \(\eta_{0}\) (Pa-s) & 1377 & 1848 & 1.2 \(\times\) 10\({}^{5}\) & 3.55 \(\times\) 10\({}^{5}\) \\ \(K\) (s) & 4.31 & 3.10 & 1191.18 & 2144.81 \\ \(m\) & 0.0644 & 0.0639 & 0.3951 & 0.3924 \\ \hline \hline \end{tabular}
\end{table}