# Phase Behavior and Rheological Properties of AES/CAPB/H\({}_{2}\)O Ternary System
## 3 Results and Discussion
### Rheological Properties  
The variations in the viscosity (\(\eta\)) versus the shear rate (\(\dot{\gamma}\)) for the three samples in Table 2 were measured in the shear rate range of 0.01-100 s\({}^{-1}\), as shown in Figure 7. It can be seen that the viscosity of all three samples decreased with increasing shear rate, exhibiting the characteristics of a shear-thinning fluid. Although the viscosities of the three samples were significantly different at low shear rates (sample 3 \(>\) sample 2 \(>\) sample 1), their viscosities were similar at high shear rates (100 s\({}^{-1}\)), and all decreased to around 1 Pa-s. The data from the three samples were further fitted and analyzed using Cross-model Equation (3), and the obtained zero-shear-rate viscosity (\(\eta_{0}\)) and R\({}^{2}\) are summarized in Table 2.  
\[\frac{\eta-\eta_{\infty}}{\eta_{0}-\eta_{\infty}}=\frac{1}{1+\left(\lambda \dot{\gamma}\right)^{n}} \tag{3}\]  
In addition, when the shear rate was in the range of 0.01-1 s\({}^{-1}\), the viscosities of both samples 1 and 2 decreased first (shear rate: 0.01-0.1 s\({}^{-1}\)) and then tended to a constant value (shear rate: 0.1-1 s\({}^{-1}\)). It indicates that the entanglement or multi-connected effect between the surfactant molecules in sample 1 and sample 2 is insufficient or uneven. Therefore, even at very low shear rates, part of the entangled or multi-connected structure is destroyed by shear, resulting in a certain degree of viscosity drop. The viscosity of sample 3 appears to be a constant value in the range of 0.01-1 s\({}^{-1}\), which is similar to the first Newtonian zone of polymer fluids, indicating that the entanglement or multi-connected effect is strong enough to overcome the destructive effect of low shear (0.01-1 s\({}^{-1}\)). Strain sweep test was performed in the shear strain range of 0.1% to 1000% with a fixed frequency of 1 Hz to investigate the changing trends of elastic modulus (G\({}^{\prime}\)) and loss modulus (G\({}^{\prime\prime}\)) with shear strain. As shown in Figure 8, with the increase in CAPB (38%) from 8 wt.% to 10 wt.%, both G\({}^{\prime}\) and G\({}^{\prime\prime}\) increased (sample 3 \(>\) sample 2 \(>\) sample 1), and the G\({}^{\prime}\) and G\({}^{\prime\prime}\) of sample 3 (10 wt.% CAPB (38%), Figure 5(c)) were much higher than that of sample 1 (8 wt.% CAPB (38%), Figure 7(a)) and sample 2 (9 wt.% CAPB (38%), Figure 7(b)). In addition, the G\({}^{\prime}\) and G\({}^{\prime\prime}\) of the three samples all decreased under the high shear strain (\(>\)100%). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & **CAPB (38\%)** & **AES (70\%)** & **H\({}_{2}\)O** & **Viscosity** & \multirow{2}{*}{**\(\eta_{0}\) ***} & \multirow{2}{*}{**R\({}^{2}\)**} \\  & **(wt.\%)** & **(wt.\%)** & **(wt.\%)** & **(Pa \(\cdot\)s)** & & \\ \hline
1 & 8 & 12 & 80 & 5.80 & 5.99 & 0.9922 \\
2 & 9 & 12 & 79 & 8.06 & 9.13 & 0.9977 \\
3 & 10 & 12 & 78 & 17.53 & 18.69 & 0.9961 \\ \hline \hline \end{tabular} All samples contained 1.2 wt.% NaCl, 0.10 wt.% EDTA-2 Na and 0.15 wt.% citric acid. * The fitting range of the shear rate is 0.1–100 s\({}^{-1}\).  
\end{table} 
Table 2: Compositions and viscosities of samples 1, 2, and 3. Figure 8: Storage modulus (G\({}^{\prime}\)) and loss modulus (G\({}^{\prime\prime}\)) versus strain sweep of the (**a**) sample 1, (**b**) sample 2, and (**c**) sample 3. Figure 7: Viscosity versus the shear rate for sample 1, sample 2, and sample 3. Interestingly, for sample 1 (Figure 8a) and sample 2 (Figure 8b), G\({}^{\prime}\) was lower than G\({}^{\prime\prime}\) over all tested shear strain ranges, indicating a dominant role of viscosity. For sample 3 (Figure 8c), when the shear strain was less than 200%, G\({}^{\prime}\) was greater than G\({}^{\prime\prime}\), and the elasticity dominated. It indicates that when CAPB (38%) reaches 10 wt.%, the entanglement or multi-connected effect of micelles is significantly enhanced to form a cross-linked structure, resulting in the formation of a gel-like state.