## 5 Test Materials and Equipment
### Construction of Rheological Models of Cement Paste Carbonated under Ultrasonication  
(1) Rheological models of cement paste carbonated under ultrasonication  
Based on the above analysis, the rheological curves of cement pastes carbonated under ultrasonication do not all conform to the traditional Bingham, M-B and H-B models as well as those carbonated under mechanical vibration. However, it can be concluded from the \(R^{2}\) values although the fitted curves have low accuracy, they do have some of the characteristics of the rheological models [26, 27, 28]. This is because the above models are ideal and cannot characterise the actual ultrasonication characteristics. On this basis, a practical rheological model that is applicable to cement-based materials carbonated under ultrasonic vibration is proposed by combining characteristics and advantages of the above three models:  
\[\tau=\tau_{0}+\varepsilon*\gamma+\eta*\gamma^{2}+\xi*\gamma^{3}+\zeta*\gamma^{4} \tag{4}\]  
where \(\tau\) is the shear stress; \(\tau_{0}\) is the yield stress; \(\gamma\) is the shear rate; and \(\varepsilon\), \(\eta\), \(\xi\) and \(\xi\) are constants.  
(2) Applications of rheological models of cement paste carbonated by ultrasonication  
To judge whether the rheological model of carbonated cement paste is applicable to the actual rheological model of cement-based materials carbonated under ultrasonic vibration,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample** & **Bingham Model** & \(\mathbf{R^{2}}\) & **MB Model** & \(\mathbf{R^{2}}\) & **H-B Model** & \(\mathbf{R^{2}}\) \\ \hline B\({}_{1}\) & \(\tau_{B1}=3.6+0.35\gamma\) & 0.883 & \(\tau_{B1}=18.75-0.323\gamma+0.0035\gamma^{2}\) & 0.926 & \(\tau_{B1}=8.78+0.0042\gamma^{1.654}\) & 0.868 \\ B\({}_{2}\) & \(\tau_{B2}=3.57+1.15\gamma\) & 0.761 & \(\tau_{B2}=44.07-0.06\gamma+0.0064\gamma^{2}\) & 0.785 & \(\tau_{B2}=28.30+0.0051\gamma^{2.1563}\) & 0.796 \\ B\({}_{3}\) & \(\tau_{B3}=7.12+1.15\gamma\) & 0.783 & \(\tau_{B3}=55.08-0.716\gamma+0.0098\gamma^{2}\) & 0.736 & \(\tau_{B3}=35.24+0.0012\gamma^{2.17}\) & 0.838 \\ B\({}_{4}\) & \(\tau_{B4}=6.21+1.59\gamma\) & 0.812 & \(\tau_{B4}=73.03-0.79\gamma+0.0125\gamma^{2}\) & 0.801 & \(\tau_{B4}=43.62+0.00302\gamma^{1.304}\) & 0.777 \\ B\({}_{5}\) & \(\tau_{B5}=18.7+1.36\gamma\) & 0.768 & \(\tau_{B5}=92.69-1.985\gamma+0.0176\gamma^{2}\) & 0.752 & \(\tau_{B5}=53.74+2.149\gamma^{1.02}\) & 0.788 \\ B\({}_{6}\) & \(\tau_{B6}=-12.22+1.38\gamma\) & 0.741 & \(\tau_{B6}=98.46-1.94\gamma+0.0175\gamma^{2}\) & 0.749 & \(\tau_{B6}=62.89+2.334\gamma^{0.959}\) & 0.767 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Rheological-model parameters of cement slurry carbonated under ultrasonic vibration.  
Figure 6: Rheological models of cement slurry carbonated by ultrasonic vibration. Note: B1–B6 represents rheology at CO\({}_{2}\) AA = 0%, 0.44%, 0.88%, 1.32%, 1.76% and 2.20%, respectively.  
rheological data under different CO\({}_{2}\) AA values were modelled again. The results are shown in Figure 7 and Table 8. Clearly, the \(R^{2}\)-values are >0.9 for all rheological curves of cement pastes with different CO\({}_{2}\) AA values, indicating that these curves meet the fitting requirements well. This proves that the rheological model of carbonated cement paste is appropriate for the actual rheological characteristics of cement-based materials carbonated under ultrasonic vibration.