## 2 Methodology
### Mix Design
#### 2.2.3 Determination of the Optimal Amount of Asphalt  
The appropriate amount of asphalt is crucial for the mixture's strength. In this study, the initial emulsion dosage was determined based on the optimal fluid volume analysis formula for the emulsified asphalt mixture, as recommended by the Cationic Emulsified Asphalt Project Collaboration Group of AASHTO. \[P=0.06A+0.12B+0.2C \tag{1}\]  
where \(P\) stands for the percentage of the mass of the emulsion and the (dry) ore; \(A\) represents the percentage of ore and total ore with a particle size of more than 2.36 mm; \(B\) indicates the percentage of 2.36-0.075 mm particle size in the total amount of ore; and \(C\) stands for the percentage of the rest of the ore in the total amount of ore. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Aggregate Mass Percentage (\%) Passing the Following Sieve Holes (mm)**} \\  & **16** & **13.2** & **9.5** & **4.75** & **2.36** & **1.18** & **0.6** & **0.3** & **0.15** & **0.075** \\ \hline Upper Limit & 100 & 100 & 85 & 68 & 50 & 38 & 28 & 20 & 15 & 8 \\ Lower limit & 100 & 90 & 68 & 38 & 24 & 15 & 10 & 7 & 5 & 4 \\ Median gradation & 100 & 95 & 76.5 & 53 & 37 & 26.5 & 19 & 13.5 & 10 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Design gradation of AC-13 asphalt mixture. Figure 2: Grading curve of AC-13 mixture. After calculation, the initial emulsion dosage was set at 8.7%. Since this dosage was not the actual optimal dosage of emulsion, four groups of emulsion dosages were designed with a difference of 0.5% on the basis of the 8.7% value, namely 7.7%, 8.2%, 8.7%, and 9. Five groups of emulsion dosages of 0.2% and 9.7% were tested, and the optimal emulsion dosage was determined by the Marshall test. For each mixture with the same amount of emulsion, three parallel specimens were prepared at the same time to reduce the error caused by the test. Taking the E20-T mixture as an example, Figure 4 shows the relative density (\(\gamma_{f}\)), Marshall stability (\(MS\)), flow value (\(FL\)), volume of void space (\(VV\)), voids in mineral aggregates (\(VMA\)), and voids filled with asphalt (\(VFA\)). The optimal asphalt content (\(OAC\)) was calculated based on the test data of \(\gamma_{f}\), \(MS\), \(VV\), and \(VFA\), and the calculation formula is shown as follows:  
\[OAC_{1}=(a_{1}+a_{2}+a_{3}+a_{4})/4 \tag{2}\]  
\[OAC_{2}=(OAC_{min}+OAC_{max})/2 \tag{3}\]  
where \(a_{1}\), \(a_{2}\), \(a_{3}\), and \(a_{4}\) are the emulsion dosages corresponding to the maximum value of \(\gamma_{f}\), the maximum value of \(MS\), the median value of \(VV\), and the median value of \(VFA\), respectively. In this study, \(a_{1}\) = 8.7%, \(a_{2}\) = 8.7%, and \(a_{3}\) = 8.1%. Since the specification requires the \(VFA\) to be larger than 65%, the \(a_{4}\) value was the median value after \(VFA\geq\) 65%, corresponding to the emulsion dosage of 9.35%. After calculation, the \(OAC_{1}\) value was 8.71%. To calculate \(OAC_{2}\), it is first necessary to make sure that the \(MS\) value, \(FL\) value, \(VV\) value, and \(VFA\) value are within the requirements of the specification. Under this premise, the emulsion dosages corresponding to the minimum and maximum values of the four indicators are \(OAC_{min}\) and \(OAC_{max}\), respectively. The maximum values of the four indicators obtained from Figure 4 are shown in Table 10. The final \(OAC_{min}\) = 8.95% and \(OAC_{max}\) = 9.7%, so the \(OAC_{2}\) value was (8.95%+9.7%)/2 = 9.33%. The \(OAC\) of the E20-T mixture obtained from the equation was \(\frac{OAC_{1}+OAC_{2}}{2}\) = 9.02%. Figure 3: Appearance of mixture at optimal water consumption. Figure 4: Marshall test indexes under different emulsion dosages. (**a**). the relative density (\(\gamma_{f}\)), (**b**). Marshall stability (_MS_), (**c**). flow value (_FL_), (**d**). volume of void space (VV), (**e**). voids in mineral aggregates (VMA), and (**f**). voids filled with asphalt (VFA). Solid line: connected line; Dotted line: Trend curve.