## 5 Void Analysis and Mortar Void Percent  
### 2D Void Analysis  
The multiscale approach to asphalt pavement often divides the asphalt mixture into coarse aggregate, mortar and voids [33; 34]. All voids are counted when designing a mixture, but for mortar, some voids are too large. In this study, a simple but effect method is introduced where only the voids smaller than the NMPS are counted as voids in mortar, while voids lager than the NMPS are counted as voids not in the mortar but in the mixture. Figure 3: Otsu segmentation of AC-20: (**a**) scanned; (**b**) coarse aggregate; (**c**) mortar; (**d**) air voids. An analysis of the 3D model followed the same rule [18]. The voids in the mortar example are shown in Figure 4.  
With image processing, each void's details were identified. By conducting the void static using the equivalent diameter, as in Equation (4), the void passing rate at an integer void size was identified. The void passing rates are introduced and shown in Table 3, and the void passing rate can be regarded as the concept of \"void gradation\", which is similar to the mixture aggregate gradation. Void gradation is defined as the passing rate of the void size, similar to the aggregate gradation, and the calculation of the void gradation of the mixture and the results are as follows: Using Table 3, we can obtain the void percent in mortar using Equation (6). The cumulative void percent followed the cumulative Weibull distribution. The cumulative value and regression value of the void gradation of the three mixtures are shown in Figure 5. With the fitting curve, the void percent of any size can be calculated. For mortar with the maximum particle of 2 mm, the AC-13 mortar had 38.6% voids for the total void content of 4.1% in the mixture, while AC-20 had 32.1% of 4.2% and AC-25 had 28.5% of 4.4%. For the mortar with the maximum particle of 4 mm, the AC-13 mortar had 81.4% of the total void content for 4.1% of the mixture, while AC-20 had 73.2% for 4.2% and AC-25 had 62.4% for 4.4%. The cumulative Weibull curve was fitted with the void passing rate and obtained relatively ideal fitting results. With this we could obtain the void percent of the mortar with a common NPMS. The regression of the cumulative Weibull curve of the three mixture's air void sizes is shown in Figure 5, and the void percent results of the commonly used sizes are shown in Figure 6. The parameter v\({}_{\rm r}\) is the void percent (%) which is the percent of air voids that exist in certain types of mortar [29]. Figure 4: Mortar with voids. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Mixture** & \multicolumn{6}{c}{**Void Passing Rate (\%)**} \\ \cline{2-7}
**Type** & **1 mm** & **2 mm** & **3 mm** & **4 mm** & **5 mm** & **6 mm** \\ \hline AC-13 & 13.2 & 38.6 & 63.4 & 81.4 & 91.8 & 96.9 \\ AC-20 & 10.8 & 32.1 & 54.8 & 73.2 & 85.8 & 93.2 \\ AC-25 & 10.9 & 28.5 & 46.6 & 62.4 & 74.9 & 84.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Void passing rate with different void sizes of the different mixtures. The void passing rate of a mixture fitting the cumulative Weibull curve is calculated using Equation (6):  
\[F(x)=\left[1-e^{-(x/\lambda)^{k}}\right]\times 100,\ x\geq 0 \tag{6}\]  
where \(x\) is the void size (mm); \(\lambda\) (mm) and \(k\) (dimensionless) are the shape parameters; and \(F(x)\) is the fitted passing rate responding to size \(x\) (%). The parameter \(\lambda\) controls the void size distribution, and the larger the value of \(\lambda\), the larger the scale of the air void size. Parameter k controls the void size concentration, and the larger the value of \(k\), the greater the concentration of air voids at the middle size. Figure 5 displays the cumulative Weibull curve and fitting curve of the three mixtures. Using these curves, the cumulative void percent can be obtained.