## 4 Study of the Microscopic Void Characteristics of PAC
### Void Characteristics Analysis
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Sample Number**} & \multicolumn{3}{c}{**Void content (\%)**} & \multirow{2}{*}{\(N\)} & \multirow{2}{*}{\(d\) (mm)} & \multirow{2}{*}{\(D_{B}\)} \\ \cline{2-2} \cline{5-6}  & **Test** & & & & \\ \hline A1 & 16.29 & 16.17 & 49.1 & 4.1 & 1.237 \\ A2 & 18.51 & 17.97 & 43.9 & 4.9 & 1.251 \\ A3 & 20.78 & 20.71 & 42.0 & 5.2 & 1.267 \\ A4 & 23.51 & 23.13 & 37.6 & 5.8 & 1.304 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Void characteristics of PAC. For samples A1-A4, Figure 5 shows the relationship between the void content determined and the void structural parameters, including the equivalent diameter and the average number of voids within each image. The measured void content is represented by the horizontal coordinate \(x\), while the double vertical coordinates \(y_{1}\) and \(y_{2}\) represent the void equivalent diameter and the average number of voids within each image, respectively. For \(y_{1}\) and \(y_{2}\), linear fits are performed and the resulting relational equation and correlation coefficient \(R^{2}\) are shown in Figure 5. The number of voids inside the PAC decreases as the void content increases but the void equivalent diameter continuously increases, which reveals that the void content of PAC increases not because the number of voids increases but because the void size increases. The relationships between the above parameters and the peak and mean values of the SAC (named \(\alpha\) in the figures) of the PAC are shown in Figure 6a,b. The results follow the same pattern as the improvement in noise reduction performance caused by increasing the macroscopic void content. Although an increase in void content reduces the number of voids, the fitted results show that the sensitivity of the SAC with the equivalent diameter is greater than that of the number of voids. This indicates that while the number of voids decreases, the equivalent diameter demonstrates a greater increase in the noise reduction effect, which proves that an increase in void diameter can increase the incident acoustic waves and provide a larger internal surface area for sound energy reflection and frictional attenuation. When the two void parameters change, the peak value of the SAC changes more than the mean value. It can be seen that as long as the PAC's design void content requirements (16-25%) are met, PAC may provide good noise reduction in most frequency bands. The geometric complexity of the void structure is described by the fractal dimension. Figure 7 shows the relationship between the void content and the average fractal dimension of all images of each sample. As the void content increases, \(D_{B}\) increases, showing that as the complexity of the void morphology improves, the structure of the internal voids of the PAC becomes more complicated and the internal roughness rises. As shown in Figure 8, the noise reduction impact improves as the fractal dimension grows, which results from a more complicated void structure, leading to a greater chance of acoustic wave reflections when the waves enter the interior void structure. Due to the increase in the fractal dimension, the interface between the acoustic waves and the void surface is increased, providing a larger friction area. This causes greater viscous loss generated between the air and the void wall, resulting in a higher SAC. The fractal dimension of the voids can explain the variation in SAC with \(d\) and \(N\). As the void content increases, \(N\) decreases, which seems to reduce the space where sound energy is lost. However, the increase in \(d\) increases the sound wave incidence and  
Figure 5: Relationship between \(d\), \(N\) and the void content. increases with increasing void content, which enhances the amount of sound energy loss inside the structure; the overall sound absorption and noise reduction performance of the structure is improved. The three characteristics, \(d\), \(N\) and \(D_{B}\), are strongly correlated with the SAC.