## 3 Experimental Methods and Model Parameters
### Critical Volume Fraction of Mineral Filler in Asphalt Mastic  
To investigate the interaction mechanism between asphalt and mineral filler, [PERSON] and [PERSON] classified the variation in the complex shear modulus (\(G^{\star}\)) of asphalt mastic with the volume fraction of mineral filler into two distinct regions: the \"dilution zone\" and the  
Figure 5: Particle size distribution of different types of mineral fillers. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Filler Type**} & \multicolumn{5}{c}{**Volume Fraction in Asphalt Mastic**} \\ \cline{2-6}  & **F/B = 0.4** & **F/B = 0.8** & **F/B = 1.2** & **F/B = 1.6** & **F/B = 2.0** \\ \hline Limestone filler & 11.89\% & 21.26\% & 28.82\% & 35.06\% & 40.30\% \\ Fly ash & 17.46\% & 29.73\% & 38.82\% & 45.83\% & 51.40\% \\ Coal gangue filler & 17.42\% & 29.67\% & 38.76\% & 45.77\% & 51.33\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Volume fraction of different fillers in asphalt mastic. \"aggregation zone\" [39]. In the dilution zone, where the volume fraction of mineral filler is relatively low, asphalt can completely encapsulate the mineral filler particles, preventing any contact between them. The reinforcement effect of the mineral filler in this region primarily results from the volumetric filling effect and the physicochemical interactions between the mineral filler and asphalt. As the volume fraction of mineral filler increases to a specific threshold, the particles, although still largely encapsulated by asphalt, begin to make contact with each other. At this point, a network structure starts to form among the particles, and the corresponding volume fraction is defined as the critical volume fraction. In the aggregation zone, when the mineral filler volume fraction exceeds the critical value, clustering of the mineral filler particles occurs. Asphalt can no longer fully encapsulate the particles, and significant interactions between the particles take place. Therefore, accurately determining the critical volume fraction of mineral filler is crucial. Additionally, when studying the interaction between asphalt and mineral filler, the mineral filler volume fraction should be maintained within the critical volume fraction range. To accurately calculate the critical volume fraction of mineral filler and establish the research range for the interaction between asphalt and mineral filler, this study employed the asphalt mastic model proposed by [PERSON] and [PERSON] [39]. This model predicts the hardening effect of mineral filler on asphalt based on the intrinsic properties of asphalt and filler. The governing equations of the model are expressed as follows:  
\[G_{\rm r}=G_{1}+a_{1}(\phi-\phi_{\rm c})+G_{2}(a_{2}-a_{1})\ln\{1+\exp[(\phi- \phi_{\rm c})/G_{2}]\} \tag{11}\]  
\[G_{\rm r}=\frac{G_{\rm c}}{G_{\rm m}} \tag{12}\]  
where \(G_{\rm r}\) is the shear modulus ratio of asphalt mastic to asphalt (dimensionless); \(G_{\rm c}\) is the complex shear modulus of asphalt mastic (kPa); \(G_{\rm m}\) is the complex shear modulus of asphalt (kPa); \(\phi\) is the volume fraction of mineral filler (%); \(\phi_{\rm c}\) is the critical volume fraction of mineral filler (%); and \(G_{1}\), \(G_{2}\), \(a_{1}\), and \(a_{2}\) are fitting parameters (dimensionless). Equations (11) and (12) served as the fitting functions for determining the critical volume fraction of mineral filler. Nonlinear fitting was performed on the experimental data of the shear modulus ratio of the asphalt mastic at various mineral filler volume fractions. The fitting results, along with the goodness of fit, are presented in Table 4.  
As shown in Table 4, the critical volume fraction (\(\phi_{\rm c}\)) of mineral filler in different types of asphalt mastic ranges from 30% to 45%, aligning well with findings from previous studies [40]. This consistency validates the reliability of the results obtained through the fitting process using Equations (11) and (12).