## 4 Results and Discussion
### Verification of Equivalent Substrates  
The comparison of evaporation fluxes on the surfaces of droplets placed on a set of equivalent substrates, as depicted in Figure 4, is presented in Figure 5. The evaporation flux distributions on the surfaces of droplets across different substrates are found to be nearly identical, indicating that the evaporation characteristics of droplets on a set of equivalent substrates are essentially uniform. The comparison of evaporation rates between the equivalent substrates and the reference substrate is shown in Figure 5. To better characterize the equivalence between the equivalent substrates and the reference substrate, a relative error was introduced, defined as the ratio of the absolute difference between the evaporation rate on the reference substrate and that on the equivalent substrate to the evaporation rate on the reference substrate, or the absolute difference between total heat transfer on the reference substrate and that on the equivalent substrate to total heat transfer on the reference substrate:  
\[\varepsilon=\left\lvert Q_{\mathrm{s}}-Q_{\mathrm{s}}{}^{\prime}\right\rvert /Q_{\mathrm{s}} \tag{25}\]  
To further investigate the properties of equivalent substrates, this study calculates the evaporation rates of droplets and the total heat transfer at the solid-liquid interface for a set of equivalent substrates, as shown in Figure 5. The relative errors in evaporation rates between the equivalent substrates and the reference substrate are found to be less than 0.0001, while the relative errors in total heat transfer at the solid-liquid interface are less than 0.02, even as the substrate thickness and relative thermal conductivity vary. Based on these results, it can be concluded that the thermal characteristics of the entire system during droplet evaporation on this set of equivalent substrates are similar. Therefore, the thickness and thermal conductivity of these substrates can be considered thermally equivalent. To verify the applicability of the equivalent substrate phenomenon to different droplet materials, Figure 6 shows the evaporation characteristics of water, methanol, and acetone droplets with an initial contact angle of 30\({}^{\circ}\) on a set of equivalent substrates. The result indicates that different kinds of liquid droplets suit the equivalent substrate phenomenon. Figure 4: (**a**) Evaporation rate lines for droplet evaporation on different substrates, (**b**) dimensionless evaporation rate lines for droplet evaporation on different substrates, and (**c**) the fitting of the data in Figure 4b. Figure 5: Evaporation of a droplet on a set of the equivalent substrates in Table 3. (**a**) Evaporation flux distribution of the equivalent substrates in column 1 of Table 2, (**b**) evaporation flux distribution of the equivalent substrates in column 2 of Table 3, (**c**) dimensionless evaporation rate of the equivalent substrates in column 1 of Table 3, (**d**) heat transfer at the solid–liquid interface of the equivalent substrates in column 2 of Table 3, (**e**) heat transfer at the solid–liquid interface of the equivalent substrates in column 1 of Table 3, and (**f**) heat transfer at the solid–liquid interface of the equivalent substrates in column 2 of Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{l_{\mathrm{R}}}\) & \(\mathbf{k_{\mathrm{RS}}}\) & \(\mathbf{k_{\mathrm{RS}}}\) & \(\mathbf{k_{\mathrm{RS}}}\) \\ \hline
0.05 & 0.06142 & 0.52417 & 0.36541 \\
0.1 & 0.01 & 1 & 0.69712 \\
0.15 & 0.016897 & 1.4345 & 1 \\
0.3 & 0.029963 & 2.5366 & 1.7693 \\
0.45 & 0.040213 & 3.4028 & 2.3722 \\
1 & 0.06308 & 5.2318 & 3.7239 \\
1.5 & 0.07349 & 6.2244 & 4.3391 \\ \hline \end{tabular}
\end{table} 
Table 3: The thickness and the relative thermal conductivity of an equivalent substrate.