## 4 Results and Discussion  
### The Influence of Substrate Heat Transfer on Droplet Evaporation  
In this study, the impact of substrate thermal conduction on droplet evaporation behavior was investigated, building on the mathematical model established in previous works [21]. Water droplets with a 30\({}^{\circ}\) contact angle were used as a model to study the variations in dimensionless surface temperature, evaporation flux, and evaporation rate \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Droplet** & **Substrate** & \(k_{\text{RS}}\) & _Ec_ \\ \hline \multirow{8}{*}{Water} & PS & 0.13245 & 0.10986 \\  & PU & 0.29801 & 0.10986 \\ \cline{1-1}  & PTFE & 0.41391 & 0.10986 \\ \cline{1-1}  & Glass & 1.5927 & 0.10986 \\ \cline{1-1}  & Ti & 36.258 & 0.10986 \\ \cline{1-1}  & Si & 246.69 & 0.10986 \\ \cline{1-1}  & Al & 392.38 & 0.10986 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensionless parameters for droplet/substrate pair. \begin{table}
\begin{tabular}{c c} \hline \hline
**Refinements** & **Numbers of Elements** \\ \hline
1 & 219,745 \\
2 & 791,510 \\
3 & 1,127,187 \\
4 & 1,788,891 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mesh refinement details. across substrates of differing thermal conductivities, with results presented in Figure 2. Across all substrates, the droplet's surface temperature gradually increased from the center to the contact line, then rose sharply near the contact line, aligning with the substrate's upper surface temperature. Higher relative substrate thermal conductivity (\(k_{\text{RS}}\)) was correlated with elevated droplet surface temperatures and evaporation fluxes, which also increased monotonically from the center to the contact line. The surface temperature distribution thus served as a key indicator of evaporation flux and rate. In the model, the liquid-gas interface acted as a low-temperature heat source due to latent heat, while the substrate's lower surface functioned as a high-temperature heat source. As \(k_{\text{RS}}\) increased, heat transfer from the substrate's lower surface to the solid-liquid interface accelerated, raising the interface temperature and leading to monotonic increases in droplet surface temperature and evaporation flux. Consequently, the evaporation rate increased until it approached that of the evaporative cooling model (Figure 2c). To quantify the substrate's thermal conduction effect, a dimensionless relative evaporation rate (HC\({}_{\text{s}}\)) was defined as the ratio of the evaporation rate in the substrate heat conduction model (\(J\)) to that in the constant-temperature substrate evaporative cooling model (\(J_{\text{EC}}\)). A smaller HC\({}_{\text{s}}\) indicated less heat transfer from the substrate to the droplet and a lower evaporation rate, whereas an HC\({}_{\text{s}}\) closer to 1 signified greater heat transfer, aligning the substrate's thermal conduction with the constant-temperature substrate evaporative cooling model. To further explore the effect of substrate thickness on droplet evaporation through simulations of water droplet evaporation on PS, PU, PTFE, glass, Ti, Si, and aluminum substrates with varying thicknesses and a constant contact angle of 30\({}^{\circ}\). PS, PU, PTFE, and glass, characterized by their extremely low relative thermal conductivities, exhibited low heat transfer rates from the substrate's lower surface to the solid-liquid interface. As the substrate thickness increased, the heat transfer path lengthened, causing a delay in heat delivery to the solid-liquid interface. This inability to compensate for the heat loss due to evaporation resulted in a rapid decrease in droplet surface temperature, as illustrated in Figure 3a. According to Fick's Law, the heat transfer quantity is inversely proportional to the length of the heat transfer path, leading to a corresponding decrease in the droplet's evaporation rate. Under these conditions, the relative evaporation rate (HC\({}_{\text{s}}\)) showed a significant decrease with an increase in the relative substrate thickness (\(h_{\text{R}}\)).