## 2 Numerical Method and Calculation Procedure
### The Bubble Volume Change Model  
In the MPS-MFAL method, the ideal gas law is adopted to describe the volume change of the bubble. During the bubble growing, the inner vapor is considered saturated, and the temperature of the interface particle can be evaluated according to the vapor's physical property:  
\[T^{nt}_{interface}=T_{sat}(P^{n}) \tag{25}\]  
The construction of the superheated liquid layer before nucleation is taken into consideration in the current research. The bubble wait time is molded by Equation (21) [4]. \[t_{w}=\frac{2}{4\pi\alpha}\bigg{\{}\frac{(T_{w}-T_{\infty})R_{c}}{T_{w}-T_{sat }[1-(2\sigma/R_{c}\rho_{b}h_{lv})]}\bigg{\}}^{2} \tag{26}\]  
where \(R_{c}\) is cavity size which is replaced by the initial bubble size. The bubble wait time refers to the time interval between the departure of the previous bubble from the nucleation site on the heating surface and the generation of the next bubble. During this period, only the liquid layer in close proximity to the superheated wall continues to absorb heat from the heat source, leading to the formation of the superheated layer. In the simulation, we only solve the energy equation (Equation (3) in the revised manuscript), while the bubble volume is not updated during this time (t < tw). At the beginning of the bubble growth, taking \(t_{w}\) = 115 \({}^{\circ}\)C as an example, the temperature distribution of the superheated liquid layer near the nucleation site is shown in Figure 4.  
Then, the heat transfer from the superheated liquid layer into the bubble can be calculated:  
\[Q_{sl}=\rho_{l}c_{p}I_{0}^{2}\sum_{i\in\textit{interface}}\left(T_{i}^{l+ \Delta t}-T_{i}^{l}\right) \tag{27}\]  
The heat transfer of the micro-layer and wicked fluid inside the nano-structure was calculated by Equations (28) and (29), respectively. \[Q_{ml}=q_{ml}^{{}^{\prime\prime}}A_{ml}\Delta T \tag{28}\]  
\[Q_{wick}=q_{wick}^{{}^{\prime\prime}}A_{ml}\Delta T \tag{29}\]  
where \(A_{ml}\) is the area of the micro-layer. Here, we assume that the area of the wicked fluid under the growing bubble is precisely as the same as that of the micro-layer, and the heat transfer of the wicked fluid is considered to be zero after the micro-layer dries out. As the calculation progresses, the micro-layer at the bottom of the bubble gradually dries up, leading to the appearance of a dry area. As shown in Figure 5, the micro-layer particles with a thickness of zero, namely the dry area particles, are represented with gray particles. Define the \(r_{f}\) as the radius of the bubble footprint and \(r_{d}\) as the radius of the dry area; then, the area of the micro-layer \(A_{m}\) in Equations (28) and (29) can be calculated by:  
\[A_{m}=\pi\left(r_{f}^{2}-r_{d}^{2}\right) \tag{30}\]  
Figure 4: Temperature distribution of the superheated liquid layer near the nucleation site when nucleation starts. Figure 5: Distribution of micro-layer and dry area during calculation. Combining those three parts of heat transfer, the volume change of the growing bubble can be obtained:  
\[\Delta V=\frac{Q_{sl}+Q_{ml}+Q_{wick}}{\rho_{v}h_{lv}} \tag{31}\]  
It should be noted that, after the micro-layer evaporates, the dry area appears under the bubble. However, the heat transfer from the dry area into the bubble is too faint to contribute to the bubble growth. Thus, the dry area heat transfer is neglected in this study. The temporal bubble pressure, which is equivalent to the interface particles' pressure, is evaluated by the ideal gas law according to the bubble volume change:  
\[t_{w}=\frac{2}{4\pi\alpha}\bigg{\{}\frac{(T_{w}-T_{\infty})R_{c}}{T_{w}-T_{sat }[1-(2\sigma/R_{c}\rho_{b}h_{lv})]}\bigg{\}}^{2} \tag{26}\]0  
where \(n\) stands for the number of the time step. Substituting the temporal interface particles' pressure \(P^{*}\) into the pressure gradient model, the position of the interface particles in the next time step can be acquired.