## 2 Numerical Method and Calculation Procedure
### The Micro-Layer Model  
The micro-layer model is one of the main modifications in current research based on [PERSON]'s MPS-MAFL method. [PERSON] et al. [29] proposed a depletable micro-layer model based on the FVM method. The heat and mass transfer part in their model is adopted in the current method, while the simulation of the fluid field inside the micro-layer is not included as indicated above. The schematic of the micro-layer is shown in Figure 1. \(\delta_{0}\) is the initial micro-layer thickness at \(r_{L}\), where \(r_{L}\) is the distance between the nucleation site and the current position. According to [PERSON] et al. 's research [27], the expression for \(\delta_{0}\) is shown as follows:  
\[\delta_{0}(r_{L})=C_{0}r_{L} \tag{12}\]  
in which \(C_{0}\) is a constant that can be only obtained by measurement or numerical experiments, and \(r_{L}\) is the distance between the nucleation site and the position of the virtual micro-layer particle. In the study carried out by [PERSON] et al. [28], they experimentally determined the slop \(C_{0}=4.46\times 10^{-3}\) for DI water in the case of the pool boiling on a heated quartz glass surface at atmospheric pressure. Considering the influence of different working fluids, surface roughness, surface wettability, material properties, and system pressure on the slop \(C_{0}\), we determined \(C_{0}\) as follows:  
(1) We carried out a first calculation with \(C_{0}=4.46\times 10^{-3}\). We compared the calculated results with the experiment. If they were in good agreement, an appropriate \(C_{0}\) was achieved. Otherwise, we went to the next step. We recalculated with the modified \(C_{0}\) and returned to the previous step. The temperature of the upper boundary of the micro-layer is the same as that of the saturated vapor inside the bubble, which is expressed as \(T_{sat}\). The temperature of the micro-layer's bottom boundary is \(T_{vm}\). In the numerical simulation, if the heated surface's temperature \(T_{w}\) is regarded as a known fixed value, the relation of the above three temperature values is equal to the Equation (13). With a known \(T_{sat}\), the \(T_{vm}\) can be calculated. \[\lambda_{\rm s}\bigg{(}\frac{T_{w}-T_{vm}}{l_{0}/2}\bigg{)}=\lambda_{l}\bigg{(} \frac{T_{vm}-T_{sat}}{\delta}\bigg{)} \tag{13}\]  
where \(\lambda_{\rm s}\) and \(\lambda_{\rm l}\) are thermal conductivity for surface and liquid, respectively, and \(l_{0}\) is the interface particle size in the MPS-MAFL method. The arrangement of the initial bubble in the current method is shown in Figure 2. Compared with [PERSON]'s MPS-MAFL method, as indicated by red circles, a set of virtual micro-layer particles are arranged at the bottom of the initial bubble. The local micro-layer thickness is recorded by those virtual particles, and the initial thicknesses are evaluated by Equation (12). The heat flux transferred from the micro-layer into the vapor bubble is calculated as follows:  
\[q_{ml}^{{}^{\prime\prime}}=\frac{T_{vm}-T_{sat}}{\frac{\delta(t)}{\lambda_{l} }+\frac{(2-\gamma)\cdot\sqrt{2\pi RT_{sat}}}{2\gamma\cdot l_{w}^{2}\rho_{c}}} \tag{14}\]  
where \(\gamma\) is the evaporation coefficient of the DI water, which is always set to be 1 [55]. \(R\) is the bubble radius. The first term in the denominator stands for the heat conduction of the micro-layer, and the second term is the evaporation resistance of the micro-layer. Figure 1: Schematic of the micro-layer. Figure 2: Arrangement of the initial bubble in current MPS-MAFL method. The mass flux change rate of the micro-layer is shown in Equation (15). \[\dot{m}^{\prime\prime}_{ml}=\frac{q^{\prime\prime}_{ml}}{h_{l\upsilon}} \tag{15}\]  
Then, the micro-layer's thickness change rate can be obtained:  
\[\frac{d\delta(t)}{dt}=-\frac{1}{\rho_{l}}\dot{m}^{\prime\prime}_{ml} \tag{16}\]  
By considering Equations (12)-(16), transition changes in the micro-layer's thickness and heat flux can be solved.