## 2 Double Distribution TLBM
### Contact Angle and Density Ratio  
In this subsection, we will show that both the contact angle and density ratio can be modified in the model. Note that contact angle and density ratio in multiphase LBM approaches have been intensively studied [21]. More recent work can be found as [44], where high density ratio (liquid to vapor) is achieved using the multiphase model of [PERSON] [45]. The main difference of our work is that temperature field is included which brings extra difficulties, for example, thermal fluid simulations are more susceptible to the numerical instability. This section shows that our TLBM approach is able to modify the contact angle and the density ratio even when the flow and temperature field are coupled. The contact angle is investigated first, which relates to the surface wettability. By changing the density of solid nodes, a different contact angle is obtained. Specifically, hydrophilic surface and hydrophobic surface are achieved by choosing \(\rho_{\rm w}\) that is close to the liquid density and vapor density, respectively. Figure 5 demonstrates how contact angle changes with \(\rho_{\rm w}\), where the temperature \(T\) is set as 0.7. It is observed that the contact angle can be modified within a large range.  
Figure 4: Evolution of the total mass in the pore and the evaporating mass flux at the outlet: the blue solid line is mass flux; the green dotted dash line is the total mass; the implementation of ghost layer boundary condition starts from time \(4\times 10^{4}\).  
Figure 3: Evolution of the evaporating mass flux at the vapor outlet: mini plots demonstrate the density distribution at different time instants; the red part is liquid and the blue part is vapor; the computational domain is \(101\times 200\), and the heat flux imposed to the walls is \(8\times 10^{-4}\).  
Different from contact angle, density ratio of liquid and vapor can be influenced by several factors, for example, \(G\), \(\alpha\) and \(T\). The dependencies of density ratio on \(G\) and \(\alpha\) are illustrated in Figure 6. It is observed that the density ratio is mainly determined by the parameter \(G\). With the increase of \(G\), density ratio increases significantly. Compared with \(G\), the parameter \(\alpha\) contributes slightly to the density ratio. However, it plays a critical role in the sharpness of liquid-vapor interface. A larger \(\alpha\) renders sharper interface, but also makes it vulnerable to numerical instability. Thus a moderate \(\alpha\) is suggested in the simulations. Another factor that affects the density ratio is the temperature \(T\). When \(T\) increases, both liquid and vapor will have smaller density. Note that the vapor is much more sensitive in the density change with respect to the temperature compared with liquid. As a result, the density ratio is also dependent on temperature.  
As a conclusion, the thermal lattice Boltzmann model used in our study is able to mimic different wettabilities. The solid surface can be hydrophilic or hydrophobic. Moreover, the density ratio of liquid and vapor ranges from less than 10 to several hundreds, depending on the parameters of the model.