## 2 Pseudopotential MRT LB Model
The parameter \(G\) is set as \(G=-1\) to ensure the whole term is inside the square root positive. The parameter \(c\) is the lattice constant, and \(c_{s}=c/\sqrt{3}\) is the lattice sound speed. The piecewise linear equation of state (EOS) proposed by [PERSON] et al. [28] is applied in the model and is given by:  
\[p(\rho)=\left\{ \begin{array}{ll}\rho\theta_{v}&if\;\rho<\rho_{1}\\ \rho_{1}\theta_{v}+(\rho-\rho_{1})\theta_{m}&if\rho_{1}<\rho<\rho_{2}\\ \rho_{1}\theta_{v}+(\rho_{2}-\rho_{1})\theta_{m}+(\rho-\rho_{2})\theta_{l}&if \;\rho>\rho_{2}\end{array} \right. \tag{9}\]  
where \(\theta_{v}=\sqrt{(\partial p/\partial\rho)_{v}}\) and \(\sqrt{\theta_{l}}=\sqrt{(\partial p/\partial\rho)_{l}}\) represent the speeds of sound the vapor phase and liquid phase, respectively; \(\theta_{m}\) is the slope in the unstable branch \((\partial p/\partial\rho<0)\). The variables \(\rho_{1}\) and \(\rho_{2}\) can be obtained by solving a set of two equations. One equation is ensured for mechanical equilibrium:  
\[\int_{\rho_{v}}^{\rho_{1}}dp=(\rho_{1}-\rho_{v})\theta_{v}+(\rho_{2}-\rho_{1}) \theta_{m}+(\rho_{1}-\rho_{2})\theta_{l}=0 \tag{10}\]and the other is ensured for chemical equilibrium:  
\[\int_{\rho_{v}}^{\rho_{l}}\frac{1}{\rho}d\rho=log\bigg{(}\frac{\rho_{1}}{\rho_{2} }\bigg{)}\rho_{v}+log\bigg{(}\frac{\rho_{2}}{\rho_{1}}\bigg{)}\theta_{m}+log \bigg{(}\frac{\rho_{l}}{\rho_{2}}\bigg{)}\theta_{l}=0 \tag{11}\]  
where \(\rho_{l}\) and \(\rho_{v}\) represent the density of vapor and liquid, respectively. With the pseudopotential \(\psi\), the pseudopotential LB model usually undergoes thermodynamic inconsistency. Subsequently, [PERSON] et al. [20] found that the thermodynamic consistency can be approximately achieved by adjusting the mechanical stability condition via an improved forcing scheme [17]  
\[\mathbf{S}=\left[ \begin{array}{c}0\\ 6\mathbf{v}\cdot\mathbf{F}+\frac{12\sigma\left|\mathbf{F_{m}}\right|^{2}}{ \psi^{2}\delta_{l}\left(\tau_{c}-0.5\right)}\\ -6\mathbf{v}\cdot\mathbf{F}-\frac{12\sigma\left|\mathbf{F_{m}}\right|^{2}}{ \psi^{2}\delta_{l}\left(\tau_{c}-0.5\right)}\\ F_{x}\\ -F_{x}\\ F_{y}\\ -F_{y}\\ 2\big{(}v_{x}F_{x}-v_{y}F_{y}\big{)}\\ v_{x}F_{y}+v_{y}F_{x}\end{array} \right] \tag{12}\]  
where \(\left|\mathbf{F_{m}}\right|^{2}=\left(F_{m,x}^{2}+F_{m,y}^{2}\right)\). The parameter \(\mathbf{F_{m}}\) in Equation (12) denotes the intermolecular interaction force, and \(\mathbf{F}\) is the total force of the system; and \(\sigma\) is used to tune the mechanical stability condition. In this work, from the investigation of the thermodynamic consistency [20,24], \(\sigma\) is set as 0.11625 in the case of cavitation bubble and 0.084 in the case of a droplet on the surface, respectively. The adhesive force at the solid surface should be considered in the pseudopotential MRT LB model. The wettability of a solid surface is decided by the adhesive force in the model. The following fluid-solid interaction [29] to mimic the adhesive force is given by:  
\[\mathbf{F_{ads}}=-G_{w}\rho(\mathbf{x})\sum_{\alpha}\omega_{\alpha}S(\mathbf{ x}+\mathbf{e}_{\alpha})\mathbf{e}_{\alpha} \tag{13}\]  
where \(G_{w}\) is the adsorption parameter, \(\omega_{\alpha}=c_{s}^{2}w_{\alpha}\) and the switch function \(S(\mathbf{x}+\mathbf{e}_{\alpha})\) is \(\psi(\mathbf{x})s(\mathbf{x}+\mathbf{e}_{\alpha})\) which is equal to 1 or 0 for a solid or a fluid phase, respectively. The total force in Equation (12) is given by \(\mathbf{F}=\mathbf{F_{m}}+\mathbf{F_{ads}}\). In this paper, the lattice units for LB are adopted. The basic units of length, time and mass are \(lu\)(lattice unit), \(ts\)(time step) and \(mu\)(mass unit), respectively. Thus, the units of velocity, pressure, and density are expressed as \(lu/ts\), \(mu/(ts^{2}\)\(lu)\) and \(mu/ts^{2}\), respectively.