## 3 Numerical Simulation and Analysis  
### Laplace Law and the Contact Angle  
Laplace law is the validation of the multi-phase model. For the case of the cavitation bubble, the Laplace law can be given as:  
\[\Delta p=p_{in}-p_{out}=\frac{\gamma}{R_{0}}, \tag{14}\]  
where the \(p_{in}\) and \(p_{out}\) are the pressures inside and outside of the bubble, respectively. \(R_{0}\) is the radius of the bubble, \(\gamma\) is the surface tension. The computational domain is 700 \(\times\) 700 lattice system.  
The periodic boundary condition is applied in two directions of the boundary. The density field is initialized as [18]:  
\[\rho(x,y)=\frac{\rho_{l}+\rho_{v}}{2}+\frac{\rho_{l}-\rho_{v}}{2}\times tanh \left[\frac{2\left(\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}}-R_{0 }\right)}{W}\right], \tag{15}\]  
where \(W\) is the prescribed width of the phase interface and it is 5 in the present work; '_tanh_' is a hyperbolic tangent function and it is set as \(tanh(x)=\left(e^{2x}-1\right)/\left(e^{2x}+1\right)\).  
To mimic the experimental value, the density ratio is set as 500 with \(\rho_{l}=500\) and \(\rho_{v}=1\). The other parameters including \(\rho_{1}\), \(\rho_{2}\), \(\rho_{v}\), \(\theta_{m}\) and \(\theta_{l}\) are set as 1.36, 481.04, 0.2133, \(-\)0.01333 and 0.33333, respectively. The lattice constant \(c\) and the time step \(\delta_{t}\) are both set as 1. The bubble is placed at the center of the computational domain. The radius \(R_{0}\) is set as 60, 65, 70, 75 and 80, respectively.  
Figure 1 shows the linear relationship between the \(\Delta p\) and \(1/R_{0}\). It indicates that the simulation results agree well with Young-Laplace law and verify the improved pseudopotential MRT LB model.  
In this paper, the wettability of the surface was measured by the achievable contact angle. It is a general practice that wettability is obtained by measuring the contact angle of a droplet on the solid surface. The contact angle \(\theta>90^{\circ}\) means the surface is hydrophobic and \(\theta<90^{\circ}\) means the surface is hydrophilic. In the computational domain, the 400 \(\times\) 400 lattice system was adopted in our simulations. The open boundary condition was employed at the top boundary. The nonslip boundary scheme [30] was applied at the button boundary \(f_{5}=f_{7}-0.5(f_{1}-f_{3})-0.25\delta_{t}\big{(}F_{x}+F_{y}\big{)}\), \(f_{6}=f_{8}+0.5(f_{1}-f_{3})+0.25\delta_{t}\big{(}F_{x}+F_{y}\big{)}\) and \(f_{2}=f_{4}\). The density field was initialized as [17]  
\[\rho(x,y)=\frac{\rho_{l}+\rho_{v}}{2}-\frac{\rho_{l}-\rho_{v}}{2}\times tanh \left[\frac{2\left(\sqrt{\left(x-x_{l}\right)^{2}+\left(y-y_{l}\right)^{2}}- R_{l}\right)}{W}\right], \tag{16}\]  
where \((x_{l},y_{l})\) is the center of the droplet and is \((200,40)\).  
The parameter \(R_{l}\) which represents the radius of the droplet is initialized as 40. The initial density ratio is 500 with \(\rho_{l}=500\) and \(\rho_{v}=1\).  
In the simulations, the intermolecular interaction force and the fluid-solid interaction force are used at the solid surface. Consequently, a contact angle of \(\theta=90^{\circ}\). can be analytically achieved when \(G_{w}\) is 0 which represents the neutral wettability. The results of contact angles corresponding to different \(G_{w}\) are shown in Figure 2. The obtained contact angle \(\theta\) is \(91.5^{\circ}\) when \(G_{w}\) is set as 0. This  
Figure 1: The validation of [PERSON] law.  
means that the numerical result was in good agreement with the analytical solution. As is shown in the results, the contact angle was approximately proportional to the parameter \(G_{w}\). It may be said that the wettability of the solid surface can be adjusted by \(G_{w}\) in the simulation.