## 5 Improvements to the Original Shan-Chen Model
### Force with a Higher Order of Isotropy (E8 Force Scheme) and Middle-Range Repulsion Force  
In the original Shan-Chen method, to compute the cohesion force, only eight neighbor nodes are considered. However, the cohesion force can include any number of neighbor nodes. By communicating only with the eight neighboring nodes in the D2Q9 lattice, the highest isotropy order that can be achieved is four (E4 force scheme). By considering the second layer (24 neighbor nodes), tensors of the eighth order can be produced (E8 force scheme) [28; 43].  
So, for the E8 force scheme, instead of using Equation (10), the force term in the D\({}_{2}\)Q\({}_{9}\) lattice is expressed by:  
\[\mathrm{F}(\mathbf{x},\mathrm{t})=-\mathrm{G}_{2}\psi(\mathbf{x},\mathrm{t}) \sum_{\gamma=1}^{24}\omega_{\gamma}\psi(\mathbf{x}+\mathbf{e}_{\gamma}\Delta \mathrm{t},\mathrm{t})\mathbf{e}_{\gamma} \tag{17}\]  
where \(\mathrm{G}_{2}\) is a negative coefficient representing the attractive force, and \(\omega_{\gamma}\) is the weighting coefficient. The direction of \(\mathrm{e}_{\gamma}\) is shown in Figure 1 and values for the weighting coefficients for the E8 force scheme are provided in Equation (18).  
\[\omega_{\gamma}=\left\{\begin{array}{ll}\dfrac{4}{21}&\gamma=1,2,3,4\\ \dfrac{4}{45}&\gamma=5,6,7,8\\ \dfrac{1}{30}&\gamma=9,10,11,12\\ \dfrac{4}{315}&\gamma=13,14,15,16,17,18,19,20\\ \dfrac{1}{2520}&\gamma=21,22,23,24\end{array}\right. \tag{18}\]  
[PERSON] et al. [44] introduced a mid-range repulsion force between fluid particles. In the original Shan-Chen model, the attractive force causes the phase separation. Without adding any repulsive force, after some iterations, only one big bubble (drop) remains in the domain.  
Basically, in Ref. [44], the authors introduced G\({}_{1}\) and G\({}_{2}\) parameters to model cohesion force. The G\({}_{1}\) is negative and represents the attractive force of the first belt of neighboring nodes, while G\({}_{2}\) is positive and represents the repulsive force of the second belt of neighboring nodes. The negative or positive signs here are, respectively, corresponding to attractive and repulsive forces between fluids.  
Therefore, Equation (10) is modified as:  
\[\mathrm{F}(\mathbf{x},\mathrm{t})=-\mathrm{G}_{1}\psi(\mathbf{x},\mathrm{t}) \sum_{\alpha=1}^{8}\omega_{\alpha}\psi(\mathbf{x}+\mathbf{e}_{\alpha}\Delta \mathrm{t},\mathrm{t})\mathbf{e}_{\alpha}-\mathrm{G}_{2}\psi(\mathbf{x}, \mathrm{t})\sum_{\gamma=1}^{24}\omega_{\gamma}\psi(\mathbf{x}+\mathbf{e}_{ \gamma}\Delta\mathrm{t},\mathrm{t})\mathbf{e}_{\gamma} \tag{19}\]  
The pressure and effective mass for both cases are defined as:  
\[\mathrm{p}=\mathrm{c}_{\mathrm{s}}^{2}\mathrm{\rho}+\dfrac{\mathrm{c}_{ \mathrm{s}}^{2}(\mathrm{G}_{1}+\mathrm{G}_{2})}{2}\psi^{2}(\mathrm{\rho}) \tag{20}\]  
and:  
\[\psi=\sqrt{\dfrac{(\mathrm{p}-\mathrm{p}\mathrm{c}_{\mathrm{s}}^{2})}{( \mathrm{G}_{1}+\mathrm{G}_{2})\mathrm{c}_{\mathrm{s}}^{2}}} \tag{21}\]  
Figure 1: Definition of directions for \(\mathrm{e}_{\gamma}\) in the two-belt lattice.  
Note that in the case of the E8 force scheme without mid-range repulsion, \(\text{G}_{1}=0\) and \(\text{G}_{2}<0\). The weighting coefficient and the direction definitions are identical to Equation (18) and Figure 1. By introducing this midrange repulsive force, phenomena, such as spray formation, and the soft-glassy system can be modeled, which is impossible to model by short-range attraction force alone as it is mentioned in Ref [44].  
In this article, the realistic EOS and E8 force scheme with midrange repulsion force were implemented to improve the Shan-Chen model for bubble-dendrite interactions during alloy solidification. To be succinct, this model will be referred to as \"the enhanced model\".