## 9 Conclusions  
This paper has compared the original Shan-Chen method, the enhanced Shan-Chen model, and the phase field lattice Boltzmann method for the problem of bubble-dendrite interaction during solidification. The original Shan-Chen model has already been studied for this problem, but the results were limited to small density ratios between the fluid and gas phases, which do not correspond to real physical properties. In this article, to attempt the simulation of higher density ratios and reduce the magnitude of the spurious current, several individual enhancements available for the original Shan-Chen model were, for the first time, used in a combined form to develop an enhanced version of the original model. It was found that the enhancements, including the use of a realistic equation of state, a high order isotropy force scheme, and a mid-range repulsion force, were able to improve the accuracy and capability of the original Shan-Chen method. Even with implementation  
Figure 12: The arrow shows the typical bubble path. (Courtesy of [PERSON] [61]).  
Figure 13: Gas porosity evolution in a directionally solidified Al-30 wt%Cu alloy. (Courtesy of [PERSON], Sintef, [62]).  
of all these enhancements, the spurious currents were still large, which makes the model unsuitable to study the Marangoni effect in its current form and the enhanced versions implemented in this manuscript. A more reliable and accurate alternative is the phase field lattice Boltzmann method. Although computationally more expensive than the Shan-Chen model, it is able to produce accurate results not affected by spurious currents.  
An enhanced model was implemented with the combination of all enhancements to validate the phase separation problem. Different phase separation scenarios regarding the bubbles coalescence were observed based on the selection of model parameters. For the phase field model validation, the Rayleigh instability problem was successfully solved with a low and high-density ratio.  
After validating the developed code, a two-dimensional lattice-Boltzmann--cellular automaton model was utilized to simulate the bubble dynamics during dendritic solidification in a binary alloy. The results showed that the enhancements in the Shan-Chen model increased the ability to simulate higher density ratios and yield more realistic results regarding the final shape and rate of coalescences of bubbles. The enhanced model can generate any contact angle, control the rate of coalescence between bubbles, reduce the spurious current, as well as predict narrow and long worm-like bubble shapes as observed in experiments. However, despite all the enhancements, the Shan-Chen models still generated a high spurious current in the order of 1 mm/s, making them unsuitable for simulation of flow phenomena, such as Marangoni convection. For this case, the more elaborate phase field lattice Boltzmann method can produce accurate results even for density ratios as high as 1000, though at the expense of a much higher computational cost.  
[PERSON], [PERSON] and [PERSON] conceived and designed the simulations; [PERSON] developed the model and performed the simulations; [PERSON], [PERSON] and [PERSON] wrote the paper.  
This research was funded by the National Aeronautics and Space Administration (NASA) grant number NNX16 AT75G.  
The authors would like to gratefully acknowledge helpful discussions with Professor [PERSON] from Cleveland State University and Dr. [PERSON] from NASA Marshall Space Flight Center. The authors also acknowledge the University of Akron and California State University, Los Angeles for their support.  
The authors declare no conflict of interest.