# Comparative Study on Snowflake Dendrite Solidification Modeling Using a Phase-Field Model and by Cellular Automaton

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Snowflake Model Based on CA

#### 2.2. Snowflake Model Based on PF

## 3. Results and Discussion

#### 3.1. Simulation Results of the CA Model

#### 3.2. Simulation Results of the PF Method

#### 3.3. Comparison of the Simulation Results of the Two Models

#### 3.3.1. Comparison of the Growth Parameters of Two Methods

#### 3.3.2. Verification of Growth Similarity of Two Methods by Growth Rate

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Symbol | Description |
---|---|

$P$ | Order parameter |

$\mathsf{\Phi}$ | Free energy functional |

$\mathrm{K}$ | Dimensionless latent heat |

m | Temperature function |

$\mathrm{T}$ | Dimensionless temperature |

μ | Supercooling coefficient |

${\mathrm{T}}_{\mathrm{e}}$ | Equilibrium temperature |

$\theta $ | Normal angle of the interface |

$\mathrm{F}$ | Double-well potential function |

$\mathsf{\epsilon}$ | Anisotropic parameter gradient energy coefficient |

$\overline{\epsilon}$ | Mean anisotropic gradient energy coefficient |

$\mathsf{\delta}$ | Anisotropic strength |

$\mathrm{j}$ | Anisotropic modulus |

$\mathsf{\tau}$ | Releasing time |

$\gamma $ | Vertical growth parameter |

$\beta $ | Background water density |

${\mathrm{X}}_{0}$ | Noise intensity |

dt | Discrete time |

## References

- Han, Y.; Yang, T.; Chen, Y. A perspective on morphology controlled synthesis of powder by tuning chemical diffusion and reaction. Adv. Powder Technol.
**2020**, 31, 922–925. [Google Scholar] [CrossRef] - Dornbusch, D.A.; Hilton, R.; Lohman, S.D.; Suppes, G.J. Experimental Validation of the Elimination of Dendrite Short-Circuit Failure in Secondary Lithium-Metal Convection Cell Batteries. J. Electrochem. Soc.
**2014**, 162, A262–A268. [Google Scholar] [CrossRef] - Tao, R.; Bi, X.; Li, S.; Yao, Y.; Wu, F.; Wang, Q.; Zhang, C.; Lu, J. Kinetics Tuning the Electrochemistry of Lithium Dendrites Formation in Lithium Batteries through Electrolytes. ACS Appl. Mater. Interfaces
**2017**, 9, 7003–7008. [Google Scholar] [CrossRef] - Liu, Q.C.; Xu, J.J.; Yuan, S.; Chang, Z.W.; Xu, D.; Yin, Y.B.; Li, L.; Zhong, H.X.; Jiang, Y.S.; Yan, J.M.; et al. Artificial Protection Film on Lithium Metal Anode toward Long-Cycle-Life Lithium-Oxygen Batteries. Adv. Mater.
**2015**, 27, 5241–5247. [Google Scholar] [CrossRef] - Tu, Z.; Nath, P.; Lu, Y.; Tikekar, M.D.; Archer, L.A. Nanostructured Electrolytes for Stable Lithium Electrodeposition in Secondary Batteries. Acc. Chem. Res.
**2015**, 48, 2947–2956. [Google Scholar] [CrossRef] [PubMed] - Cheng, X.B.; Zhang, R.; Zhao, C.Z.; Zhang, Q. Toward Safe Lithium metal Anode in Rechargeable Batteries: A review. Chem. Rev.
**2017**, 117, 10403–10473. [Google Scholar] [CrossRef] - Nakaya, U. Snow Crystal: Natural and Artificial; Harvard University Press: Cambridge, MA, USA, 1954. [Google Scholar]
- Ball, P. Material witness: Close to the edge. Nat. Mater.
**2016**, 15, 1060. [Google Scholar] [CrossRef] - Libbrecht, G. The physics of snow crystals. Prog. Phys.
**2005**, 68, 855–895. [Google Scholar] [CrossRef] [Green Version] - Mason, B.J. Snow crystal, natural and man-made. Contemp. Phys.
**1992**, 33, 227–243. [Google Scholar] [CrossRef] - Nelson, J. Growth mechanisms to explain the primary and secondary habits of snow crystals. Philos. Mag. A
**2001**, 81, 2337–2373. [Google Scholar] [CrossRef] - Neumann, V. Theory of Self-Reproducing Automaton; University of Illinois Press: Champaign, IL, USA, 1966. [Google Scholar]
- Reiter, C.A. A local cellular model for snow crystal growth. Chaos Solut. Fractals
**2005**, 23, 1111–1119. [Google Scholar] [CrossRef] - Liu, J.; Dai, J.; Han, C.; Zhang, J.; Ai, J.; Zhai, C.; Liu, X.; Sun, W. Simulation of the Crystallization Process based on Cellular Automaton—Snowflake Formation from Pure Water System. Comput. Aided Chem. Eng.
**2020**, 48, 187–192. [Google Scholar] - Liang, L.; Qi, Y.; Xue, F.; Bhattacharya, S.; Harris, S.J.; Chen, L.-Q. Nonlinear phase-field model for electrode-electrolyte interface evolution. Phys. Rev. E
**2012**, 86, 051609. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wang, S.-L.; Sekerka, R.; Wheeler, A.; Murray, B.; Coriell, S.; Braun, R.; McFadden, G. Thermodynamically consistent phase-field models for solidification. Phys. D Nonlinear Phenom.
**1993**, 69, 189–200. [Google Scholar] [CrossRef] - Kobayashi, R. Modeling and numerical simulations of dendritic crystal growth. Phys. D Nonlinear Phenom.
**1993**, 63, 410–423. [Google Scholar] [CrossRef] - Kim, S.G.; Kim, W.T.; Suzuki, T. Phase-field model for binary alloys. Phys. Rev. E
**1999**, 60, 7186–7197. [Google Scholar] [CrossRef] - Chen, M.; Ou, Z.; Zhou, Y. Optimization of correlative parameters in numerical simulation of ice crystal growth by phase-field. J. Nanjing Univ. (Nat. Sci.)
**2014**, 50, 873–882. [Google Scholar] - Demange, G.; Zapolsky, H.; Patte, R.; Brunel, M. A phase field model for snow crystal growth in three dimensions. npj Computat. Mater.
**2017**, 3, 15. [Google Scholar] [CrossRef] [Green Version] - Barrett, J.; Garcke, H.; Nürnberg, R. Numerical computations of faceted pattern formation in snow crystal growth. Phys. Rev. E
**2012**, 86, 011604. [Google Scholar] [CrossRef] [Green Version] - Akyurt, M.; Zaki, G.; Habeebullah, B. Freezing phenomena in ice–water systems. Energy Convers. Manag.
**2002**, 43, 1773–1789. [Google Scholar] [CrossRef] - Godunov, S.; Bohachevsky, I. Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb.
**1959**, 47, 271–306. [Google Scholar]

**Figure 4.**Water phase diagram [22].

**Figure 5.**A schematic node-ordering representation for a two-dimensional finite-difference grid [23].

**Table 1.**PF parameters [17].

Parameter | Description | Value |
---|---|---|

$\tau $ | Releasing time | 0.0003 s |

$\overline{\epsilon}$ | Mean anisotropic gradient energy coefficient | 0.01 J m^{−2} |

$K$ | Dimensionless latent heat | 1.8 |

$\delta $ | Anisotropic strength | 0.02 |

$j$ | Anisotropic modulus | 6.0 |

μ | Supercooling coefficient | 10.0 |

$\alpha $ | Temperature coefficient | 0.9 |

${T}_{e}$ | Equilibrium temperature | 1.0 K |

${\theta}_{0}$ | Initial angle | 0.5 |

${X}_{0}$ | Noise intensity | 0.01 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dang, Y.; Ai, J.; Dai, J.; Zhai, C.; Sun, W.
Comparative Study on Snowflake Dendrite Solidification Modeling Using a Phase-Field Model and by Cellular Automaton. *Processes* **2022**, *10*, 2337.
https://doi.org/10.3390/pr10112337

**AMA Style**

Dang Y, Ai J, Dai J, Zhai C, Sun W.
Comparative Study on Snowflake Dendrite Solidification Modeling Using a Phase-Field Model and by Cellular Automaton. *Processes*. 2022; 10(11):2337.
https://doi.org/10.3390/pr10112337

**Chicago/Turabian Style**

Dang, Yi, Jiali Ai, Jindong Dai, Chi Zhai, and Wei Sun.
2022. "Comparative Study on Snowflake Dendrite Solidification Modeling Using a Phase-Field Model and by Cellular Automaton" *Processes* 10, no. 11: 2337.
https://doi.org/10.3390/pr10112337