# Numerical Simulation of Microstructure Evolution in Solidification Process of Ferritic Stainless Steel with Cellular Automaton

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Nucleation Model

^{2}) corresponding to the mold wall and in the bulk liquid, respectively, and ${P}_{n}$ is nucleation probability in the cell. During the time step, nucleation will be activated in a liquid cell that needs to meet two conditions—first, the total undercooling in the cell is greater than the critical nucleation undercooling, and second, the nucleation probability is greater than a random number that ranges from 0 to 1. After nucleation, the cell becomes active and grows with a preferential direction corresponding to the randomly crystallographic orientation.

#### 2.2. Solute Field Calculation

#### 2.3. Interface Growth Kinetics

#### 2.4. Calculation of the SL Interface Curvature and the Capture Rules

#### 2.5. The Simulation Principle of Technique

## 3. Results and Discussion

#### 3.1. Equiaxed Dendritic Growth in Undercooled Melt

#### 3.2. The Evolution of Microstructure

^{2}was meshed into CA cells with a size of $\u2206x=2$μm and was initially full of the solute concentration of ${C}_{0}$ at the liquidus temperature. The transient thermal conditions of the microscopic CA grid are interpolated from the result calculated by ANSYS Fluent 17.0, and the dendritic evolution under the vibration-excited liquid metal nucleation technology is predicted with the 2D parallel CA–FD model, as shown in Figure 4.

#### 3.3. Influence of the Surface Undercooling Degree

#### 3.4. Influence of the External Vibration Effect

## 4. Conclusions

- With the increase of surface undercooling on the crystal nucleus generator, the columnar dendrites regions are more developed, and the equiaxed dendrites regions decrease visibly, the number of new-born equiaxed dendrites decreases, and the average grain size increases. The quantitative analysis of the temperature field distribution in the melt shows that the temperature gradient at the columnar crystal front played a critical role in the nucleation and growth of the center equiaxed grain;
- With the vibration frequency increases, the area of the equiaxed dendrites regions is obviously extended, and the refined grains and homogenized microstructure are achieved. With the vibration intensified, the possibility of dendrite fragmentation is increased by increasing the probability of nucleation in the melt; the convection stirring ability of the melt is improved, resulting in uniform temperature gradient distribution and making a large area in the melt to reach the critical nucleation undercooling.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

- Huh, M.Y.; Engler, O. Effect of intermediate annealing on texture, formability and ridging of 17%Cr ferritic stainless steel sheet. Mater. Sci. Eng. A.
**2001**, 308, 74–87. [Google Scholar] [CrossRef] - Park, S.; Kim, K.; Lee, Y.; Park, C. Evolution of microstructure and texture associated with ridging in ferritic stainless steels. ISIJ Int.
**2002**, 408, 1335–1340. [Google Scholar] [CrossRef] [Green Version] - Gan, Y.; Zhao, P.; Wang, M.; Zhang, H. Physical analogue of liquid metal original position nucleation stirred by vibration. J. Iron Steel Res.
**2006**, 8, 9–13. [Google Scholar] [CrossRef] - Zhang, H.; Tao, H.; Li, F.; Wang, M.; Huang, W. Research on mechanism of nucleation in liquid metal excited by vibration. Iron Steel
**2008**, 8, 20–24. [Google Scholar] [CrossRef] - Wang, W.; Chang, X.; Xu, R. Effect of wetting angle on crystal grains nucleation and detachment of vibration chilling surface. Foundry Technol.
**2013**, 34, 1682–1685. [Google Scholar] - Asta, M.; Beckermann, C.; Karma, A.; Kurz, W.; Napolitano, R.; Plapp, M.; Purdy, G.; Rappaz, M.; Trivedi, R. Solidification microstructures and solid-state parallels: Recent developments, future directions. Acta Mater.
**2009**, 57, 941–971. [Google Scholar] [CrossRef] [Green Version] - Liu, B.C.; Xu, Q.Y.; Jing, T.; Shen, H.F.; Han, Z.Q. Advances in multi-scale modeling of solidification and casting processes. JOM
**2011**, 63, 19–25. [Google Scholar] [CrossRef] - Nakajima, K.; Zhang, H.W.; Oikawa, K.; Ohno, M.; Jönsson, P. Methodological progress for computer simulation of solidification and casting. ISIJ Int.
**2010**, 50, 1724–1734. [Google Scholar] [CrossRef] [Green Version] - Reuther, K.; Rettenmayr, M. Perspectives for cellular automata for the simulation of dendritic solidification-A review. Comput. Mater. Sci.
**2014**, 95, 213–220. [Google Scholar] [CrossRef] - Stefanescu, D.M. Microstructure evolution during the solidification of steel. ISIJ Int.
**2006**, 46, 786–794. [Google Scholar] [CrossRef] [Green Version] - Stefanescu, D.M. 30 years of modeling of microstructure evolution during casting solidification. Adv. Mater. Res.
**2007**, 23, 9–16. [Google Scholar] [CrossRef] - Zhu, M.F.; Pan, S.Y.; Sun, D.K.; Zhao, H.L. Numerical simulation of microstructure evolution during alloy solidification by using cellular automaton method. ISIJ Int.
**2010**, 50, 1851–1858. [Google Scholar] [CrossRef] [Green Version] - Smolin, A.Y.; Shilko, E.V.; Astafurov, S.V.; Konovalenko, I.S.; Buyakova, S.P.; Psakhie, S.G. Modeling mechanical behaviors of composites with various ratios ofmatrixeinclusion properties using movable cellular automaton method. Def. Technol.
**2015**, 11, 18–34. [Google Scholar] [CrossRef] [Green Version] - Psakhie, S.G.; Shilko, E.V.; Popov, M.V.; Popov, V.L. Key role of elastic vortices in the initiation of intersonic shear cracks. Phys. Rev. E
**2015**, 91, 063302. [Google Scholar] [CrossRef] [PubMed] - Zhu, M.F.; Hong, C.P. A modified cellular automaton model for the simulation of dendritic growth in solidification of alloys. ISIJ Int.
**2001**, 41, 436–445. [Google Scholar] [CrossRef] [Green Version] - Zhu, M.F.; Kim, J.M.; Hong, C.P. Modeling of globular and dendritic structure evolution in solidification of an Al-7mass%Si alloy. ISIJ Int.
**2001**, 41, 992–998. [Google Scholar] [CrossRef] [Green Version] - Kurz, W.; Fisher, D.J. Dendrite growth at the limit of stability: Tip radius and spacing. Acta Metall.
**1981**, 29, 11–20. [Google Scholar] [CrossRef] - Kurz, W.; Giovanola, B.; Trivedi, R. Theory of microstructural development during rapid solidification. Acta Metall.
**1986**, 34, 823–830. [Google Scholar] [CrossRef] - Zhu, M.F.; Dai, T.; Lee, S.Y.; Hong, C.P. Modeling of solutal dendritic growth with melt convection. Comput. Math. Appl.
**2008**, 55, 1620–1628. [Google Scholar] [CrossRef] [Green Version] - Zhu, M.F.; Lee, S.Y.; Hong, C.P. Modified cellular automaton model for the prediction of dendritic growth with melt convection. Phys. Rev. E
**2004**, 69, 061610. [Google Scholar] [CrossRef] - Zhu, M.F.; Stefanescu, D.M. Virtual front tracking model for the quantitative modeling of dendritic growth in solidification of alloys. Acta Mater.
**2007**, 55, 1741–1755. [Google Scholar] [CrossRef] - Pan, S.Y.; Zhu, M.F. A three-dimensional sharp interface model for the quantitative simulation of solutal dendritic growth. Acta Mater.
**2010**, 58, 340–352. [Google Scholar] [CrossRef] - Zhu, M.F.; Li, Z.Y.; An, D.; Zhang, Q.Y.; Dai, T. Cellular automaton modeling of microporosity formation during solidification of aluminum alloys. ISIJ Int.
**2014**, 54, 384–391. [Google Scholar] [CrossRef] [Green Version] - Yin, H.; Felicelli, S.D. Dendrite growth simulation during solidification in the LENS process. Acta Mater.
**2010**, 58, 1455–1465. [Google Scholar] [CrossRef] - Han, R.; Dong, W.; Lu, S.; Li, D.; Li, Y. Modeling of morphological evolution of columnar dendritic grains in the molten pool of gas tungsten arc welding. Comput. Mater. Sci.
**2014**, 95, 351–361. [Google Scholar] [CrossRef] - Han, R.; Lu, S.; Dong, W.; Li, D.; Li, Y. The morphological evolution of the axial structure and the curved columnar grain in the weld. J. Cryst. Growth
**2015**, 431, 49–59. [Google Scholar] [CrossRef] - Ao, X.; Xia, H.; Liu, J.; He, Q. Simulations of microstructure coupling with moving molten pool by selective laser melting using a cellular automaton. Mater. Des.
**2020**, 185, 108230. [Google Scholar] [CrossRef] - Liu, S.; Shin, Y.C. Integrated 2D cellular automata-phase field modeling of solidification and microstructure evolution during additive manufacturing of Ti6Al4V. Comput. Mater. Sci.
**2020**, 183, 109889. [Google Scholar] [CrossRef] - Rai, A.; Markl, M.; Körner, C. A coupled cellular automaton-lattice boltzmann model for grain structure simulation during additive manufacturing. Comput. Mater. Sci.
**2016**, 124, 37–48. [Google Scholar] [CrossRef] - Beltran-Sanchez, L.; Stefanescu, D.M. Growth of solutal dendrites: A cellular automaton model and its quantitative capabilities. Metall. Mater. Trans. A
**2003**, 34, 367–382. [Google Scholar] [CrossRef] - Beltran-Sanchez, L.; Stefanescu, D.M. A quantitative dendrite growth model and analysis of stability concepts. Metall. Mater. Trans. A
**2004**, 35, 2471–2485. [Google Scholar] [CrossRef] - Dong, H.B.; Lee, P.D. Simulation of the columnar-to-equiaxed transition in directionally solidified Al–Cu alloys. Acta Mater.
**2005**, 53, 659–668. [Google Scholar] [CrossRef] - Luo, S.; Zhu, M.Y. A two-dimensional model for the quantitative simulation of the dendritic growth with cellular automaton method. Comput. Mater. Sci.
**2013**, 71, 10–18. [Google Scholar] [CrossRef] - Wang, W.; Lee, P.D.; McLean, M. A model of solidification microstructures in nickel-based superalloys: Predicting primary dendrite spacing selection. Acta Mater.
**2003**, 51, 2971–2987. [Google Scholar] [CrossRef] - Wang, W.L.; Luo, S.; Zhu, M.Y. Development of a CA-FVM model with weakened mesh anisotropy and application to Fe-C alloy. Crystal
**2016**, 6, 147. [Google Scholar] [CrossRef] [Green Version] - Lin, X.; Wei, L.; Wang, M.; Huang, W.D. A cellular automaton model with the lower mesh-induced anisotropy for dendritic solidification of pure substance. Mater. Sci. Forum
**2010**, 654–656, 1528–1531. [Google Scholar] [CrossRef] - Wei, L.; Lin, X.; Wang, M.; Huang, W.D. A cellular automaton model for the solidification of a pure substance. Appl. Phys. A Mater.
**2011**, 103, 123–133. [Google Scholar] [CrossRef] - Wei, L.; Lin, X.; Wang, M.; Huang, W.D. Orientation selection of equiaxed dendritic growth by three-dimensional cellular automaton model. Physica B
**2012**, 407, 2471–2475. [Google Scholar] [CrossRef] [Green Version] - Wei, L.; Cao, Y.Q.; Lin, X.; Wang, M.; Huang, W.D. Quantitative cellular automaton model and simulations of dendritic and anomalous eutectic growth. Comput. Mater. Sci.
**2019**, 156, 157–166. [Google Scholar] [CrossRef] - Reuther, K.; Rettenmayr, M. A comparison of methods for the calculation of interface curvature in two-dimensional cellular automata solidification models. Comput. Mater. Sci.
**2019**, 166, 143–149. [Google Scholar] [CrossRef] - Wei, L.; Cao, Y.Q.; Lin, X.; Chang, K.; Huang, W.D. Globular to lamellar transition during anomalous eutectic growth. Model. Simul. Mater. Sci. Eng.
**2020**, 28, 065014. [Google Scholar] [CrossRef] - Wang, W.L.; Chen, J.; Li, M.M.; Wang, A.L.; Su, M.Y. Numerical simulation of temperature and fluid fields in solidification process of ferritic stainless steel under vibration conditions. Crystals
**2019**, 9, 174. [Google Scholar] [CrossRef] [Green Version] - Thévoz, P.; Desbiolles, J.L.; Rappaz, M. Modeling of equiaxed microstructure formation in casting. Metall. Mater. Trans. A
**1989**, 20, 311–322. [Google Scholar] [CrossRef] - Wei, L.; Lin, X.; Wang, M.; Huang, W.D. Low artificial anisotropy cellular automaton model and its applications to the cell-to-dendrite transition in directional solidification. Mater. Discov.
**2016**, 3, 17–28. [Google Scholar] [CrossRef] [Green Version] - Gueyffier, D.; Li, J.; Nadim, A.; Scardovelli, R.; Zaleski, S. Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comput. Phys.
**1999**, 152, 423–456. [Google Scholar] [CrossRef] [Green Version] - Sergei, N.; Vladimir, S.; Sergey, K.; Alexey, G.; Victor, G. Formation mechanism of micro- and nanocrystalline surface layers in titanium and aluminum alloys in electron beam irradiation. Metals
**2020**, 10, 1399. [Google Scholar] [CrossRef] - Mizukami, H.; Suzuki, T.; Umeda, T.; Kurz, W. Initial stage of rapid solidification of 18-8 stainless steel. Mat. Sci. Eng. A
**1993**, 173, 361–364. [Google Scholar] [CrossRef] - Yang, M.B.; Pan, F.S. Analysis about forming mechanism of equiaxed crystal zone for 1Cr18Ni9Ti stainless steel twin-roll thin strip. J. Mater. Process. Tech.
**2009**, 209, 2203–2211. [Google Scholar] [CrossRef]

**Figure 1.**The schematic diagram of the vibration-excited liquid metal nucleation experimental apparatus and the simulation principle: 1-cylindrical mold, 2-melt, 3-crystal nucleus generator, 4-water circulation cooling structure, 5-vibration instrument, and 6-power source and transmission. (

**a**) Two-dimensional temperature field distribution along the A–A’ section at different times. (

**b**) Three-dimensional fluid flow trace diagram. (

**c**) Modeling domain for microstructure formation. (

**d**) The full equiaxed dendrites formed at different stages. (

**e**) The process of columnar dendrites and equiaxed dendrites competitive growth.

**Figure 2.**The process of a single equiaxed dendrite growing at the undercooled melt with different undercooling. (

**a**–

**c**) dendrite growing from at $\u2206T=5\mathrm{K}$ after 0.24 s, 0.28 s, 0.32 s. (

**d**–

**f**) dendrite growing from at $\u2206T=10\mathrm{K}$ after 0.12 s, 0.16 s, 0.2 s.

**Figure 3.**Characteristics of the solute distribution and the growth rate of dendrite tip obtained by simulation, at the conditions of $\u2206T=5\mathrm{K}$ and mesh size $\u2206x$= 0.5 μm. (

**a**) Solute distribution in the centerline of the dendrite arm at different times. (

**b**) Partial enlargement of picture (

**a**). (

**c**) The calculated tip growth velocity as a function of time.

**Figure 4.**Simulated the process of dendritic structures formed in the melt; different colors repre-sent grains with different crystallographic orientations. The calculation time is 5.5 s, 6.3 s, 7.2 s, 8.5 s, and 11.0 s, from (

**a**–

**e**), respectively. The coordinate axis in the figure represents the number of grids, and same meaning is expressed in the following pictures.

**Figure 5.**The grain structures obtained by simulation results and experiment. (

**a**) Simulated result. (

**b**) Columnar grain observed by experiment. (

**c**) Equiaxed grain observed by experiment.

**Figure 6.**Simulated dendritic structures formed in the melt under different degrees of undercooling on the surface of the crystal nucleus generator. The degrees of undercooling are 200, 300, 400, 500, 700 K from (

**a**–

**e**), respectively. Different colors represent grains with different crystallographic orientations.

**Figure 7.**Quantitative analysis of temperature field distribution characteristics and grain size information. (

**a**) The temperature gradient distribution between two A and B points, as illustrated in Figure 4a. (

**b**) The cooling rate curve of point A. (

**c**) The cooling rate curve of point B. (

**d**) The average grain size under different degrees of undercooling on the surface of the crystal nucleus generator.

**Figure 8.**The solute field during the dendrite growth process when the undercooling of the crystal nucleus generator is 300 K. The solidification time from (

**a**–

**c**) is 7.2, 7.5, and 7.8 s, respectively.

**Figure 9.**The solute field during the dendrite growth process when the undercooling of the crystal nucleus generator is 500 K. The solidification time from (

**a**–

**c**) is 6.8, 7.0, and 7.2 s, respectively.

**Figure 10.**Simulated the dendritic structures formed in the melt under different vibration frequencies of the crystal nucleus generator. The vibration frequency is 500, 800, 1000, 1500, 1800 Hz from (

**a**–

**e**), respectively. Different colors represent grains with different crystallographic orientations.

**Figure 11.**Quantitative analysis of temperature field distribution characteristics and grain size information. (

**a**) Temperature gradient distribution between A and B positions under different vibration frequencies. (

**b**)The average grain size under different vibration frequencies.

Definition and Symbols | Values |
---|---|

Liquidus temperature, ${T}_{liq}$ (K) | 1781 |

Solute composition, ${C}_{0}$ (wt.%) | 17.0 |

Partition coefficient, ${k}_{e}$ | 0.83 |

Liquidus slope, ${m}_{l}$ (K∙wt%^{−1}) | −2.27 |

Gibbs–Thomson coefficient, $\Gamma $ (K∙m) | 3.0 × 10^{−7} |

Liquid diffusion coefficient, ${D}_{l}$ (m^{2}·s^{−1}) | 5.0 × 10^{−9} |

Solid diffusion coefficient, ${D}_{s}$ (m^{2}·s^{−1}) | 3.0 × 10^{−13} |

Maximum density of nuclei, ${n}_{max}$ (m^{−3}) | 5.5 × 10^{9} |

Mean nucleation undercooling,$\u2206{T}_{n}$ (K) | 7.0 |

Standard deviation of undercooling, $\u2206{T}_{\mathsf{\sigma}}$ (K) | 0.1 |

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

© 2021 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, W.; Shi, Q.; Zhu, X.; Liu, Y.
Numerical Simulation of Microstructure Evolution in Solidification Process of Ferritic Stainless Steel with Cellular Automaton. *Crystals* **2021**, *11*, 309.
https://doi.org/10.3390/cryst11030309

**AMA Style**

Wang W, Shi Q, Zhu X, Liu Y.
Numerical Simulation of Microstructure Evolution in Solidification Process of Ferritic Stainless Steel with Cellular Automaton. *Crystals*. 2021; 11(3):309.
https://doi.org/10.3390/cryst11030309

**Chicago/Turabian Style**

Wang, Wenli, Qin Shi, Xu Zhu, and Yinhua Liu.
2021. "Numerical Simulation of Microstructure Evolution in Solidification Process of Ferritic Stainless Steel with Cellular Automaton" *Crystals* 11, no. 3: 309.
https://doi.org/10.3390/cryst11030309