# Cellular Automaton Simulation of the Growth of Anomalous Eutectic during Laser Remelting Process

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## Abstract

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## 1. Introduction

_{3}Sn, which means the morphologies of discontinuous α-Ni plays an important role in the formation of the anomalous eutectic. In order to avoid the influence of β-Ni

_{3}Sn nucleation, the anomalous eutectic growth of Ni-30wt.%Sn alloy was investigated. This work numerically simulated anomalous eutectic growth during laser remelting of a Ni-30wt.%Sn powder bed [21]. Simulation of anomalous eutectic growth is difficult, requiring many trial and error simulations. The CA model, which has high computational efficiency and relatively simple physical principles, has large potential for scientific and engineering simulations. Compared to the PF model, the computational efficiency in the CA model is the key aspect for us to use the CA model to simulate anomalous eutectic growth. Due to the fact that the remelting-based anomalous growth models are infeasible for anomalous eutectic growth [17], the growth mechanism of the anomalous eutectic is still unknown, and very few simulations of the growth of anomalous eutectic were published in the literatures. The temperature filed during solidification from undercooled melts is extremely complicated, that is why very few simulations are shown for this experimental condition. In the present article, the temperature field at the bottom of melt pool is derived from the thermal simulations, providing better conditions for numerical simulation, which can be simplified into directional solidification. It is shown that the cooling rate has a significant effect on the growth of the anomalous eutectic.

## 2. Materials and Methods

_{l}is the solute concentration in liquid, t is time, D

_{l}is the solute diffusion coefficient in liquid, k

_{α}and k

_{β}are the partition coefficients of α and β phases, respectively, and f

_{s}

_{,α}and f

_{s}

_{,β}are the solid fractions. The solute diffusion in solid phase is neglected in present CA model. The governing equation is solved by an explicit finite difference method.

^{∗}is SL interface temperature, the subscript i = α, β, T

_{E}is the eutectic temperature, m is the liquidus slope, C

_{E}is the eutectic concentration, Γ is the Gibbs–Thomson coefficient, κ is the SL interface curvature, f (φ, θ

_{0}) is representing the SL interface energy anisotropy, φ is the angle between the SL interface normal and the x axis, and θ

_{0}is the angle of crystal orientation to the x axis.

_{E}, m, C

_{E}, and Γ are physical parameters, shown in Table 1. The κ and f (φ, θ

_{0}) are calculated from solid fractions. For isothermal solidification, the SL interface equilibrium composition ${C}_{l}^{*}$ is the only unknown variable in Equation (2), which should be solved during each time step. The local actual liquid composition of each SL interface cell C

_{l}is updated to the local equilibrium composition ${C}_{l}^{*}$. In order to achieve mass conservation, a quantity of mass is provided, resulting in the increment of solid fraction ∆f

_{s}, which is governed by:

## 3. Results

#### 3.1. Anomalous Eutectic Morphologies

_{3}Sn. The α-Ni morphologies can be summarized into four types: globular, globular with a tail, lamellar eutectic, and 1λO pattern, as shown in Figure 1. Compared to eutectic colony solidified from undercooled melt, the anomalous eutectic morphologies during laser remelting process were sandwiched between lamellar eutectic at the bottom of the melt pool after laser remelting Ni-30wt.%Sn powders twice, as seen in Figure 1. The microstructure evolution at the bottom of melt pool can be divided into two processes: one is the lamellar to anomalous transition (LAT), which indicates the anomalous growth from the lamellar eutectic substrate; the other is the anomalous to lamellar transition (ALT), after which, the final microstructure becomes coarser lamellar eutectic. The LAT process is beyond the expected epitaxial growth from the lamellar eutectic substrate. This is contrary to epitaxial dendrite growth from the dendritic substrate. The ALT process is also remarkable in that, as the solidification process continues, lamellar eutectic growth is more competitive than anomalous eutectic.

#### 3.2. Temperature Filed at the Bottom of Melt Pool

_{L}= 2 mm. We selected a lineout along the x axis at the bottom of melt pool, the temperature distribution of which is shown in Figure 2b. It can be seen that the peak point represents the deepest position of the melt pool, on the right of which is heating and remelting, and on the left of which is cooling and solidifying.

_{m}, when Vt < x

_{m}, the cooling rate is R < 0 (heating); when Vt = x

_{m}, R = 0; and when Vt > x

_{m}, R > 0 (cooling). So steady state directional solidification cannot describe the thermal condition here.

#### 3.3. CA Simulation of Anomalous Eutectic Growth

^{6}K/m. The pulling velocity, V, was linearly increased from 0 µm/s to 2000 µm/s in 0.015 s representing a linearly increased cooling rate. The lamellar spacing in experiment was quite small, approximately 0.5 µm, as shown in Figure 1. And the curvature calculation in the present CA model needs at least several grids. So we used small mesh size of 0.01 µm to give quantitative results. Thanks to massively parallel GPU, the simulation finished within 10 h for 4,000,000 steps (≈0.015 s). The eutectic Ni-Sn alloy was used in the present simulations for a more general understanding of anomalous eutectic growth.

_{3}Sn. It can be seen that α-Ni underwent overgrowth of β-Ni

_{3}Sn. After that, the growth of cellular β-Ni

_{3}Sn resulted in the enriched Ni solute in front of the SL interface. Nucleations of α-Ni occurred, as seen in Figure 3a. Therefore, the α-Ni phase was discontinuous at the bottom of melt pool, however the β-Ni

_{3}Sn phase was continuously grown upward, which agreed with the electron back-scatter diffraction pattern (EBSD) analysis [21].

## 4. Discussion

_{3}Sn overgrows α-Ni and subsequently α-Ni particulates nucleate. Why is it not epitaxial growth from the lamellar eutectic substrate? The reason should be that the α-Ni phase has higher SL interface curvature undercooling than the β-Ni

_{3}Sn phase, because the volume faction of α-Ni in the eutectic is f

_{α}= 0.318 [26]. The smaller the volume faction turns the higher the SL interface curvature undercooling. As we mentioned before, the lamellar spacing of the substrate was quite small, approximately 0.5 µm, which was solidified under rapid cooling rate. So when the very fine lamellar eutectic grows at a much smaller cooling rate, it is reasonable to observe the α-Ni overgrowth by β-Ni

_{3}Sn, as shown in Figure 3a. It is indicated that the rapidly changing cooling rate R has great influence on the growth of anomalous eutectic.

_{3}Sn rapidly. Some α-Ni nucleations grew into lamellar eutectic coupled with β-Ni

_{3}Sn. The larger the α-Ni nucleation size, the more they tended to be were wrapped. The smaller α-Ni nucleations became the origin of lamellar eutectic. The growing size of α-Ni nucleations is proportional to the distance from the nucleation position to β-Ni

_{3}Sn SL interface. For instance, the first nucleation in Figure 3a was almost the largest α-Ni particulate, which was nucleated far from the β-Ni

_{3}Sn SL interface.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Backscattered electron images of anomalous eutectic morphologies at the bottom of melt pool after laser remelting Ni-30wt.%Sn powders twice, and its enlarged view of typical morphologies of anomalous α-Ni phase.

**Figure 2.**Numerical simulation of thermal field and lineout at the bottom of melt pool. (

**a**) temperature distribution of melt pool; (

**b**) the temperature lineout at the bottom of melt pool.

**Figure 3.**CA simulations of Ni-Sn anomalous eutectic growth under various cooling rates, and the comparison to experiment results: (

**a**) temperature gradient G = 1.5 × 10

^{6}K/m, pulling velocity V linearly increased from 0 µm/s to 2000 µm/s in 0.015 s; (

**b**) temperature gradient G = 1.5 × 10

^{6}K/m, pulling velocity V = 2000 µm/s; (

**c**) typical anomalous eutectic morphologies of α-Ni.

Constant | Value |
---|---|

Eutectic temperature (T_{E}) | 1403 K |

Eutectic concentration (C_{E}) | 18.7 at.% |

α liquidus slope at T_{E} (m_{α}) | −21 K/at.% |

β liquidus slope at T_{E} (m_{β}) | 37 K/at.% |

α partition coefficient (k_{α}) | 0.57 |

β partition coefficient (k_{β}) | 1.21 |

Diffusion coefficient of solute (D_{l}) | 5.0 × 10^{−9} m^{2}/s |

α Gibbs-Thomson coefficient (Г_{α}) | 2.98 × 10^{−7} mK |

β Gibbs-Thomson coefficient (Г_{β}) | 2.1 × 10^{−7} mK |

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**MDPI and ACS Style**

Wei, L.; Cao, Y.; Lin, X.; Huang, W.
Cellular Automaton Simulation of the Growth of Anomalous Eutectic during Laser Remelting Process. *Materials* **2018**, *11*, 1844.
https://doi.org/10.3390/ma11101844

**AMA Style**

Wei L, Cao Y, Lin X, Huang W.
Cellular Automaton Simulation of the Growth of Anomalous Eutectic during Laser Remelting Process. *Materials*. 2018; 11(10):1844.
https://doi.org/10.3390/ma11101844

**Chicago/Turabian Style**

Wei, Lei, Yongqing Cao, Xin Lin, and Weidong Huang.
2018. "Cellular Automaton Simulation of the Growth of Anomalous Eutectic during Laser Remelting Process" *Materials* 11, no. 10: 1844.
https://doi.org/10.3390/ma11101844