Phase-field lattice Boltzmann simulations of multiple dendrite growth with motion, collision, and coalescence and subsequent grain growth

doi:10.1016/j.commatsci.2018.02.004

Computational Materials Science

Volume 147, May 2018, Pages 124-131

https://doi.org/10.1016/j.commatsci.2018.02.004 Get rights and content

Highlights

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Growth, motion, collision, and coalescence of multiple dendrites and subsequent grain growth are modeled.
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A phase-field lattice Boltzmann model is used for modeling.
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The developed model is validated through simulations.
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Multiple dendrite growth simulations are performed.

Abstract

In the formation of a typical equiaxed structure during the solidification of metals and alloys, multiple equiaxed dendrites typically grow with motion, collision, and coalescence and subsequently grain growth occurs after the formation of grain boundaries. In this study, we develop a phase-field lattice Boltzmann model that can simulate these complex formation processes involving equiaxed structures. In this model, multiple dendrites are represented by employing multiple phase-field variables, and the formation of grain boundaries is modeled by simply introducing an interaction term between the phase-field variables. Liquid flow is computed using the lattice Boltzmann method, and the motion of a solid is described by solving the equations of motion. Collision-coalescence representation in the present model was validated by performing simulations of collisions between two circular objects. Furthermore, grain growth was validated through static and dynamic conditions in a simple three-grain system. Good agreements with theoretical solutions were obtained for both cases. Finally, using the developed model, a series of formation processes of multiple-dendrite growth with motion, collision, and coalescence and the subsequent grain growth are successfully performed for the first time.

Graphical abstract

Introduction

A typical solidification structure consists of columnar and equiaxed polycrystalline structures [1], [2], [3]. It is essential to accurately control and predict the equiaxed structures, especially because they determine the mechanical and other properties of the cast materials. In the formation process of an equiaxed structure, the solids nucleate in a supercooled liquid and they grow into equiaxed dendrites. Importantly, they move in the liquid as a result of forced convection caused by pouring and/or natural convection due to the difference in density between the solid and liquid. During growth of equiaxed dendrites, therefore, they collide with each other and coalesce into a single solid particle. The grain boundaries are then formed and grain growth begins. Finally, the entire region becomes a polycrystalline solid and grain growth proceeds.

The formation of an equiaxed structure has been simulated using a phase-field method [4], which is the most accurate model for describing dendritic growth, in both two dimensions (2D) [5], [6] and three dimensions (3D) [7], [8]. In such simulations, the motion of dendrites was not taken into account. Because the melt convection necessarily occurs in terrestrial solidification, the isolated equiaxed dendrites can move [9], [10], [11], [12]. Considering this fact, some studies on dendritic growth with motion have been presented in recent years [13], [14], [15], [16], [17], [18]. Rojas et al. succeeded in modeling this phenomena by coupling the phase-field method, lattice Boltzmann method, and equations of motion [15]. Besides, through accelerated computations of the model using a graphical processing unit (GPU), Takaki et al. showed that the model can simulate the long-distance motion of growing dendrite with rotation [17]. Although good predictions have been obtained, this model focus on the growth of a single dendrite. Thus, Qi et al. modeled the multiple dendrite growth with motion [18]. A more realistic situation, however, involves the interaction of multiple dendrites undergoing growth, motion, collision, coalescence, and the subsequent grain growth. This complex equiaxed polycrystalline microstructure has not been predicted so far.

In this study, by extending the model developed by Rojas et al. [15] to multiple dendrites, we develop a new phase-field lattice Boltzmann model that can describe the formation process of an equiaxed polycrystalline structure with growth, motion, collision, and coalescence of multiple dendrites and subsequent grain boundary formation and grain growth.

Section snippets

Model

In this study, a 2D isothermal solidification of a binary alloy is assumed unless otherwise stated. The dendrite growth and subsequent grain growth are modeled using the phase-field method [19], [20], the liquid flow is computed using the lattice Boltzmann method, and the motion of a solid is expressed by the equations of motion. The combination of these models has some advantages including the easy implementation on Cartesian grids and the capability of parallel computation.

Validations

Compared to the previous model [15], [17], the collision and coalescence of two solids and the subsequent grain growth are newly introduced in the present model. Here, the validity of the model is assessed. First, we validate the motion after coalescence of two circular solids. Second, we investigate the grain growth in a three-grain system.

Simulations of multiple dendrite growth

By employing the developed model, the isothermal growth of five dendrites within a shear flow and the polycrystalline solidification under continuous cooling are simulated here. An SCN–1 wt% acetone binary alloy with the properties listed in Table 1 is used as a sample material.

Conclusions

We developed a new numerical model that can express the growth of multiple dendrites with motion, collision, and coalescence and subsequent grain growth by extending the previous single-dendrite growth model with motion [15] to the case of multiple dendrites. In this model, the growth of multiple dendrites was expressed by employing multiple phase-field variables, process of grain boundary formation was modeled by introducing an interaction term between the multiple phase-field variables,

Acknowledgements

This research was partly supported by KAKENHI, Grant-in-Aid for Scientific Research (A) 17H01237. This research also used computational resources of the HPCI system provided by Tokyo Institute of Technology through the HPCI System Research Project (Project ID: hp170184). It was also supported in part by MEXT as a social and scientific priority issue (Creation of new functional devices and high-performance materials to support next-generation industries) to be tackled using the post-K computer.

References (40)

Y. Chen et al.
Comput. Mater. Sci.
(2015)
B. Böttger et al.
Comput. Mater. Sci.
(2015)
Z. Guo et al.
Comput. Phys. Commun.
(2015)
C. Yang et al.
Comput. Mater. Sci.
(2017)
B. Appolaire et al.
Acta Mater.
(1998)
A. Badillo et al.
J. Cryst. Growth
(2007)
M. Do-Quang et al.
J. Comput. Phys.
(2008)
D. Medvedev et al.
Procedia Comput. Sci.
(2013)
R. Rojas et al.
J. Comput. Phys.
(2015)
H.G. Lee et al.
Phys. A
(2015)

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Phase-field lattice Boltzmann simulations of multiple dendrite growth with motion, collision, and coalescence and subsequent grain growth

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Model

Validations

Simulations of multiple dendrite growth

Conclusions

Acknowledgements

Comput. Mater. Sci.

Comput. Mater. Sci.

Comput. Phys. Commun.

Comput. Mater. Sci.

Acta Mater.

J. Cryst. Growth

J. Comput. Phys.

Procedia Comput. Sci.

J. Comput. Phys.

Phys. A

J. Comput. Phys.

J. Cryst. Growth

J. Cryst. Growth

Acta Mater.

J. Cryst. Growth

Phys. D

Comput. Mater. Sci.

Acta Mater.

J. Cryst. Growth

Acta Mater.