Large-scale three-dimensional phase-field simulation of multi-variant β-Mg₁₇Al₁₂ in Mg–Al-based alloys

Highlights

• Three-dimensional phase field model for multi-variant precipitation of Mg–Al-based alloy.

• Micrometer scale simulation of precipitates with different orientations.

• Parallel computation and numerical techniques for dealing with phase interface and mesh anisotropy.

Abstract

In this study, a micrometer-scale simulation of the precipitation of multi-variant β-Mg₁₇Al₁₂ phase in Mg–Al based alloy was executed for the first time to explore the possibility of using three-dimensional phase-field models for the design of high-strength magnesium alloys and for the optimization of the heat treatment schedules. To improve the scale and efficiency of the simulation, a method coupling a special interface treatment with parallel coding techniques was adopted to implement the phase-field model. The length scale in the simulation can reach 2.22 μm × 2.56 μm × 2.08 μm. The model showed that the evolution of complex multi-variant precipitates with different orientations in a grain can be obtained. The morphological characteristics, average size, and growth kinetics of the multi-variant precipitates were in good agreement with experimental observations, which is of great significance for the prediction of the mechanical properties of Mg–Al based alloys.

Introduction

The phase-field model based on thermodynamic driving force and ordering potential has emerged as a versatile and powerful tool for the simulation of microstructure evolution, including solidification, precipitate growth and coarsening, martensitic transformations, and grain growth [1], [2], [3]. The phase-field model introduces a set of conserved and non-conserved phase-field variables that are continuous functions of spatial coordinates and time to describe the interface migration better than the front-tracking method without explicit intervention [1], [2], [3], [4]. The phase-field model has a long history in physics and mathematics, and phase-field simulations are often of great interest to physicists because they reveal physical laws. In addition, engineers tend to pay more attention to the tangible products of phase-field simulations than to the laws revealed by models [5], [6]. To actually design a material with the required properties, the processing-microstructure-property relationships must be established in the phase-field simulation to obtain results applicable to real systems, which requires a large length scale. However, in order to maintain diffusive characteristics of the interface in the simulation, its width should not be larger than the sizes of the microstructural elements. Thus, the interface width must be on the nanometer scale, which makes phase-field simulations of realistic physical processes very difficult from the computational point of view [4], [5], [6]. This is the most challenging obstacle preventing large-scale simulations with direct engineering applications.

On one hand, it is important to develop phase-field formulations that allow us to change the interface width without affecting the physical process. To increase the length scale of the simulated system and reduce the computational burden, the interface width must be artificially large. There have been many attempts [7], [8], [9], [10], [11] to extend the interface width and thus the length scale of the simulations. The most prevalent models were proposed by Karma and Rappel [7] and Kim et al. [8]. In particular, the thin-interface model developed by Karma and Rappel allows the interface width to vary depending on the radius of curvature of the interface, and this model has been successfully used for realistic quantitative simulations of solidification. Meanwhile, in the Kim–Kim–Suzuki (KKS) model, the interface region is assumed to be a mixture of the matrix and the precipitate phase, which have identical chemical potentials but different compositions. In KKS simulations, there is no chemical free energy component in the total interface energy. Instead, the interface energy and interface width are entirely determined by a double-well potential and the gradient energy coefficient in the phase-field model. This allows a larger interface width to be used and thus can increase the length scale of the simulation. This model has been widely used for simulations of solidification and has recently been extended to solid-state transformations [9], [10], [11], [12], [13].

On the other hand, increasing the number of numerical elements in the simulation is a straightforward way to expand the scale of the simulated system. However, maintaining diffusive behavior of the interface requires a very fine numerical mesh [3]. Therefore, simulations of real systems are often not feasible because of excessive computation times and insufficient computer memory. Parallel computing and the adaptive mesh technique have been adopted in many phase-field simulations to reduce the computational load and save computer memory. The adaptive mesh technique is often employed for simulations of solidification. However, this technique is usually not appropriate for solid-state transformations because of the strain effects, which need to be resolved using a Fourier-spectral method [3], [5]. Nevertheless, parallel computing can still be used to increase of the size of the simulated system.

Studies of precipitation transformations in magnesium alloys utilizing phase-field models have attracted increasing attention in recent years [13], [14], [15], [16], [17], [18]. Mg alloys are important lightweight materials widely used in the automobile and aerospace industries and thus have been intensively researched using both experiments and models. In the modeling work, the precipitation in Mg–RE alloy has been studied intensively using phase-field simulation [14], [15]. For Mg–Al-based alloy, related research has also been carried out rather recently to reveal the morphology evolution of the β-Mg₁₇Al₁₂ precipitate and reveal the effects of certain factors on the precipitation behavior [13], [16], [17], [18]. All this modeling work on Mg–Al alloys suggests that large-scale three-dimensional (3D) phase-field simulation of multi-variant precipitates in Mg–Al-based alloys is possible and could provide useful information for prediction of the mechanical properties of Mg–Al-based alloys. However, to the best of our knowledge, no such simulation has been reported, and there remains a large gap between the reported phase-field simulations and real systems. Increasing the length scale of the simulated system is also vitally important for understanding the processing-microstructure-property relationships for actual engineering applications. It is also worth noting that using KKS interface treatment to enlarge the length scale in phase field simulation has been achieved in modeling evolution of precipitates in Ni-based alloy [19], and parallel computation is indeed a mature and widely-used method. However, for Mg–Al-based alloy the related researches coupling the two methods have not been reported yet. Coupling the interface treatment with parallel computing to enlarge the length scale of precipitation simulation of Mg–Al-based alloy is a useful attempt to promote the potential application of phase field simulation of this alloy.

Usually, different grains have different orientations. In this study, a 3D phase-field model was developed to simulate the evolution of multi-variant precipitates in a grain in Mg–Al alloy. To increase the size of the simulated system and accelerate the computation, a method of the β-Mg₁₇Al₁₂/α-Mg interface treatment and parallel computing was adopted. The simulation results on the morphology evolution, average size, and growth kinetics of multi-variant precipitates were compared with experimental observations.