Phase field simulating grain refinement of magnesium alloy by thin strip second phase particles

The study has investigated the grain growth of AZ31 magnesium alloy matrix with fine strip second phase particles, by the phase field model in the real space and time, through introducing free energy equation, and compared with the simulated results containing spherical particles. The results showed that both the thin strip ellipsoidal particles and the cuboid particles have grain refinement effect on the microstructure, moreover, when the content of the second phase particles is the same, the refining effect of thin strip particles could be better than the spherical particles on the matrix microstructure. This study provides the real scale phase field models to research the grain refinement by second phase particles.


Introduction
Magnesium alloy is a very environmentally friendly alloy.As the lightest alloy at present, it has many excellent properties.However, Due to its crystal structure characteristics, its plasticity is poor at room temperature.At the same time, it is very sensitive to stress concentration and has low yield strength, which also makes it unable to be directly used as structural materials.
As is well known, improving material properties could be achieved by refining grain size.such as plastic deformation for grains refinement, however, it has strict requirements for the size of the samples.Second phase particles also could be introduced for grains refinement, and they have the pinning effect at grain boundaries, preventing their migration, thereby enhancing the creep resistance and strength of magnesium alloys [1][2][3][4] .
References [5][6][7][8] pointed out that the spatial orientation of the second phase particles affected the strengthening effect with low concentration, and the experiments showed that the grain boundaries of small grains are more likely to be nailed by the parallel particles.In addition, the existing Zener pinning rule indicates that the larger the particle size, the better the pinning effect of the individual particle on the grain boundaries [9] .
In the paper, The phase field models were built to investigated the grain growth of AZ31 magnesium alloy matrix with fine strip second phase particles, and compared with the simulated results containing spherical second phase particles.The study provides theoretical reference for the establishment of microstructure refinement models containing second phase particles for real alloys.

Model-building principle
The phase field method was based on classical thermodynamic and kinetic theory.Under the comprehensive action of ordering potential and thermodynamics driven, the phase field equation is constructed to complete the model of the system evolution dynamics.The microstructure evolution is determined by the Ginzburg-Landau equation and the Cahn-Hilliard equation: where L is the dynamic coefficient of the interface motion; M is the coefficient of the diffusion mobility; t meas the time; r is the position; p  is the ordering parameters.According to the literature [10] , p is set as 32; c (r, t) is the component field variable; F is the total free energy of the system, it is used as: where K2 is the gradient term coefficient; f0 is the local free energy density function.The literature [10] proposes to introduce a visual function   r  when constructing the free energy density function to describe the particles in the microstructure.
where ϕ is the parameter to describe the second phase particle distribution, and taking 1 inside the second phase particle, and otherwise 0. The expression of f0 in this model is [10] : where c (r, t) is the component of Al; cl is the component content at the lowest position of the free energy component curve at a specific temperature; A, A1 and A2 are the constants related to the free energy of the system; B1 and B2 are the coefficient; and K1 is the intercoupling coefficient [10][11][12][13] .

Initial conditions of the model experiments
In the study, AZ31 magnesium alloy was used as an example.The alloy composition was w (Al) =3%, w (Zn) =1%, and the allowance was Mg.The annealing temperature of the simulation was 350℃.The simulation system was a two-dimensional system, the calculated area was discretized into a square grids, periodic boundary conditions were chosen, the concept of grain boundary range has been chosen in the models [14] .The total size of the simulation was 150 μm×150 μm with 512×512 cells.The recrystallization nucleation is treated by the phenomenological method, The time step of the simulation was taken as 0.3s.

Analysis of the simulation results
Existing experiments [15] have shown that there can be multiple shapes of second phase particles in the microstructure, as is shown in figure 1.

Figure 1.
Al-Mn, Al-Ce-Mn, and Al-Ce phases observed with SEM-EDS line scanning in AZ31+Ce alloy [15] From the figure 1, it is seen that there are second phase particles with elliptical, rod-shaped, and irregular shapes in the microstructure.Therefore, particles with different shapes have been investigated in this study .

Microstructure evolution with ellipsoidal particles
Ellipsoid particles are introduced to explore the influence on the grain growth of magnesium alloy matrix.The results are shown as figure 2. The microstructure of magnesium alloy matrix introducing ellipsoid particles is shown in figure 2. With the passage of annealing time, the numbers of grain boundaries are constantly decreasing, while the average size of grains is constantly increasing, and the particles constantly slide to the grain boundaries and maintain the same spatial orientation.In the later stage of grain growth, the growth rate slows down and the changes in grain size decrease.

Microstructure evolution with cuboid particles
Cuboid second phase particles were introduced to explore their influence on the microstructure.The simulation results are shown in figure 3. It is shown from figure 3 that the trend of grain growth is similar to figure 2: the number of grains is decreasing and the average size is increasing, the spatial orientation of the cuboid particles remains consistent.The changes of grains size are more obvious than that of ellipsoidal particles.

Comparison of thin strip particles and spherical particles
In order to further investigate the effect of particles with different shapes on grain growth, this study compared the microstructure of particles with different shapes at the same annealing time, as shown in figure .4. In figure 4, it is found that, the particles could indeed play the role of refining the grains, which can significantly slow down the growth of grains and reduce the phagocytic behavior between grains, by comparing the microstructure with and without second phase particles.As shown in Fig. 5, the average grain size increases as the annealing time prolongs, and the refinement effect of thin strip particles is better than spherical particles.The refinement effect of cuboid thin strip particles is better than that of ellipsoid particles, at the early stage of growth.When the evolution time reaches 70 min, the grain growth curves containing ellipsoid and cuboid particles intersect, indicating that the two types of particles have a closer effect on grain refinement.

Summary
In the study of the phase field methods for the refinement effect of thin strip particles on the microstructure of magnesium alloys, these conclusions are obtained: 1．Both the ellipsoid thin strip of particles and the cuboid thin strip of particles have grain refinement.
2．The refinement effect of cuboid thin strip particles is better than that of ellipsoid particles, at the early stage of growth.
3．the ellipsoid second phase particles is better than the cuboid thin particles at later stages.4．when content of the particles is the same, the refinement effect of the thin strip second phase particles can better than the spherical particles.

Figure 4 .
Figure 4. Comparison of thin strip second phase particles and spherical particles when t = 40 min: (a) no second phase particles, (b) spherical particles, (c) cuboid particles, (d) ellipsoid particles.

Figure 5 .
Figure 5. Average size of grains with different shape particles.