Influence of second-phase particles on grain growth in AZ31 magnesium alloy during equal channel angular pressing by phase field simulation

Phase field methods are based on thermodynamics and kinetics. The temporal evolution of the microstructure can be determined by solving the time-dependent Allen–Cahn equation [27] and Cahn–Hilliard diffusion equation [28] as follows:

$$\begin{eqnarray}&&\frac{\partial {{\eta}_{p}}(r,t)}{\partial t}=-L\frac{\delta F}{\delta {{\eta}_{p}}(r,t)},p=1,2,3\ldots n,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\frac{\partial c(r,t)}{\partial t}=M{{\nabla}^{2}}\frac{\delta F}{\delta c(r,t)},\end{eqnarray} \tag{ 2 }$$

where L and M are structural relaxation and chemical mobility parameters, respectively, η_p(r, t) is the long-range orientation parameter, c(r, t) is the concentration field variable and F is the system's total free energy. The concentration field variable is not independent to the orientation parameter at any position, i.e. the gradient form of the concentration is a function of the gradient form of the orientation parameter. It is equivalent to choose any one of the gradient forms as zero and we choose the concentration one being zero, the same as in [29]. Therefore the expression of an isotropic single-phase system is given by

$$\begin{eqnarray}&&F={{{\int}^{}}_{V}}\left[\frac{1}{2}{{K}_{2}}\underset{p=1}{\overset{n}{\sum}}\,{{\left(\nabla {{\eta}_{p}}\right)}^{2}}+{{f}_{0}}\left(c(r,t),{{\eta}_{1}},{{\eta}_{2}}\ldots {{\eta}_{n}}\right)\right]\text{d}r,\end{eqnarray} \tag{ 3 }$$

where f₀ is the local free energy density function and K₂ is the gradient energy coefficient.

ECAP is such a complicated physical process that any attempt to simulate real experimental results gives a huge unrealistic task. However, a phase field model is a flexible tool for investigating the effect of a mechanism on the process by choosing a proper local free energy density function from the experimental data to fix other mechanisms. Therefore, the local free energy density function f₀ is the key in the phase field model to simulate a specific real industrial alloy. It requires that f₀ should have p degenerate minima located at (η₁, η₂, ..., η_n)  =  (1, 0, ..., 0), (0, 1, ...,0), ..., (0, 0, ..., 1). The following equation for f₀ is suggested by adding an extra term, $f_{0}^{p}=\varepsilon \Phi(r){\sum}_{p=1}^{n}\eta _{p}^{2}(r,t)$, as proposed by Moelans et al, to describe second-phase particles [19, 20]:

$$\begin{eqnarray}&&{{f}_{0}}=A+{{A}_{1}}{{\left(c(r,t)-{{c}_{l}}\right)}^{2}}+\frac{{{A}_{2}}}{4}{{\left(c(r,t)-{{c}_{l}}\right)}^{4}}\\ &&-\frac{{{B}_{1}}}{2}{{\left(c(r,t)-{{c}_{l}}\right)}^{2}}\underset{p=1}{\overset{n}{\sum}}\,\eta _{p}^{2}(r,t)+\frac{{{B}_{2}}}{4}{{\left(\underset{p=1}{\overset{n}{\sum}}\,\eta _{p}^{2}\right)}^{2}}+\frac{{{K}_{1}}}{2}\underset{p=1}{\overset{n}{\sum}}\,\underset{p\ne q}{\overset{n}{\sum}}\,\eta _{p}^{2}(r,t)\eta _{q}^{2}(r,t)+\varepsilon \Phi(r)\underset{p=1}{\overset{n}{\sum}}\,\eta _{p}^{2}(r,t)\end{eqnarray} \tag{ 4 }$$

where c(r, t) is the concentration of all elements in the α-Mg matrix. However, the composition of Zn and O is very low compared with Al and its variation contributes are very small to the total free energy, so we selected concentration of Al only to be calculated for simplification. Therefore the c(r, t) is the concentration of Al and c_l is the concentration of Al at the lowest point of the free energy curve as a function of temperature, and K₁ is the coefficient of coupling between η_p and η_q. Parameter n is the possible number of grain orientations in the system and it was taken as 32, as suggested in the literature [29]. The Φ(r) is a spatially dependent function and independent with time. Inside the particle, the Φ(r) is equal to 1 where the free energy density is a minimum at η_p  =  0; whereas, outside the particle the Φ(r) is equal to 0 where the free energy density is a minimum at η_p  =  1 [20]. The value of ε is set to characterize the ability of different types of particles to hinder grain growth [20]. Here, Al₂O₃ particles are treated as immobile and incoherent particles without coarsening or dissolution and have a strong hindering effect, therefore, we define ε  =  1.

To keep f₀ minimal at (η₁, η₂, ..., η_p)  =  (1, 0, ..., 0), (0, 1, ..., 0), (0, 0, ..., 1), the derivative of equation (4) with respect to η_p ($\partial {{f}_{0}}/\partial {{\eta}_{p}}$), must be zero and the relation between B₁ and B₂ must satisfy

$$\begin{eqnarray}&&{{B}_{2}}={{B}_{1}}{{\left(c(r,t)-{{c}_{l}}\right)}^{2}}.\end{eqnarray} \tag{ 5 }$$

All constants in equation (4) require real physical values to enable the simulation of real industrial alloy AZ31. Therefore, the initial phase could be considered as a solid amorphous state and its free energy may be calculated by the basic crystal free energy plus the volume grain boundary energy. This scheme results in the following equation:

$$\begin{eqnarray}&&\frac{{{E}_{\text{gb}}}}{R}=f\left(\underset{q=1}{\overset{n}{\sum}}\,{{\eta}_{q}}(r,t)=0\right)-f\left(\underset{q\ne p}{\overset{n}{\sum}}\,{{\eta}_{q}}(r,t)=0,{{\eta}_{p}}(r,t)=1\right)\\ &&=\frac{{{B}_{1}}}{2}{{\left(c(r,t)-{{c}_{l}}\right)}^{2}}-\frac{{{B}_{2}}}{4},\end{eqnarray} \tag{ 6 }$$

where E_gb is the grain boundary energy and R is the grain boundary range [25, 26]. By solving equation (6) together with equation (5), where E_gb  =  0.55 J · m⁻² [30] and R  =  0.11 μm, B₁ and B₂ can be determined. The calculated results are B₁  =  106.12 J · mol⁻¹ and B₂  =  3414 J · mol⁻¹.

The free energy concentration curve of Al can be obtained using the Mg–Al–Zn ternary system database in THERMOCALC (Thermo-Calc Software AB, Stockholm Technology Park, Stockholm, Sweden) at different temperatures based on the corresponding experimental data, the same as in previous work [25, 26]. According to the curve, c_l is obtained to equal 0.17 at 523 K but 0.2 at 573 K and 623 K, respectively. The best values for A, A₁, and A₂ were obtained by fitting the curve with equation (4) when ${{\eta}_{p}}\left(\mathbf{r},t\right)=1$ and ${\sum}_{q\ne p}^{n}{{\eta}_{q}}\left(\mathbf{r},t\right)=0$, and the values of the parameters shown in table 1 were used for different temperatures.

The grain boundary range of AZ31 alloy has been suggested to be between 0.1 and 1.2 μm according to its physical characteristics and experimental microstructure measurement reported in the literature [25, 26]. The results of the present simulation indicate that the range should take a value of 1.1 μm for consistency with the experimental results; therefore, K₂ must take a value of 3.0  ×  10⁻¹² J · m² · mol⁻¹ according our simulations. Both K₁ and K₂ substantially affect grain boundary energy determined by simulation [24–26]. The random grain boundary energy was measured to be between 0.5 and 0.6 J · m⁻² for most previously investigated polycrystalline metal systems [30]; thus, here, 0.55 J · m⁻² was set as the target value. By our simulation, we determined that K₁  =  4.0  ×  10⁻² J · mol⁻¹ if the calculated boundary energy satisfies a target value.

The L in equation (1) is the kinetic constant related to the balance effect of deformation and recrystallization on grain growth during ECAP. We used the same method in our previous works [24–26] to obtain the value of L by matching the simulated results with the experimental data [31] for grain growth without Al₂O₃ particles during ECAP. Therefore, the values of L represent the effect of all physical mechanisms on grain growth under experimental conditions, with the exception of the Al₂O₃ particle effect. The values of L were set to 0.61  ×  10⁻⁷, 1.12  ×  10⁻⁷, and 1.72  ×  10⁻⁷ m² · mol · J⁻¹ · s⁻¹ for temperatures of 523 K, 573 K, and 623 K, respectively. The diffusion mobility parameter M in equation (2) was approximated as 3.87  ×  10⁻²⁰ m² · mol · J⁻¹ · s⁻¹ based on literature [32]. Eventually, the physical values of all necessary parameters for the present model were found and are summarized in table 1.

The nucleation of crystallization was simplified by a phenomenological method, and the well-defined grain microstructure was artificially formed after a short time. The initial nucleation state was given as 4dx  ×  4dx unit grids that were evenly distributed in the simulated area, and the radius of each unit of a crystal nucleus was a random value between 0 and 2 grid sizes. In the simulation, we did not introduce second-phase particles during grain nucleation and initial grain growth. The particles were then randomly introduced over the entire AZ31 matrix when average matrix grains grew to a small specific grain size of approximately 3 μm. Therefore, our simulation considers only the grain growth character and ignores the effect of grain nucleation. The initial concentration of Al was set according to the local initial η value. If η  =  1, the local initial concentration was set as 0.03, and the initial concentration at other positions was set as 0.031 to simulate element boundary segregation. Our simulations were conducted in 1024  ×  1024 2D uniform grids. The overall size of the simulation cell was 102.4 μm  ×  102.4 μm; specifically, the size of each cell was 0.1  ×  0.1 μm. Time-step values were selected as 0.2 s in this model to balance between convergence and efficiency. The boundary condition of the phase field was defined as the periodic boundary.

All physical parameters in the present model were determined by physical analysis or experimental as described above so that the microstructure evolution can be simulated in real length and time scales to carry out a comparison with actually measured grain morphology and size.

The shade of gray in the simulated microstructure represents the value of ${\sum}_{p=1}^{n}\eta _{p}^{2}(r)$; i.e. larger values are represented by lighter shades. Therefore, figure 1 shows our simulation results with experimental optical microscopy images reported in the literature [14] at 503 K. The simulated and experimental morphology results clearly match well over a range of particle contents. Furthermore, figure 1(c) shows that the grain size statistical distribution of the simulation and that of the observation match remarkably with the same curves and peaks; such agreement is rarely reported in similar modeling work. This agreement provides strong evidence of the reliability of our model and represents progress in the phase field simulation technique.

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To quantitatively study grain size variation, we simulated the average grain size versus processing time with different Al₂O₃ particle fractions; the results are plotted in figure 2, along with limited experimental data [14] for comparison.

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Figure 2 shows the average simulated grain size versus processing time in the presence of Al₂O₃ particles for several weight fractions. The kinetics of grain growth of the average grain diameter is given by the power growth law as follows:

$$\begin{eqnarray}&&{{D}^{n}}-D_{0}^{n}=kt,\end{eqnarray} \tag{ 7 }$$

where t is the processing time, D is the average grain size in diameter, D₀ is the initial average grain size, n is grain growth exponent and k is a temperature-dependent rate constant. The grain growth during ECAP in this simulation essentially obeyed the power law at n  =  2 in the particle-free case and at n  =  3 in containing 2 wt.% Al₂O₃ particles. Also, it shows relatively better agreement was achieved between the simulations and experiments with low particle contents. However, the simulation results deviate from the experimental data in the case of 2 wt.% particle content; in addition, the average simulated grain diameter is 4.73 μm, but the experimental value is 4.46 μm. The larger grain diameter obtained in the simulation defined by a random, even distribution of the particles implies that Al₂O₃ particles are not randomly distributed in the real alloy but are preferentially located at grain boundaries, which would result in a stronger pinning effect at the boundaries. Figure 2 also indicates that the addition of 2 wt.% Al₂O₃ particles strongly affects grain refinement, reducing the grain size by 28.7% compared to that of the alloy without particles. Furthermore, the results indicate that if the fraction of Al₂O₃ particles is greater than approximately 4 wt.%, any further increase of the particle content will have little effect on grain refinement.

Because of the lack of an accurate expression for grain size variation, a parameter (Δd) has been proposed to describe it: Δd  =  d_max  −  d_min [25], where d_max represents the maximum characteristic grain size, which is the average size of the largest 5% of the grains in a system, and d_min is the minimum characteristic grain size, which is the average size of the smallest 5% of the grains. Grain size variation simulated using experimental results versus the processing time is plotted in figure 3 for the four particle fractions. Notably, grain size variation here is not the deviation of the grain size from the corresponding value on the common normal distribution of the grain size. The variation here is used to represent the degree of grain size fluctuation due to abnormal grain growth. Figure 3 shows that grain size fluctuation increases with increasing processing time and decreases with increasing Al₂O₃ particle fraction. However, grain size fluctuation becomes severe when the particle fraction is below a critical fraction of 1 wt.%, consistent with existing experimental data [14] shown in figure 3 for AZ31 Mg alloy.

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The effect of the second-phase particle size on the grain growth kinetics of AZ31 alloy during ECAP was studied by simulation using two particle volume fractions of 3% and 10%; the selected processing temperature was 623 K. The diameters of the examined particles were categorized into five grades: 100 nm, 300 nm, 500 nm, 800 nm, and 1.3 μm; all particles were randomly introduced in the alloy as an initial condition. The simulated average grain diameter of the alloy is plotted in figure 4 as a function of the processing time for several sizes of particles.

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In AZ31 Mg alloy, it was found that second-phase particles should have a critical size of 0.5–0.8 μm for the grain refinement effect to occur appropriately. If the particle size is smaller than the critical size, large particles will strongly hinder grain growth; in contrast, if the size is larger than the critical size, large particles will exhibit a weaker hindering effect than small particles. This suggests that the particles have to be big enough compared to the critical size to have a full effect to pin up the grain boundary, so that the large particle size has a better effect to refine grain size of the alloy even for less particle numbers. However, if the particle size is over the critical, further increase of the size will contribute nothing in the ability to pin up the boundary so that the alloy with large second-phase particles shows much coarser grains after the ECAP process, because an increase in the particle size for a fixed volume fraction means a decrease in the number of particles, which will result in poor hindrance of grain growth. The critical second-phase particle size on grain refinement is a new finding. This can explain a common puzzle that the enormous number of tiny precipitates in light metals can increase strength remarkably but cannot refine the grain size to contribute to plasticity.

Zener first proposed a model to describe the effect of second-phase particles on polycrystalline grain growth [12]. This model established a quantitative relation among the steady grain size, second-phase particle size, and volume fraction as

$$\begin{eqnarray}&&R=\frac{4r}{3{{f}_{v}}},\end{eqnarray} \tag{ 8 }$$

where R is the steady grain radius, r is the average particle radius, and f_v is the volume fraction of particles. We plotted the ratio of the average grain radius to the particle radius (R/r) as a function of the particle fraction ( f_v) for several sizes of particles using our simulation, as shown in figure 5. As evident in the figure, simulations for micron-scale particles more closely match the Zener model curve at low particle fractions; whereas simulations for nanoscale particles better match the curve at high fractions. These results suggest that nanoscale particles cannot effectively pin up grain boundaries at volume fractions as high as 10%, because the particle distance will be too small to distinguish number difference at a boundary during it moving. However, the simulation results are consistent with the Zener relation shown in figure 5 to support the theoretical argument.

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The distribution of in situ synthesized second-phase particles differs in various matrices. Al₈CeMn₄ rod-shaped particles produced via in situ synthesis in AZ31 were mostly distributed in grain boundaries, Al₈Mn₅ oval-shaped particles were distributed inside grains, and lamellar or oval Al₂Ca particles were distributed inside grains or in grain boundaries [33]. Therefore, an investigation of the effect of different distributions of second-phase particles on the grain growth of AZ31 alloy by simulation is interesting.

The simulations were conducted for two particle volume fractions of 2 and 8% and for spherical particles with diameter of 0.5 μm. The processing temperature was selected as 623 K, and the particles were randomly introduced in AZ31 alloy but with 10%, 30%, 50%, 70%, and 100% of the total particles distributed in the grain boundaries. A simulation result for 150 min of grain growth is presented in figure 6. A comparison of figure 6(a) with figure 6(b) reveals that the presence of more particles distributed in grain boundaries results in finer and more uniform grains of AZ31 alloy after ECAP.

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Additional simulation results are presented in figure 7. With increasing fraction of particles distributed in grain boundaries, the average grain size of the alloy not only decreases but also arrives at a stable grain size much quicker. This result is obtained because only the particles distributed in grain boundaries exert a pinning effect when the grain boundary moves, resulting in a small average grain size. As shown in figure 7(b), when the total particle volume fraction is as high as 8%, the average grain size remains stable and small, although the increase in the number of particles at the grain boundaries can slightly refine the grains. However, as shown in figure 7(a), in the case of a low total particle volume fraction, the average grain size effectively decreases with increasing amount of particles distributed in the grain boundaries; in contrast, in the case of 70% of the particles distributed in the grain boundaries, any further increase in the amount of these particles negligibly affected grain refinement. The reason for this behavior may be that when the amount of particles at the boundary is small, additional available particles located in boundaries improved the pinning effect and resulted in better grain refinement, whereas, when the amount exceeds 70%, many particles are already distributed in boundaries, as a consequence, grain refinement is insensitive if the amount of particles at grain boundaries is further increased.

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To make the simulation closer to experimental conditions, we set three particles with different shapes: spherical particles with a diameter of 1.5 μm, oval-shaped particles with major axis of 2 μm and a minor axis of 0.8 μm, and rod-shaped particles with a width of 0.5 μm and a length of 3.5 μm, where each particle having the same volume. Figure 8 shows that different grain growth rates are obtained for the three types of particles present in the microstructure of AZ31 Mg alloy.

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More simulation results are presented in figure 9 with different particle volume fractions of 2, 4, and 8% for the three particle shapes. The average grain size versus processing time with different shapes and volume fractions of second-phase particles are plotted in figure 9 to reveal the shape effect. When the volume fraction of particles is 8%, rod-shaped particles exert the strongest pinning force, whereas spherical particles exert the weakest pinning force, which is in agreement with another simulation work [22]. However, our simulations showed that spherical particles exert a stronger pinning effect than rod-shaped particles when the volume fraction is low (2%). This result is interesting, but further experiments are needed to confirm it.

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The phase-field model has been established to simulate the effect of Al₂O₃ second-phase particles on grain growth in AZ31 alloy during ECAP. This model is sufficiently flexible to be adapted to simulate other alloy systems during other processes, such as rolling and extrusion, by changing the value of parameters and the expression of f₀. Moreover, the results in this study can be considered as general regularity and can be applied in other alloys.