Estimation of the Critical Value of the Second-Phase Particles in the Microstructure of AZ31 Mg Alloy by Phase-Field Methods

Abstract

In this study, phase-field models were employed to simulate the effects of second-phase particles (SPPs) on grain growth of the AZ31 Mg alloy, under realistic spatial and temporal scales, at 350 °C, during annealing. The particle sizes ranged from 0 to 7 μm, and the particles with large volume fractions were used in the paper. The results reveal that the volume fractions and sizes of the SPP affect grain growth and that the volume fractions and sizes of the SPP on pinning exhibited critical values. When the SPP volume fraction is f = 5%, the SPP is at the maximum critical size, $r{\mathsf{\mu }\mathrm{m}}_{max}$; when the SPP size is $r=1 \mathsf{\mu }\mathrm{m}$, the SPP minimum critical volume fraction is f_min = 0.25% and the maximum critical volume fraction is f_max = 20%. The critical values increase with the increase of the sizes or volume fractions of the second-phase particles. Finally, the average grain size, particle size, and particle volume fraction obtained from the simulation were fitted according to the Zener relationship, and the obtained results showed that the fitting indices were in the range of 0.33–0.50. The results were compared with the experimental results. The simulation results obtained in this study will provide an important academic reference for understanding the mechanism and law of grain growth, an important reference for accurate control of grain size and properties of the material, a reference for the development of the annealing treatment process of Mg alloy, and a theoretical guide for the use of recrystallization process to control the microstructure of Mg alloy and improve the plastic-forming properties.

1. Introduction

Mg and its alloys, as the light metals at present, are known as the “modern green engineering materials” [1,2]. So, the vigorous development and application of Mg alloys for the alleviation of the shortage of domestic Fe, Al, and other metal mineral resources not only has great significance [3,4,5,6], but it will also play an irreplaceable important role in promoting the realization of the national strategy of becoming carbon neutral [7,8]. Among many Mg alloys, the AZ system is the earliest and most widely used Mg alloy, among which AZ31 Mg alloy is the most typical. However, the crystal structure of Mg is HCP, and there are few slip systems at room temperature, resulting in insufficient strength, stiffness, plasticity, and wear resistance, as well as an especially high temperature resistance, which limits their application.

As we know, the microstructure of the Mg alloy is one of the key factors that determine the mechanical properties according to the Hall–Petch formula [9]:

$${\sigma }_s={\sigma }_0+K{d}^{-1/2}$$

(1)

where ${\sigma }_s$ is the material yield strength; ${\sigma }_0$ is the constant, the yield strength of a single crystal independent metal; $d$ is the average diameter of the grain; and $K$ is a constant related to the property and grain size of materials. The strength and plasticity of the material are significantly improved with the reduction of the grain size. On the other hand, the crystallographic symmetry of the Mg alloy is low; while $K_{\mathrm{Mg}}\approx 280\sim 320 \mathrm{MPa}/\mathsf{\mu }\mathrm{m}$, it is much larger than that of BCC and FCC metals. Therefore, grain refinement can significantly improve the mechanical properties of Mg alloys.

Previously, Zener [10,11,12] investigated the effect of randomly distributed immobile hard SPPs on grain growth and derived the Zener theoretical model. The model [13] had spherical, mono-sized, and randomly distributed SPPs, which exhibited grain refinement. So, using SPPs as a reinforcing phase to prepare Mg-based composites is considered to be one of the most feasible methods to improve the comprehensive mechanical properties of Mg alloys.

Wang et al. fabricated pure Ti and Al₂O₃ particles–reinforced AZ31-based composites by using the friction stir processing, and the microstructure and tensile properties of these AZ31/Al₂O₃ and AZ31/Ti composites were examined and compared; the grain refinements and the corresponding hardening mechanisms were also analyzed and discussed. He et al. studied the grain growth of Mg alloys by using phase field models, and the authors found that, when the volume fraction of the second-phase particles is large, increasing the volume fraction of the particles will not further significantly refine the matrix grains; however, they did not further analyze its critical value.

It is well-known that it is very costly and expensive to rely solely on experiments, as it is often difficult to obtain accurate results due to the large number of trial-and-error experiments and the limitations of experimental conditions and equipment. Simulation can solve this problem very well. The common simulation methods are the front-tracking method [14,15], Monte-Carlo method [16,17], and phase field simulation method [18,19,20,21,22]. Compared with the other two simulation methods, the phase field method does not need to track the grain boundary position in the simulation of grain growth, and it is easy to deal with the problems of solute aggregation and second-phase precipitation on the grain boundary, so the phase field method was chosen for the study.

Studies using phase field models to simulate the second-phase particles’ pinning have also been reported. Lai et al. [21] studied the morphology and kinetics evolution of the Ni3Al phase during re-dissolution in Ni-15% Al alloys; the changes of the volume fraction and average particles’ radius of the Ni3Al phase were clarified with the phase field simulation. Sato et al. [22] studied the retardation effect of dispersed inert particles on δ—γ interface migration in carbon steel, during isothermal δ to γ transaction, using two-dimensional phase field simulations; they focused on the effects of the particles’ radius, particle spacing, and initial carbon concentration of δ phase, and the important findings are summarized. However, these studies either did not achieve real timescale and space-scale simulations, or the simulated timescales were too small to reach the industrial scale.

In this study, multi-order parametric phase-field models were used to simulate the grain growth of AZ31 Mg alloy containing micron-sized second-phase field on real space scales and timescales. The effects of the second-phase-field size and volume fraction on the microstructure, as well as the corresponding critical values of these two factors to achieve fine-grain strengthening through pinning, were investigated. The simulation results obtained in this study provide important academic references for understanding the mechanism and law of grain growth containing the second-phase particles and provide important reference values for accurate control of grain size and characteristics of micron materials.

2. Model Description

The phase-field model [23,24] is based on thermodynamics, which constructs a model of phase-field equations to describe the dynamics of system evolution by considering the combined effects of the ordering potential and thermodynamic driving force. The core idea [25] of the model is to transform the sharp interface problem in the process of tissue transformation into a diffuse interface problem by introducing one or more continuously varying order parameters, as doing so avoids the complex and tedious mathematical processing problems associated with real-time tracking of interfaces and greatly reduces the workload. The temporal evolution of the microstructure can be determined by the evaluation of the time-dependent Allen–Cahn equation and Cahn–Hilliard diffusion equation as follows [26]:

$$\frac{\partial {\eta }_p\left(\mathit{r},t\right)}{\partial t}=-L\frac{\delta F}{\delta {\eta }_p\left(\mathit{r},t\right)},(p=1,2,3,\dots ,n)$$

(2)

$$\frac{\partial c\left(\mathit{r},t\right)}{\partial t}=M{\mathsf{\nabla }}^2\frac{\delta F}{\partial c\left(\mathit{r},t\right)}$$

(3)

where L is the interfacial energetic coefficient; M is the diffusion mobility coefficient; t is time; r is the position; ${\eta }_p\left(\mathit{r},t\right)$ is the ordering parameter, which represents the grain orientation in this simulation, and p is set to 32 in this system, as suggested in a previous study [14]; and $c\left(\mathit{r},t\right)$ is the component field variable. According to Reference [14], the model is simplified, and only the diffusion at a higher content of Al composition field is taken into account without considering the diffusion of other alloying elements. Therefore, there is only one composition field variable in the model. F is the total free energy of the system. In this study, the system is assumed to be isotropic, so F expressed as follows [27]

$$F={{\displaystyle \int }}_V[\frac{K_2}{2}{\displaystyle \sum }_{p=1}^n{(\mathsf{\nabla }{\eta }_p(\mathit{r},t))}^2+f_0\left(c,{\eta }_1,{\eta }_2,\dots ,{\eta }_p\right)]d\mathit{r}$$

(4)

where r is the position vector, and $d\mathit{r}=dx\times dy\times dz$; $K_2$ is the gradient energy coefficient and $f_0$ is the local free-energy density function. In order to introduce the second phase into the models, the following term, $f_0^p$, could be added to $f_0$:

$$f_0^n=\epsilon \phi {\displaystyle \sum }_{p=1}^n{\eta }_p^2\left(\mathit{r},t\right)$$

(5)

where $\varphi$ is used to describe the particle distribution. When $\varphi =1$, there is a particle, and when $\varphi =0$, there is no particle, while $\varphi$ is considered to be constant with the time. Moreover, $\epsilon$ is the coefficient related to the interaction between the matrix and the particles. Since the interaction between the matrix and the particles have not been considered, the value of $\epsilon$ is 1 J mol⁻¹.

The local free-energy density function used in this study can be expressed as follows [28]:

$$\begin{array}{l}{f}_0(c,{\eta }_1,{\eta }_2,\cdots {\eta }_p)\\ =A_0+\frac{A_1}{2}{(c(\mathit{r},t)-c_l)}^2+\frac{A_2}{4}{(c(\mathit{r},t)-c_l)}^4-\frac{B_1}{2}(c(\mathit{r},t)\\ {-c_l)}^2{\displaystyle {\displaystyle \sum }_{p=1}^n}{\eta }_p^2(\mathit{r},t)\\ +\frac{B_2}{4}{({\displaystyle {\displaystyle \sum }_{p=1}^n}{\eta }_p^2(\mathit{r},t))}^2+\frac{K_1}{2}{\displaystyle {\displaystyle \sum }_{p=1}^n}{\displaystyle {\displaystyle \sum }_{q\ne p}^n}{\eta }_p^2(\mathit{r},t){\eta }_q^2(\mathit{r},t)+\epsilon \phi {\displaystyle \sum _{p=1}^n}{\eta }_p^2(\mathit{r},t)\end{array}$$

(6)

where $c_l$ is the composition at the lowest point of the free-energy composition curve at a certain temperature of the Al in the Mg alloy; n is the number of possible ordering parameters in the system; A₀, $A_1$, and $A_2$ are the constants associated with the free energy of the system; $B_1$ and $B_2$ are the coefficients for restored energy; and $K_1$ is the coupling term coefficient between ${\eta }_p$ and ${\eta }_q$.

In the study, the grain boundary range is chosen in the model, which is different from the traditional grain boundary. Generally, the grain boundary refers to the structure with 3 or 4 atoms in the crystal that is different from other parts. However, the grain boundary range represents the energy difference between the interface and the matrix, and it reflects the distribution range of the interface energy in the direction perpendicular to the interface. At the same time, the calculation [20] shows that this range also corresponds to the interface segregation range of alloy elements. The width of the grain boundary scope is selected as a 4 grid size according to previous work [20].

This study was performed by using the AZ31 Mg alloy, and the annealing temperature was 350 °C. The alloy composition is w(Al) = 3%, w(Zn) = 1%, and Mg is the remainder. The added SPPs are set as insoluble particles. The simulation system is a two-dimensional system with a square grid as the calculation area and periodic boundary conditions. The total area of the simulation area is 150 μm × 150 μm, with 512 × 512 cells; that is, the width of each cell dx is 0.293 μm. The time step is typically selected to be relatively small to converge the results, but if it is too small, additional computational time will be required to form the required microstructure; hence, the time step selected for this simulation is 0.3 s. The shape of the SPP is circular. Since the recrystallization nucleation process is quite complex, the study treats the nucleation process by the phenomenological method and takes it as the initial condition, and the average simulation area, 4dx × 4dx, is used as the initial unit, and the radius of the grain nucleus in each unit is a random value between 0 and 2 grids. The initial composition is set according to the local initial $\eta$ value; when $\eta =1$, it is in the new recrystallized grains, the composition c is 0.03, and the composition in other regions is 0.031. The parameters in the model are set as follows [29]: $c_1=0.2$, A₀ = −25.01 kJ mol⁻¹, $A_1=22.02{\mathrm{kJ}\mathrm{mol}}^{-1}$, $A_2=18.30{\mathrm{kJ}\mathrm{mol}}^{-1}$, $B_1=3.54{\mathrm{kJ}\mathrm{mol}}^{-1}$, $B_2=92.86{\mathrm{kJ}\mathrm{mol}}^{-1}$, $K_1=141.24{\mathrm{kJ}\mathrm{mol}}^{-1}$, $K_2=35.37\times {10}^{-13}{\mathrm{J}\mathrm{m}}^2{\mathrm{mol}}^{-1}$, and $L=1.15\times {10}^{-2}{\mathrm{mol}\mathrm{J}\mathrm{s}}^{-1}$. Different from the models proposed in References [30,31], the parameters in the models are all with physical meanings in order to achieve the realistic spatiotemporal simulation in this work.

3. Simulation Results of the AZ31 Mg Alloy

The evolution of the microstructure of AZ31 magnesium alloy with time was simulated by the constructed phase field model, and the simulation results are shown in Figure 1.

It can be seen from Figure 1 that the microstructures simulated by the composition field and ordering parameters correspondence well, which proves that the free energy density function constructed in this paper is correct and proves the validity of the constructed phase field model. It can been seen from Figure 1b,d that the distribution of Al components is relatively uniform near the straight interface, while it is very uneven near the curved interface, leaving traces of the previous grain evolution. The simulation results are also in good agreement with the experimental results about the microstructure evolution [32].

The evolution of the microstructure of AZ31 Mg alloy containing SPP with time was simulated, and the results are shown in Figure 2.

Figure 2 shows the evolution of the microstructure of the AZ31 Mg alloy containing SPPs with a grain size of r = 1 μm and volume fraction of f = 5%. It can be found from Figure 2 that the number of grains in the system shows a trend of gradual decrease to basic stability, indicating that the grains are gradually growing. The simulated microstructure morphologies are consistent with the experimental observations [33].

It can also be found, due to the addition of SPP, that there is the occurrence of abnormal grain-growth phenomenon, as shown by the circle in Figure 2. The reason for this phenomenon is the uneven pinning of the SPP. It is shown in the circle that there are multiple SPPs on some grain boundaries and no SPPs on some grain boundaries, so during the grain-growth process, there is a difference in the migration rate of each grain boundary, and the unpinned grain boundaries migrate faster and easily merge with the surrounding grain boundaries to cause abnormal grain growth.

In order to further investigate the effect of the SPPs on the evolution of the microstructure, the change of the average grain size with time and with and without SPPs is shown in Figure 3.

It can be seen from Figure 3 that the average grain size is significantly refined when the AZ31 Mg alloy contains SPPs at the same time, indicating that the addition of SPPs can refine the grains to achieve fine-grain strengthening. This is because the grain-growth process is related to grain-boundary migration; however, particles were trapped in the grain boundaries and increased resistance to grain-boundary migration, leading to grain refinement.

The effect of the volume fraction of SPPs on the AZ31 Mg alloy microstructure was further investigated. Figure 4 shows the results.

Figure 4 shows the results of the effect of particle volume fraction on the microstructure in the AZ31 Mg alloy, and the simulation results at t = 100 min were selected for comparison. Figure 4 shows that when the particle size is certain, the number of grains in the system gradually increases with the increase of the particle volume fraction, and an obvious grain-refinement phenomenon appears; when the particle volume fraction is small (as shown in Figure 4b), the number of grains in the system is smaller, the spacing between the grains is larger, and the overall pinning effect that can be played on the grain boundaries is weaker, so most of the grains can grow normally. When the volume fraction of particles is larger (Figure 4d), which means that the spacing between particles also decreases, the overall pinning effect on the grain boundary is enhanced, and the movement of the grain boundary is hindered more seriously, so the number of grains in the system increases, and the grain-refinement phenomenon is more significant. The effect of the size of the SPPs on the AZ31 Mg alloy microstructure was further investigated, as shown in Figure 5.

Figure 5 shows the simulation results of grain growth with different particle sizes. With the increase in the size of the added particles, the grain-refinement effect is reduced. According to the classical Zener model, the resistance to grain growth of the second phase at the grain boundary F is as follows:

$$F=\frac{3f}{2r}\sigma$$

(7)

where $\sigma$ is the specific interfacial energy ($\mathrm{J}/{\mathrm{m}}^2$); $r$ and $f$ are the SPP size and volume fraction, respectively. It can be seen that, the smaller the particle size and the more the volume fraction, the stronger the overall pinning effect of the particles on grain growth.

Overall, the addition of particles leads to limited grain growth, and the refining effect is influenced by the particle size, as well as the volume fraction.

4. Analysis and Discussion

4.1. Effect of SPP on the Grain Growth of Mg Alloy

According to the results of the effects of SPPs on the microstructure of the Mg alloy in Figure 4 and Figure 5, the quantitatively calculated curves of the average grain size evolution with the annealing time are plotted in Figure 6.

It can be seen in Figure 6 that the average grain size during the initial stage of grain growth did not differ significantly between cases t < 10 min. With the increase in the simulation time, the grains gradually expand, the average size of the grains increases, and the difference of the change in the volume fractions and sizes of the grains become clear. This result is related to the fact that particles are randomly distributed, with few particles located at the grain boundary in the early stages of grain growth, and that grains with a large radius of curvature are present in the early stages of growth; hence, it is difficult to effectively impede grain growth due to the pinning force generated by particles. With the progress of evolution, grain-boundary migration leads to the gradual increase in the number of particles located at grain boundaries, the overall pinning effect of particles increases, and the effect of the difference in grain size and volume fraction gradually manifests itself; hence, the average grain-size curves gradually differentiate in the later stages.

Overall, particles mostly impede grain growth in the latter period of grain growth because their pinning effect on the grain boundaries is greater than that in the former stages.

4.2. Critical Value of SPP for Grain Boundary Pinning

The pinning effect of particles on grains is mainly manifested in the hindrance of grain growth. According to the grain-growth kinetic equation [34], we have the following:

$$R_{\mathrm{ave}}=k{t}^{1/m}+R_0$$

(8)

where R_ave is the average grain size, and it increases rapidly with the progress of evolution; m is the grain growth index; k is a constant related to time, and its unit is related to m; and R₀ is a constant.

Moreover, m is obtained by fitting the growth-index equation and simulation results. Clearly, the higher the grain growth index, m, the slower the grain growth. With the progress of grain growth, the number of particles on the grain boundaries progressively increases, but the growth rate of the matrix grains gradually decreases and ultimately stabilizes.

It can be seen from Figure 6 that, the fewer the second-phase particles, the weaker their pinning force to the grain boundary. Therefore, the critical values of the second-phase particles were considered in the investigation, as they affect the grain growth.

4.2.1. Critical Volume Fraction of SPP for Grain Boundary Pinning

The critical values of the particle volume fraction are determined by the phase-field models. For example, the minimum critical volume fraction is determined by the comparison of the average grain size curve without the second-phase particle, and it is considered to be reached when two curves approximately coincide. When the particle volume fraction is less than the minimum critical volume fraction, the pining impact of particles on grain boundaries is extremely weak, approaching negligible.

Figure 7 shows the average grain size, R_ave, as a function of time for each volume fraction of the particles at r = 0.5 μm, 1.0 μm, 1.5 μm, and 2.0 μm, and the minimal critical volume fraction, f_min, and maximum critical volume fraction, f_max, were obtained and are shown in

Table 1. Comparison of the critical volume fractions for different second-phase particle sizes.

SPP Size, r (μm)	0.5	1	1.5	2
Minimal critical volume fraction, fmin	0.20%	0.25%	0.30%	0.33%
Maximum critical volume fraction, fmax	11%	20%	35%	40%

.

As can be observed from Table 1, the critical value of the volume fraction of the corresponding particles increases with the increase in the particle size.

According to the fitting of the curves in Figure 7 in terms of Equation (10), the grain-growth law was obtained and is shown in

Table 2. Variation of the parameters in the grain-growth-index equation with SPP volume fraction.

		${\mathit{R}}_{\mathbf{ave}}=\mathit{k}{\mathit{t}}^{1/\mathit{m}}+{\mathit{R}}_{\mathbf{0}}$
		$\mathit{k}$	${\mathit{R}}_{\mathbf{0}} \left(\mathsf{\mu }\mathbf{m}\right)$	m	R2
r = 0 μm	f = 0%	2.0	1.1	2	0.99
r = 0.50 μm	fmin = 0.20%	1.6	1.8	2	0.99
r = 1.00 μm	fmin = 0.25%	1.4	2.3	2	0.99
r = 1.50 μm	fmin = 0.30%	1.7	1.49	2	0.99
r = 2.00 μm	fmin = 0.33%	2.4	0.7	2	0.99

. The fitting results are consistent with the simulation results [35], as well as experimental results [36].

It could be seen from Table 2 that when the volume fraction of the SPP takes the minimum value, the grain growth index is relatively close under different SPP sizes, and this, to a certain extent, justifies the determined maximum critical size. The fitting degree, R², is 0.99, indicating that the fitting is reasonable.

4.2.2. Critical Size of SPP for Grain Boundary Pinning

The particle size also exhibits the minimum critical size and the maximum critical size at a certain volume fraction. The simulation results are shown in Figure 8.

Figure 8 shows a graph of the average grain size as a function of time for the critical size of particles for f = 1.0%, 3.0%, 5.0%, and 7.0%. As can be observed in Figure 8a, the grain-growth curve for particle size r = 3.9 μm overlaps well with that without the addition of particles, indicating that r = 3.9 μm is the maximum critical size for the volume fraction of particles. Similarly, the critical value of the largest second-phase particle sizes is also found in Figure 8b–d, making the growth curve of the grains close to the case in which there are no SPPs. The results are shown in

Table 3. Comparison of critical sizes for different SPP volume factions.

SPP Volume Fraction, f	1%	3%	5%	7%
Maximum critical size, rmax	3.9 μm	6.0 μm	6.5 μm	7.0 μm

.

Table 3 shows the critical particle size value for each volume fraction. With the increase in the volume proportion of particles, the maximum critical size critical value of the corresponding particles increases (Table 3), and the minimum critical size is not observed.

According to the fitting of critical curves in Figure 8 and the Equation (10), the grain-growth law is obtained in

Table 4. Variation of parameters in the grain growth index equation with second-phase particle size.

		${\mathit{R}}_{\mathbf{ave}}=\mathit{k}{\mathit{t}}^{1/\mathit{m}}+{\mathit{R}}_{\mathbf{0}}$
		$\mathit{k}$	${\mathit{R}}_{\mathbf{0}} \left(\mathsf{\mu }\mathbf{m}\right)$	m	R2
f = 0%	r = 0 μm	2.0	1.1	2	0.99
f = 1.0%	rmax = 3.9 μm	2.1	1.1	2	0.99
f = 3.0%	rmax = 6.0 μm	1.6	1.9	2	0.99
f = 5.0%	rmax = 6.5 μm	3.3	−0.7	3	0.99
f = 7.0%	rmax= 7.0 μm	4.4	−2.3	3	0.99

. As can be observed from Table 4, the growth index is between 2 and 3, which means that the grain growth law is basically in line with the traditional grain-growth theory at this time, indicating that the SPPs have little influence on the grain growth, and the fitting degree, R², is 0.99, too.

4.3. Zener Relationship Verification

The effect on the limited radius of grains, $R_{\lim }$, is another manifestation of the pinning of the second-phase particle on grain boundaries. Zener posits the effect of particles on the limiting radius of the grain, assuming no particle growth [37]. In the Zener formula [38,39], we have the following:

$$\frac{R_{\lim }}{r}=K\frac{1}{f^b}$$

(9)

where K is parameter, and b is the fitting index. Previous lots of simulated and experimental results [40,41] reveal that the fitting indices are in the range of −0.55 to −0.30. It is also known from Reference [42] that the experimental results based on thin aluminum strips showed that the curve-fitting index is around −0.5. The simulation results obtained in this simulation: the average grain size, second-phase particle size, and second-phase particle volume fraction were fitted by the Zener relationship, and the resulting curves are shown in Figure 9.

The curves were fitted by using least squares via the Zener relationship. The corresponding relationship is expressed by the following equations:

$$\frac{R_{\lim }}{r\frac{1}{f^b}\left\{\begin{array}{c}4.0\times \frac{1}{f^{0.40}},\left(r=0.5 \mathsf{\mu }\mathrm{m},R^2={0.99}^{}\right)\\ 3.8\times \frac{1}{f^{0.41}},\left(r=1.0 \mathsf{\mu }\mathrm{m},R^2=0.99\right)\\ 3.5\times \frac{1}{f^{0.37}},\left(r=1.5 \mathsf{\mu }\mathrm{m},R^2=0.99\right)\\ 3.1\times \frac{1}{f^{0.34}},\left(r=2.0 \mathsf{\mu }\mathrm{m},R^2=0.99\right)\end{array}\right.}$$

(10)

The simulation results obtained in this study show that the fitting indices are in the range of 0.33–0.50.

On the other side, Yang et al. investigated the effect of second-phase particles on the microstructure of the laser-melted column during solidification by using the phase field method; the average grain radius, particle radius, and particle volume fraction were obtained from the simulations; and the fit index was 0.38, according to the Zener relationship. Huang et al. [43] investigated the second-phase particle precipitating at grain boundaries by the phase field method, and the simulation results showed that the effect of the second-phase particles on the limit grain sizes follows the Zener relationship, and the corresponding fitting index is 0.49. Du et al. [44] used the phase-field method to numerically simulate the pinning effect of differently shaped second-phase particles on the grain growth in polycrystalline structures; particles of spherical, cubical, ellipsoidal, and cuboidal shapes were added to the microstructure separately, with different volume fractions. The results obtained from the simulation were fitted to the Zener relationship, and the pinning forces were calculated based on the Zener theory. The results obtained from the simulation that the fit indices were 0.4–0.55. Olgeerd et al. [41] investigated the grain growth in polycrystalline calcite by the addition of Al₂O₃ particles, and the stable grain size reached during heat treatment was inversely proportional to l/f ^b, where b varied from 0.3 to 0.55. Thus, the simulation results obtained in this study are consistent with the simulation results [24,43,44] and experimental results [41,42].

5. Conclusions

In the simulations, the critical values of the particle size and volume fraction of the SPP that affect the growth of matrix grains during recrystallization are determined by the real space–time phase field model.

The critical value of the second-phase particle volume fraction contains the maximum and minimum critical volume fractions, which increase with the increase in the second-phase particle size.

Different from the critical volume fraction, only the maximum critical size increases with the increase in the second-phase particle volume fraction.

The values of the grain growth index, m, in the grain index equation and the average grain size curve confirm the validity of the critical value of the SPP determined in this study.

The simulation results obtained by fitting the Zener relationship for four second-phase particle sizes are compared to the relevant simulation results and experimental data, and the simulated results are in good agreement, indicating that the model is reliable.