Simulation on Changes in Microstructure and Texture during Normal Grain Growth of Steel Sheet by Two-dimensional Local Curvature Multi-vertex Model

Abstract

The two-dimensional local curvature multi-vertex model was applied to the normal grain growth of actual steel sheets for examination of the effect of the respective misorientation dependencies of grain boundary energy and mobility on grain growth and for comparison with experimental results. The simulation result revealed that the grain boundary energy had a major influence on the change in misorientation distribution with grain growth, whereas the grain boundary mobility did not have such a large influence. The simulation considering the misorientation dependency on grain boundary energy and mobility, in particular, accounting for Σ1 and high angle boundaries was constructed and was effective for reproducing the experimental results. Simulated microstructures were similar to the experimental ones; however, the detailed standard deviation of grain size distribution was smaller in the calculation than that in the experiment. The texture change with grain growth in the simulation was weaker than that in the actual steel sheets. As a whole, the developed model described the experimental grain growth well, and the difference in the results between the simulation and the experiment is probably attributable to the difference in dimension; i.e., two-dimension in simulation and three-dimension in experiment and the inaccuracy of the grain boundary conditions such as grain boundary energy and mobility in the model.

1. Introduction

Grain growth is one of the most important phenomena in materials and it determines the properties that we require in practical applications. In the case of steel, for instance, grain size and texture evolution during grain growth are the dominant factors in mechanical and magnetic properties. Thus, the prediction and control of grain size, boundary character and texture evolution during annealing are essential for material developments.

Topological simulation models of grain growth have been developed by many researchers and can be classified into two categories. One category is the stochastic model in which the Monte-Carlo Potts models are the most commonly investigated.^1,2) The other category is a deterministic approach, in which the phase field method, the curvature model³⁾ and the vertex model^4,5,6,7) have been developed, and a two-dimensional local curvature multi-vertex model considering local curvatures on boundaries and boundary tension at the triple junctions has been proposed.^8,9,10) The stochastic model handles the grain boundary characteristics and pinning mathematically. Therefore, it cannot sufficiently express the physical image of grain growth. The phase field method essentially requires expressing the grain boundaries by finite width and hence, the grain boundaries blur. Also recently, a mixed model, stochastic vertex model¹¹⁾ has been developed.

In our previous work,¹⁰⁾ the advantage of the two-dimensional local curvature multi-vertex model was presented. For instance, the grain growth of artificial texture was simulated under the different grain boundary conditions in which energy and mobility were changed simultaneously. The results revealed that the grain growth by simulation obeyed the one-second power law of time which was derived from the classical curvature model. And also, the grain growth velocity and misorientation distribution were changed by the grain boundary characteristics.

The present work aims to apply the two-dimensional local curvature multi-vertex model to the normal grain growth in actual steel sheets under different misorientation dependencies of boundary energy and mobility. The effect of the respective misorientation dependencies of boundary energy and mobility on grain growth was examined. The influences of mobility and energy on grain growth were discussed. And the simulated results of actual steel sheets were compared with the experimental results. Two-dimensional simulation was performed for the actual three-dimensional phenomenon. However, the two-dimensional experimental data for the actual phenomena were obtained experimentally by EBSD analysis, because it is not always easy to obtain accurate three-dimensional experimental topologies. Regarding the comparison between the results of experiment and simulation, the change in misorientation of neighboring grains, grain size distributions, and texture with grain growth were analyzed and discussed.

2. Simulation Model

The two-dimensional local curvature multi-vertex model shown in Fig. 1 has already been proposed by the present authors.^8,9,10) This model has solved the following problems associated with the curvature model and the vertex model. The curvature model cannot physically describe the migration of the triple junction, which is mathematically adjusted so that the three grain boundary tensions are in balance. The vertex model approximates the grain boundary with a curved line by a straight line. Therefore, the proposed model adopts the vertex model at the triple junction and the local curvature model on the grain boundary to overcome the aforementioned problems. In the local curvature model, the virtual vertices (double junctions) are put on the grain boundary, and the local curvature at a virtual vertex is determined by the virtual vertex and two adjacent junctions. Local grain boundaries, or virtual vertices on the grain boundary, move with velocity ${\overrightarrow{v}}_{gb,i}$ .   

$${\overrightarrow{v}}_{gb,\hspace{0.17em}i}=M_{gb,\hspace{0.17em}i}{\overrightarrow{F}}_{gb,i}$$(1)

  

$${\overrightarrow{F}}_{gb,\hspace{0.17em}i}={\gamma }_i{\overrightarrow{\kappa }}_{gb,\hspace{0.17em}i}$$(2)

where M_gb,i denotes mobility of the grain boundary to which virtual vertex p_i belongs; ${\overrightarrow{F}}_{gb,\hspace{0.17em}i}$ , driving force per unit length acting on the grain boundary; γ_i, grain boundary energy per unit length; ${\overrightarrow{\kappa }}_{gb,i}$ , local grain boundary curvature at virtual vertex p_i which is determined by three junctions of virtual vertex p_i and two adjacent junctions. In contrast, triple junctions move with velocity ${\overrightarrow{v}}_{tp,\hspace{0.17em}i}$ .   

$${\overrightarrow{v}}_{tp,\hspace{0.17em}i}=M_{tp,\hspace{0.17em}i}{\overrightarrow{F}}_{tp,i}$$(3)

  

$${\overrightarrow{F}}_{tp,\hspace{0.17em}i}=\sum _{j=1}^3{\gamma }_{ij}\frac{{\overrightarrow{q}}_{ij}}{\Vert {\overrightarrow{q}}_{ij}\Vert }$$(4)

where M_tp,i denotes mobility of the triple junction p_i; ${\overrightarrow{F}}_{tp,\hspace{0.17em}i}$ , driving force acting on triple junction p_i; γ_ij, grain boundary energy per unit length to which triple junction p_i and its neighboring junction p_j belong; ${\overrightarrow{q}}_{ij}$ , vector from triple junction p_i to its neighboring junction p_j.

Fig. 1.

Two-dimensional local curvature multi-vertex model, in comparison with the curvature model and vertex model.

3. Experiment and Simulation Procedure

3.1. Experiment 1: Sample Preparation for Examination of the Effect of Grain Boundary Characteristics on Grain Growth

Before the comparison between the experimental and the simulation results of grain growth, the effect of grain boundary characteristics on grain growth was examined. The simulations, considering the misorientation dependency of grain boundary energy and mobility, were performed using the texture shown in Fig. 2 as an initial condition. The specimen in Fig. 2 was prepared as follows. Hot-rolled 3.2 wt.% Si steel sheets with a thickness of 2.0 mm were cold-rolled to a thickness of 0.2 mm and annealed for 120 s at 1123 K. The microstructures and textures were obtained at one-tenth of the thickness from the surface. Details of the microstructures and textures measurement are described in 3.2.

Fig. 2.

a) Initial microstructure, and b) misorientation distribution of neighboring grains for preliminary examination of grain growth simulation. Cold-rolled and annealed 3.2Si steel (wt.%) was used.

3.2. Experiment 2: Sample Preparation for the Comparison between Experimental and Simulation Results of Grain Growth

An ingot of Fe with 0.002 wt.%C, 0.5 wt% of Si and Mn was produced in a vacuum melting furnace in the laboratory. After reheating for 1 hour at 1423 K, hot-rolled sheets with a thickness of 2.3 mm were produced. The steel sheets were then cold-rolled to a thickness of 0.5 mm and annealed for 30 s at 1003 K. This is the initial sample with an average grain diameter of 16.1 μm. In order to obtain samples with larger grain size, they were again annealed for 30 s at two different temperatures, 1003 and 1053 K. The average grain sizes obtained were 23.5 and 28.2 μm at 1003 and 1053 K, respectively.

Then, in order to examine the microstructures and textures, mechanical and chemical polishing was performed at the center of the thickness, and the microstructure and orientation maps were measured with an electron back-scattered diffraction pattern (EBSD) system by TSL Solutions equipped in a SEM, JSM6400. In each sample, four maps were taken with each map having an area of 600×1800 μm² with square grids of 3 μm followed by analytical modification whereby one grain has one orientation. Entire textures were manually measured with a thousand representative points with EBSD in each sample.

3.3. Procedure of Simulation

Regarding the simulation of grain growth, the local curvature multi-vertex model was implemented for the initial EBSD map. The initial EBSD data had orientation information in each grid, but for the implementation of simulation, the grid data were transformed to the line data (boundary data) with multiple vertices.

As for grain boundary characters, simulations with different grain boundary conditions were implemented as shown in Fig. 3 for examination of the effect of grain boundary characteristics and in Fig. 4 for the comparison between experimental and simulation results. The preliminary examinations in Fig. 3 considered only the Σ1 grain boundary, and assumed the various combinations of misorientation dependencies of grain boundary mobility and energy. The grain boundary conditions in Fig. 4 are as follows: (A) energy and mobility are independent of the misorientation angles, (B) Σ1 boundary with low grain boundary energy and mobility was taken into account, (C) Σ1 and high angle boundaries close to 60 degrees (hereafter, denoted as HAB) with low grain boundary energy and mobility were considered. The Σ1 boundary was set to be consistent with Brandon’s criteria.¹²⁾ HAB was defined from the analogy of Brandon’s criteria of Σ3; however, the minimum value was set as 30% of the maximum values. The energy and mobility profiles are shown in Figs. 3 and 4. Read-Shockley type grain boundary energies were considered and the mobility curve against misorientation was set to be sigmoidal.¹³⁾ The equations of both energy and mobility were modified as follows in order to describe energy and mobility whose minimum values were not zero.   

$$\begin{array}{l}{\gamma }_i={\gamma }_{min}+\left({\gamma }_{max}-{\gamma }_{min}\right)\frac{\left|\theta -{\theta }_0\right|}{{\theta }_m}\left[1-ln\frac{\left|\theta -{\theta }_0\right|}{{\theta }_m}\right],\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\hspace{0.17em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{when}\mathrm{\hspace{0.17em} }\left|\theta -{\theta }_0\right|\le {\theta }_m\end{array}$$(5)

  

$${\gamma }_i={\gamma }_{max},\hspace{1em}\text{when}\mathrm{\hspace{0.17em} }\left|\theta -{\theta }_0\right|>{\theta }_m$$(6)

  

$$\begin{array}{l}{M}_{gb,\hspace{0.17em}i}=M_{min}+\left(M_{max}-M_{min}\right)\left[1-e^{-B{\left(\frac{\left|\theta -{\theta }_0\right|}{{\theta }_m}\right)}^n}\right],\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} } \text{when}\mathrm{\hspace{0.17em} }\left|\theta -{\theta }_0\right|\le {\theta }_m\end{array}$$(7)

  

$$M_{gb,\hspace{0.17em}i}=M_{max},\hspace{1em}\text{when}\mathrm{\hspace{0.17em} }\left|\theta -{\theta }_0\right|>{\theta }_m$$(8)

where γ_max and γ_min denote maximum and minimum values of grain boundary energy; M_max and M_min, maximum and minimum values of grain boundary mobility; θ₀, misorientation angle of Σ1 or Σ3 which is 0 or 60 degrees; θ_m, misorientation range; B and n, sigmoidal parameters. The parameters were set as follows for preliminary examinations in Fig. 3; γ_max = 1, γ_min = 0, M_max = 1, M_min = 0, B = 5, n = 4 in all the conditions; θ_m = 15 in the condition of narrow misorientation range and θ_m = 30 in the condition of wide misorientation range. In Fig. 4, the parameters were set as follows: γ_max = 1, M_max = 1, B = 5, n = 4 in all the conditions; γ_min = 0 and M_min = 0 in the conditions except HAB in Fig. 4(c); γ_min = 0.3 and M_min = 0.3 for HAB in Fig. 4(c); θ_m = 15 degrees for Σ1 and θ_m = 8.7 degrees for HAB. The mobility at the triple junction was set to be the average of the three grain boundary mobilities, because of less data on mobility at the triple junction.

Fig. 3.

The subtracted misorientation distributions of neighboring grains. Difference in frequency before and after the one-and-half times grain growth was plotted as a function of misorientation. The influence of misorientation on grain boundary energy and mobility was evaluated. Cold-rolled and annealed 3.2Si steel (wt.%) was used.

Fig. 4.

Profiles of boundary energy and mobility as a function of misorientation; (A) no special boundaries, (B) Σ1 boundary and (C) Σ1 and high angle boundaries (HAB) close to 60 degrees were taken into account.

The details on how to determine the simulation condition are described in the Ref. 9). The simulation conditions on space and time in the present study are as follows. The distance between neighboring vertices was 0.7 μm to 1.5 μm. The time interval of simulation was 0.1 (arb. units). The edge condition of the EBSD map was set to be mirror symmetrical.

4. Results and Discussions

4.1. Grain Boundary Condition Effect

The subtractions of the misorientation distributions between the initial material shown in Fig. 2 and the material whose average grain size is one-and-a-half times larger than the initial one are also shown in Fig. 3. The preliminary simulation result in Fig. 3 revealed that the grain boundary energy had a major influence on the change in misorientation distribution with grain growth, whereas the grain boundary mobility did not have such a large influence. Since the migration of grain boundary (virtual vertex) is determined by the product of mobility, energy and curvature as described in Eqs. (1) and (2), the influences of mobility and energy on the migration of grain boundary (virtual vertex) are expected to be equal. However, the influences of mobility and energy on the migration of the triple junction are expected to be different as described in Eqs. (3) and (4). When mobility at the triple junction is not zero, the triple junction connecting to one grain boundary of small energy and to two grain boundaries of large energies is able to migrate, so that the grain boundary of small energy tries to be longer and the other two grain boundaries of large energies try to be shorter as compared with the state where apex angles are 120 degrees. The result revealed that the frequency of the small-energy grain boundary was expected to increase. The blanks in Fig. 3 denote that simulations were not performed, but do not denote that simulations cannot be performed.

4.2. Misorientation Distributions

First, the misorientation distributions of neighboring grains before and after the grain growth in the experiment and simulations were compared as shown in Fig. 5. The initial average diameter was 16.1 μm. As for the grain growth experiment, the sample after annealing at 1053 K for 30 s was analyzed, in which the average diameter was 28.2 μm. This evaluation after grain growth was performed in different areas from the initial observation before grain growth. Simulations were performed with the aforementioned three different boundary conditions until their grain sizes became 28.2 μm. Before grain growth, the simulated misorientation distributions of the case (A1), (B1) and (C1), dependent on boundary conditions, were slightly modified for implementation of the simulation from the experimental one (E1), but the initial four distributions were basically the same.

Fig. 5.

Misorientation distributions of neighboring grains; (E) is the experimental result, and (A), (B) and (C) are the simulation results for the boundary conditions as indicated in Fig. 4. (1), (2) and (3) mean the results before and after grain growth and the subtraction of the two, respectively.

In the experiment, (E2), after grain growth, there was no such large change of the profile. However, based on the subtracted misorientation distribution before and after grain growth, (E3), the boundaries with misorientation higher than 48 degrees increased. On the contrary, the simulated distribution with constant boundary energy and mobility, (A2), was close to the Mackenzie distribution¹⁴⁾ and the lower angle boundaries decreased largely as shown in (A3). When the Σ1 boundary was taken into account, the frequencies of Σ1 boundaries increased as shown in (B2), and the decrease of the lower angle boundaries was diminished as shown in (B3). When the tendency of (E3) and (B3) was compared, the discrepancy in higher angle boundaries remained. Then, in order to evaluate the effect of higher angle boundaries, the simulation considering Σ1 and HAB was performed. As shown in (C2) and (C3), the frequencies of the higher angle boundaries increased and the tendency in subtracted misorientation distribution became closer, though the extent of increase in HAB was larger in (C3) than in (E3). From the results above, it was speculated that, without considering any special boundaries with low energy, the misorientation distribution after grain growth became close to the Mackenzie distribution, but in the actual materials, the Σ1 boundary and HAB were more difficult to migrate than other boundaries due to their low grain boundary energy so that after grain growth the misorientation distribution became close to that of (E2). The distinction of the effect between HAB and Σ3 is a future topic for investigation. Simulations assuming case (C) are discussed in detail below.

4.3. Grain Size Distributions

Figure 6 shows the microstructures and the grain size distributions of the experiment and simulation (C) with the same average diameters. Average grain size, standard deviation and grain numbers are denoted in the figure. The areas in the experiment were taken at different sample sites, and therefore, the microstructures (E2) and (E3) were completely different from those of (C2) and (C3), respectively. Irrespective of the experiment and simulation, the size distributions appear comparatively homogeneous after grain growth, but the experimental microstructures seem to be less homogeneous than the simulated ones; a larger number of small grains are observed in the experimental microstructures. Considering the four areas of EBSD mapping in the experiment, the grain numbers were almost four times as high as the simulated numbers. The difference between the experiment and the simulation is that the frequencies of small and large grains in the experiment were larger than those in the simulations. This is also indicated in the values of standard deviations after grain growth; standard deviations in (E2) and (E3) were larger than those in (C2) and (C3), respectively.

Fig. 6.

Microstructures and their grain size distributions obtained by experiment and simulation; (E) and (C) denote the results by experiment and simulation, respectively. The grain boundary condition (C) in Fig. 4 was assumed for simulation. Average grain diameters, their standard deviations and numbers of grains are described in the figure.

The reason why the experimental grain size distribution had higher frequencies in small and large grains than the simulation has not been clarified yet. However, it might be due to the difference in dimension between simulation (two) and experiment (three). In the experiments, we observed two-dimensional sections of three-dimensional phenomenon.^15,16) In general, the grain size distribution in a two-dimensional section is shifted to a smaller size. The comparison of the results between two-dimensional sections of a three-dimensional simulation and a two-dimensional simulation should be investigated in future. Another likely reason is the inaccuracy of the grain boundary characteristics such as grain boundary energy and mobility. The effect of grain boundary characters, for example, coincident site lattice boundaries, requires meticulous consideration in future.

4.4. Textures

Figure 7 shows the texture change with grain growth. The average grain size of simulation (C) was set to be the same as that of the experiment by adjusting the time for grain growth. The initial textures, (E1) and (C1), are different on the grounds that (E1) is the texture of the entire area of the sample (12×10 mm²) while the origin of (C1) texture was taken from a local area (600×1800 μm²). However, the two are basically the same. During grain growth, the experimental texture changed largely compared to the simulated texture; in the experiment, the intensity of {111} orientation became stronger and that of {100} became weaker; in contrast, no remarkable change was observed in the simulation. However, the same tendency in terms of increase in ND//<111> and decrease in ND//<100> with grain growth was confirmed between the experiment and the simulation.

Fig. 7.

Changes in texture with grain growth with grain diameter 16, 24 and 28 μm, based on the experiment and the simulation assuming the condition (C). The subtracted ODFs of the specimens with 16 and 28 μm are also represented. ODF in ϕ₂ 45 degrees section are presented. The contour of intensity in ODF is 1-2-3-4-5-6. The contour of intensity in the subtracted ODF is shown in the figure.

The reason for the larger texture change in the experiment was analyzed. Table 1 indicates the average grain diameters in three orientations in the same area used for simulation (C1). Average grain diameters of the grains with {111}<112> and {111}<110> orientations were larger, and that of {100}<001> orientation was smaller than the entire average grain diameter so that, during grain growth, {111} grains became stronger and {100} grains became weaker due to the size effect.¹⁵⁾ Therefore, the present experiment was a reasonable result for the normal grain growth and no special phenomena occurred in the texture evolution during grain growth.

Table 1. Average grain diameters of the grains with three orientations.

Orientation	Whole grains	{111}<112>*	{111}<110>*	{100}<001>*
Average diameter (μm)	16.2	16.5	17.1	15.3
Grain numbers	4042	646	520	169

*  Orientation within 15 degrees

The probable reason for the weak texture change in the simulation compared to that in the experiment is again due to the difference in grain growth dimension; the dimension of the simulation and the experiment were two and three, respectively. In two-dimensional grain growth we can easily obtain the equilibrium at the vertices because of the simple topology; if all energy boundaries have the same values and the areas are formed with an array of the same shape of hexagons, each boundary angle at the vertices maintains 120 deg and no boundary curvatures occur. On the contrary, in three-dimensional grain growth, even if all the space is filled with regular tetrakaidecahedra, boundaries must curve to obtain equilibrium at the vertices.^16,17) This means that in three-dimensional grain growth the size effect may become stronger as compared to the two-dimensional case. This subject requires clarification in a future study.

Since there is some difference between two-dimensional and three-dimensional simulation, three-dimensional simulation is required for higher accuracy. The three-dimensional model is speculated to be quite complex compared to the two-dimensional model. However, it is possible to calculate the migration of the grain boundary face by the construction of the grain boundary face of triangle meshes and by the calculation of curvature of the two orthogonal directions at each point of the triangle mesh.

5. Conclusions

The developed two-dimensional grain growth simulation method by the local curvature multi-vertex model was applied to the actual steel sheets for examination of the effect of the respective misorientation dependencies of boundary energy and mobility on grain growth and for comparison with the experimental results. The simulation result revealed that the grain boundary energy had a major influence on the change in misorientation distribution with grain growth, whereas the grain boundary mobility did not have such a large influence. The simulation considering the misorientation dependence on grain boundary energy and mobility, in particular, accounting for Σ1 and high angle boundaries was constructed and was effective for reproducing the experimental results. The simulated microstructures were similar to the experimental ones; however, the standard deviation of grain size distribution was smaller in the simulation than in the experiment. The texture change with grain growth in the simulation was weaker than in the experiment. As a whole, the developed model described the experimental normal grain growth well. The difference between the simulation and the experiment is presumably due to the difference in dimension; i.e. two-dimension in simulation, and three-dimension in experiment, and the inaccuracy of the grain boundary characteristics such as grain boundary energy and mobility in the model.

References

• 1)   M. P.  Anderson,  D. J.  Srolovitz,  G. S.  Grest and  P. S.  Sahni: Acta Metall., 32 (1984), 783.
• 2)   A .L.  Garcia,  V.  Tikare and  E. A.  Holm: Scr. Mater., 59 (2008), 661.
• 3)   H. J.  Frost,  C. V.  Thompson,  C. L.  Howe and  J.  Whang: Scr. Metall., 22 (1988), 65.
• 4)   A.  Soares,  A. C.  Ferro and  M. A.  Fortes: Scr. Metall., 19 (1985), 1491.
• 5)   K.  Kawasaki,  T.  Nagai and  K.  Nakashima: Philos. Mag. B, 60 (1989), 399.
• 6)   F. J  Humphreys: Scr. Metall. Mater., 27 (1992), 1557.
• 7)   D.  Weygand,  Y.  Brechet and  J.  Lepinoux: Philos. Mag. B, 78 (1998), 329.
• 8)   T.  Tamaki,  K.  Murakami,  H.  Homma and  K.  Ushioda: Mater. Sci. Forum, 715–716 (2012), 551.
• 9)   T.  Tamaki,  K.  Murakami and  K.  Ushioda: Tetsu-to-Hagané, 101 (2015), No. 3, 211.
• 10)   T.  Tamaki,  K.  Murakami and  K.  Ushioda: Nippon Steel Tech. Rep., 102 (2013), 31.
• 11)   K.  Piekos,  J.  Tarasiuk,  K.  Wierzbanowski and  B.  Bacroix: Comput. Mater. Sci., 42 (2008), 36.
• 12)   D. G.  Brandon: Acta Metall., 14 (1966), 1479.
• 13)   F. J.  Humphreys: Acta Mater., 45 (1997), 4231.
• 14)   J. K.  Mackenzie: Biometrika, 45 (1958), 229.
• 15)   M.  Hillert: Acta Metall., 13 (1965), 227.
• 16)   C. S.  Smith: Metal Interfaces, ASM, Cleveland, OH, (1952), 65.
• 17)   H. V.  Atkinson: Acta Metall., 36 (1988), 469.