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Monte Carlo simulation of microstructure evolution based on grain boundary character distribution
Materials Science and Engineering A281 (2000) 176 – 188 www.elsevier.com/locate/msea
Monte Carlo simulation of microstructure evolution based on grain boundary character distribution H.N. Lee a, H.S. Ryoo b, S.K. Hwang b,*,1 b
a Agency for Defense De6elopment, Yuseong, 305 -600, South Korea Inha Uni6ersity, Department of Metallurgical Engineering, Incheon, 402 -751, South Korea
Received 14 May 1999; received in revised form 22 October 1999
Abstract A Monte Carlo simulation of microstructure evolution was conducted with emphasis on grain growth anomaly and texture formation arising from grain boundary characteristics. Microstructures consisting of strategically varied grain size, grain boundary mobility and texture components were generated. It was found that the initial grain size was of the most significant parameter that determines the topographical properties of the final microstructure. Contrary to some observations and expectations, high-energy grain boundaries of high mobility did not necessarily vanish first but sometimes persisted and even increased in areal fraction depending on the local distribution of characteristics. In a microstructure with two texture components, an abnormal gain growth occurred when a large grain with high mobility selectively grew consuming surrounding small grains. From the analysis of the variation of grain boundary fraction and associated energy reduction, it was concluded that minimization of the total energy of the system, instead of the local instability, governed the evolution characteristics of the microstructure. Published by Elsevier Science S.A. Keywords: Monte Carlo; Microstructure; Grain boundary; Mobility; Energy; Texture; Grain size
1. Introduction Grain boundary character distribution (GBCD) has been the subject of many recent research efforts concerning texture evolution [1,2]. Attempts have been made to correlate the frequency of particular grain boundaries with the development of Goss texture in Fe-3%Si alloy. For example, Harase et al. [3,4] claimed that the texture development in this alloy was caused by the growth of the grains with the Goss orientation, which were associated with special low CSL (S5 or S9) boundaries during growth. These boundaries, therefore, were concluded to be of high mobility on the ground that they were relatively immune to segregation of solute atoms or secondary phases. Hayakawa et al. [5,6], on the other hand, claimed that the boundaries * Corresponding author. Tel.: +82-328607537; fax: +82328625546. E-mail address: [email protected] (S.K. Hwang) 1 Jointly appointed by the Center for Advanced Aerospace Materials. 0921-5093/00/$ - see front matter Published by Elsevier Science S.A. PII: S 0 9 2 1 - 5 0 9 3 ( 9 9 ) 0 0 7 2 5 - X
with the misorientation angle in the range of 20–45° were of high mobility on the ground that they were of high energy. Precipitate coarsening on these boundaries would reduce the pinning force against abnormal growth of the Goss grains. These apparently conflicting observations were made on the basis of the frequency of particular boundaries, and as such, the disagreement arises from imperfect GBCD information on individual grains. Most of all, there is no a priori reason for the quick disappearance of the mobile boundaries. Evolution of particular boundaries must satisfy the local as well as the system average energy minimization criterion. Therefore, a comprehensive set of information concerning the grain itself as well as the surroundings is necessary to account for in detail the path of microstructural and texture evolution. In the present study we attempted to develop a methodology to define the stages of microstructural evolution from the standpoint of a comprehensive information system. In a simulated microstructure, the orientation and the grain boundary characteristics of each grain were individually assigned and their evolu-
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tion stages were monitored. In the end it will be shown that the boundaries of high mobility do not necessarily disappear quickly and sometimes they persist even longer than low-energy, low mobility boundaries. It will also be shown that the criterion of local energy minimization requires the initial grain size, rather than the grain boundary mobility per se, as the crucial factor to control the microstructure distribution and texture.
2. Monte Carlo simulation method The method to simulate grain growth was according to the one developed by the Exxon group [7,8]. In this method, a system is composed of a triangular lattice, which has a periodic boundary condition at each end. A system contained a total of 40 000 (200× 200) lattice points. Each lattice point is made to have a crystal orientation by assigning a number, Q. A grain is defined as an aggregate of the lattice points with an identical Q number. A grain boundary is defined as the interface between two lattice points of different Q number. Since each grain is assigned with its own Q number ranging from 1 to 988 (Qmax), coalescence of grains with the same orientation is prohibited in this method. Elapse of time is measured as a unit of MCS during which 40,000 flipping attempts of the lattice points occur in a system. Orientation change of lattice points means grain boundary movement. Therefore, the grain boundary mobility can be assigned by adjusting the probability of the orientation change. In the present study, two methods were used to assign mobility to individual grain boundaries. The first method, which was proposed by Rollett et al. [8], defines the grain boundary mobility, M, as a factor modifying the probability of the orientation change. Thus: P=
1 for DE 50 m*
P=
exp(−DE/kBT) for DE \ 0 m*
(1)
where m* is mobility factor (M =1/m*), DE is the energy change associated with the orientation change of a lattice point, kB is the Boltzmann constant and T is absolute temperature. Isotropic mobility corresponds to m*=1. Anisotropy of the mobility can be introduced by assigning each boundary individual value of m* according to the degree of misorientation. This method was used for the study of two-component texture systems. Another method of assigning anisotropy to the grain boundary mobility, newly used in the present work for the multi-component texture systems, was as follows:
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Qi − Qj (2) Qmax − 1 where dQ is the misfit parameter between adjacent grains. In this scheme, orientation switching of a lattice point is allowed only if M is larger than a random number so that the larger the grain boundary misorientation the higher the mobility. In the present work, attempt of a lattice point switching into a different orientation is limited to one nearest neighbor orientation [9]. Therefore, a grain with a new orientation is not allowed to nucleate within an existing grain. M= dQ =
3. Result of Monte Carlo computer simulation
3.1. System with two-component texture The initial microstructure used in the computer simulation was made by the Voronoi mosaic method. The characteristics of this microstructure are explained elsewhere [10]. Distribution of the grain size in the initial microstructure, as determined by the number of lattice points, fits a gamma distribution function as shown in Fig. 1 with the average grain size A( being 40.5. However, topological frequency, as determined by the number of sides of grains, fits a log-normal distribution function. The average number of sides of grains was 6 and the grain size increased linearly with the number of sides. Microstructural evolution was studied in the matrix textured with two components, A and B. Four different types, designated as S1, S2, S3 and S4, of two-component textures were generated. Each type had either different areal proportions of the two texture components or different grain size for a particular texture component. For types S1, S2 and S3, the areal fraction of A-type grains (grains with A component) texture was set as 0.25 and the average grain size was varied. In the S1 type microstructure, all A-type grains were smaller than any B-type grain whereas the case for S2 was opposite. In the S3 type microstructure, the grain size of either texture component was set as random. S4 had the same property as that of S3 except that the areal proportion of A-type and B-type grains was 1:1. These categories of microstructures were designed to study the effect of the initial grain size and grain boundary mobility between the texture components on subsequent growth. Anisotropy of grain boundary mobility was assigned in such a way that the boundaries of grains of the same texture component (either AA or BB) was made slow (M= 0.1) and the others (AB) fast (M= 1.0). This arrangement resulted in mostly fast boundaries around small grains in S1 and large grains in S2. Computer simulation of growth was conducted for three different microstructures for each category at T :0 and ensemble averages are presented.
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Fig. 1. Grain size distribution of the initial microstructure generated by Voronoi method.
Fig. 2. Evolution of microstructure in the systems with two-component texture: (a) S1; (b) S2 and (c) S4. In all microstructures, A-type grains appear as white whereas B-type grains as gray. The black boundary lines and the red lines indicate grain boundaries of low mobility and of high mobility, respectively.
Conditions
S1 S2 S3 S4 a b
AA
0.25 0.25 0.25 0.50
A A:A B (t= 0 MCS)
0.56A0:1.36A0 2.05A0:0.85A0 0.97A0:1.01A0 1.05A0:0.95A0
A A/A0 (t =2000 MCS)
150 10 12 –
tr
1300 MCS 120 MCS 250 MCS –
t =0 MCS
t = 2000 MCS
G H, %
GL AA, %
GL BB, %
G H, %
GL AA, %
GL BB, %
9.02a (33.5)b 6.18 (22.9) 9.77 (36.3) 12.84 (47.7)
4.54 1.91 1.94 6.73
13.37 18.84 15.22 7.37
1.16 (25.8) 0 (0) 0.61 (7.9) 2.73 (38.3)
1.61 8.45 6.99 1.33
1.72 (38.4) 0 (0) 0.02 (0.3) 3.08 (43.1)
(16.8) (7.1) (7.2) (25.0)
(49.7) (70.0) (56.5) (27.3)
(35.8) (100) (91.8) (18.6)
In columns of t= 0 and 2000 MCS, the number is the fraction of lattice sites of grain boundaries between different texture components to total lattice sites in a system. In columns of t= 0 and 2000 MCS, the number within ( ) means the fraction of grain boundaries between different texture components to total grain boundaries in the system.
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Table 1 Analysis result of microstructures before and after grain growth in two-component texture systems
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Table 1 summarizes the result of computer simulation of microstructural evolution up to 2000 MCS. Graphical presentation of grain growth stages of representative microstructures of each category is given in Fig. 2. The one for S3 was omitted since it was very similar to that of S2. Grain growth rates of each texture component and that of the total system are shown in Fig. 3. All the microstructures in the categories S1, S2 and S3 showed abnormal grain growth (AGG), characterized by a bimodal grain size distribution, and a texture reversal. Characteristics of these developments, however, considerably differ depending on the starting size of A-type grains. In the case of S1, where the starting size of A-type grains was small, most of the A-type grains vanished early, before 100 MCS. Only a few of the A-type grains survived, which began to grow abnormally at around 500 MCS, becoming about 150 times the initial size at 2000 MCS. In the case of S2, where the starting size of A-type grains was large, A-type grains grew into abnormal grains in the very early stage of growth, at about 40 MCS, and consumed B-type grains at 500 MCS. Beyond this point there were no more high-mobility grain boundaries, thus normal grain growth set in. During the growth, there appeared a bimodal distribution of grain sizes as shown in Fig. 2(b) and Fig. 3(b). In the case of S3, where the sizes of A-type or B-type grains were random, the growth pattern was very similar to that of the S2 case except for a late texture reversal and slightly larger size of fully grown A-type grains compared to those in case S2. This is because of the increased distance of collisions of the large high-mobility grains that were formed by annihilation of small grains of high mobility. Therefore if the areal fraction of A-type grains were lower than 0.25 then there would be more abundant AGG in S2 than that in S1. In the case of S4, where the two types of texture components were balanced with random size distribution, only normal grain growth was observed. During the normal growth, the time exponent [11], n, of the growth rate was 0.47, excluding the initial transition. Variation of the number fraction of grain boundaries, of high mobility and low mobility, during microstructural evolution was examined as a function of time. The number fraction of grain boundaries was calculated by counting the lattice points at boundaries such as AA-type (G LAA), BB-type (G LBB) or AB-type (G H). The former two are low mobility boundaries and the latter one is the high mobility boundary. Fig. 4 shows the number fraction of grain boundaries varying with time for the four categories of microstructure. Due to the large areal fraction of B-type grains, BB-type grain boundaries are initially predominant in S1, S2 and S3. Most of the AA and AB-type grain boundaries quickly disappear in S1, yielding to BB-type
boundaries, due to the small initial grain size of A-type grains; however, they increased slightly again due to the abnormal growth of several surviving grains. In contrast, AB-type grain boundaries quickly became annihilated along with BB-type boundaries in S2, yielding to AA-type boundaries. This behavior was repeated in S3 except that some AB and BB-type boundaries persisted. In the case of S4, all the grain boundaries showed a monotonic decrease. Overall, it must be emphasized that highly mobile AB-type boundaries do not always vanish first. The energy of the total system, as defined by the following equation decreased with time as shown in Fig. 5: J N nn Et = % % (1−dij ) 2 i j
(3)
In the above equation, Et is the energy of the total system consisting of N lattice sites, J is the unit energy, dij is the Kronecker delta and nn is the number of nearest neighbor sites. As expected from the variation of the fraction of grain boundaries (Fig. 4), the reduction rate of the total energy was most significant in the early stage of growth. System S1 showed a rather peculiar trend in that its energy decreased fast in the beginning; maintained the highest energy level in the intermediate stage; then decreased fast again in the final stage. AGG was responsible for this development.
3.2. Systems of multi-component texture (Random texture) Using identical initial microstructures to those of the two-component texture system, systems of random texture were generated by assigning each grain an individual orientation as well as individual mobilities. According to Eq. (2), then, grain boundaries with a variety of mobility exist in a microstructure. To simplify analysis, grain boundaries were categorized into five different groups: G1 for 0BM50.2, G2 for 0.2B M50.4, . . . and G5 for 0.8 BM5 1.0, with an increasing order of mobility. Two different systems of microstructures were generated; N1, with most grains of similar orientations being neighbored, and N2, with those randomly mixed orientations. In system N1, therefore, low mobility grain boundaries were predominant (some areas as clustering) whereas in N2, grain boundaries of different mobility were evenly distributed. Five simulations were performed for each condition of T:0 and ensemble averages are presented. The results of microstructural evolution for systems N1 and N2 are shown in Fig. 6(a) and (b), respectively. In system N1, a few initially large grains surrounded by high mobility grain boundaries grew abnormally to
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Fig. 3. Grain growth rates of systems with two-component texture. See text for the description of microstructure of (a) S1, (b) S2, (c) S3 and (d) S4.
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eight times the average size at 3000 MCS. In system N2, on the other hand, a normal growth was observed evenly for most of grains.
Variations with time of the fraction of grain boundaries in multi-component texture microstructure are summarized in Table 2 and Fig. 7. The trends of grain boundary annihilation or reinforcement shown in Fig. 7 are considerably different from those shown in Fig. 4. All the grain boundary areas gradually diminished along with grain growth, except for a slight departure of the highest mobility grain boundaries, G5, due to abnormal growth of several grains in local areas in N1 system. Fig. 8 shows the rate of energy reduction of the system with grain growth in microstructures with multicomponent textures. The two systems, N1 and N2, showed a linear (in the log–log plot) decrease of energy with grain growth. Thus: −
dEt R( =k dt R( 0
2a
(4)
where k and a are constants, R( 0 and R( are the average radii of grains at t= 0 and t=t, respectively. From the slopes of the curves in Fig. 8 it was found that a= − 1.25 and n= 0.67 for N1 and a= − 1.39 and n=0.56 for N2. During the initial growth stage the energy reduction rate of system N2 was higher than that of N1; however, in the later stage of growth the two systems showed a similar rate of energy reduction. This phenomenon was due to the grain boundary characteristics. In system N2 there are more high mobility grain boundaries, which efficiently annihilate low mobility grain boundaries in the initial stage. In the later stage, however, the rate of annihilation is reduced due to collisions of high mobility grain boundaries. In the case of N1, there are mostly low mobility grain boundaries; therefore, the annihilation process by a few high-mobility grain boundaries takes more time [see Fig. 7(a)]. Comparison of the n values substantiates this interpretation.
4. Discussion
Fig. 4. Time-dependent variation of the fraction of grain boundaries L in the microstructures with two-component texture. The G L AA, G BB and G H stand for boundaries of A-A type, B-B type and A-B type, respectively. See text for the description of microstructure of (a) S1, (b) S2, (c) S3 and (d) S4.
Whether or not grain boundaries of high mobility always disappear early in the grain growth stage is an important question in the field of grain size control and texture control. Among many studies made so far, the following observations are noteworthy. Martikanen and Lindroos [12] reported an increasing frequency of low-mobility low-energy grain boundaries during growth of low-carbon, low-nitrogen 25wt%Cr-1wt%Mo ferritic stainless steel. In case of Fe-3wt%Si where grain growth was controlled with AlN and MnS particles, increasing frequency of S1 boundaries was found after secondary recrystallization [3,4]. In a similar alloy system, the boundaries with misorientation angle of 20– 45° disappeared quickly [5,6]. A few previous computer
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Fig. 5. Variation with time of the system energy in microstructures with two-component texture.
simulation works [13,14] also asserted the quick disappearance of the high-mobility boundaries regardless of texture. These interpretations, however, have not presented the rationale for the lack of the effect of the texture of the entire system. In principle, it must be possible that grains of particular boundary characteristics continually form during growth due to the special orientation distribution of all the gains contained in the system, thus the phenomenon should be texturedependent. The increasing frequency of low energy grain boundaries during grain growth appears to be consistent with the thermodynamical concept of minimization of the system free energy. However, the total energy of a system is not determined solely by the ‘frequency’ of particular boundaries but by the sum of all grain boundary energies, which depends on the distribution of grain boundaries. Therefore, although the total energy of the system gradually decreases with time as shown in Fig. 8, the initial microstructure may evolve into a microstructure apparently consisting of frequent high-energy boundaries. It is possible for some local high-energy boundaries that the rate of decrease in energy is temporarily negative. This is because the main factor governing the system energy reduction is the total grain boundary area instead of preferential disappearance of some high-energy boundaries. During the process of grain growth, there may be a transient energy increase in local areas due to annihilation of small grains with low-energy boundaries by
large grains with high-energy boundaries. This was predicted by a geometrical model study [15] and was also experimentally observed in 316L stainless steel [9], pure Al [16] and Nimonic alloy [17]. Boundaries of large grains in these cases were found to have a random high-energy structure. Many evidences (for example, Figs. 5 and 8) found in the present work support the hypothesis of the total energy criterion for microstructural evolution. There is no a priori reason why high-energy high-mobility grain boundaries should disappear faster than low energy boundaries do. Contradictory results reported in the earlier works, therefore, must be carefully re-analyzed. For example, encounters of the grains of minor texture component were quoted as the reason for the increase of low-mobility grain boundaries in the later stage of growth [14]. However, it is not highly likely that grains of minor texture component should meet frequently, forming low-energy, low-mobility boundaries. In an earlier work [18] the present authors showed that the texture component consisting of small grains (AB 0.6A) vanishes quickly regardless of its volume fraction. This phenomenon is easily understood from the standpoint of the total energy of the system as well as from the von Neumann–Mullins analysis [19,20] that states the predominance of the growth stage by large grains having more than six nearest neighbor grains. Since the energy of a two-dimensional system is proportional to the fraction of the grain boundaries per
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area the system consisting of small grains experiences a high driving force for growth, which requires sacrifice of small grains. In the present work, however, it was shown [Fig. 2(a)] that only several grains of a minor texture component can grow abnormally and become the major
texture of the system. This indicates that although the mobility of grain boundaries is important, the grain size effect is even more important. Compared at the same volume fraction of the minor texture component, small grains vanished whereas large grains survived.
Fig. 6. Evolution of microstructures in the systems composed of multi-component textures with grain boundaries of individually different mobility: (a) N1 and (b) N2.
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Table 2 Analysis result of grain boundary fractions before and after grain growth in multi-component systems Boundary
G1 G2 G3 G4 G5
Mobility
0BM50.2 0.2BM50.4 0.4BM50.6 0.6BM50.8 0.8BM51.0
N1
N2
t =0 MCS,%
t=3000 MCS, %
t=0 MCS, %
t= 3000 MCS, %
10.41a (38.6)b 7.18 (26.7) 5.22 (19.4) 3.07 (11.4) 1.07 (3.9)
3.52 0.81 0.71 0.70 0.86
5.25 5.90 5.23 4.34 6.22
1.56 1.27 1.03 1.09 1.36
(53.3) (12.3) (10.8) (10.6) (13.0)
(19.5) (21.9) (19.4) (16.1) (23.1)
(24.7) (20.1) (16.3) (17.3) (21.6)
a In Columns of N1 and N2, the number is the fraction of lattice sites of grain boundaries between different texture components to total lattice sites in a system. b The number within ( ) means the fraction of grain boundaries between different texture components to total grain boundaries in the system.
The effect of size on grain growth behavior was more clearly visualized by tracing several grains in a special environment. An arrangement was made such that Btype grains surrounded ap small grain of A-type in system S1. Surrounding B-type grains were made to have low-mobility grain boundaries except for the sides in contact with the A-type grain. Growth rates of all these grains are shown in Fig. 9. The A-type grain grew rapidly during the early stage due to the high mobility, reaching about four times the initial size at about 500 MCS. However, this particular grain then started to shrink due to the growth of large neighboring B-type grains. Therefore, the grain size effect is more important than the mobility effect for the microstructure evolution since the direction of grain boundary movement is mainly affected by the local curvature induced from the difference in grain size, i.e. grain topology. Another evidence of the grain size effect was found in the multi-component texture system. Fig. 10 shows the frequency of grains found at t = 3000 MCS in five microstructures of each ensemble as a function of their initial size. Microstructures N1, N2 and N3 were generated with differently assigned grain boundary mobilities, ranging from : 0.001 to 1 (Qmax =988), even though individual grains have the same local size environment. Microstructure N3 was defined to have similar conditions as those of N2 in that the areal fraction of anisotropic grain boundaries was same but grain orientation was differently assigned. Despite the significant difference in the grain boundary mobility, initially large grains definitely tend to survive longer. Most of the surviving grains are larger than the average size of the system. This result is consistent with those experimentally found in Al – Mn alloy [21], commercial purity Al [22], high purity Cu [23], and Fe-3wt.%Si alloy [24,25], where selective growth of large grains governed the final texture [26,27]. AGG is a competitive growth process between a few abnormal grains and the majority of normal grains. To incur AGG, therefore, either an extra driving force should exist for the abnormal grains or a restraining
force for the normal grains. The nature of the extra driving force was suggested [28] to be the anisotropic mobility of grain boundaries due to texture. According to the result of the present work, however, the anisotropic movement alone did not cause AGG. For the case of the restraining force, it was suggested [28] that the source could be either solute atoms or secondary phases. According to Srolovitz’s computer simulation [29], however, the restraining action of second phase
Fig. 7. Variation with time of the fraction of grain boundaries of different mobility during grain growth. Misorientation among grains of the initial microstructure was made small in (a) N1, whereas it was made as random in (b) N2.
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Fig. 8. Rate of system energy reduction due to grain growth in microstructures with multi-component textures.
Fig. 9. Growth behavior of a small A-type grain with high mobility boundaries (black dots) and its neighboring B-type grains in system S1.
particles alone is insufficient to allow AGG. Therefore, we think that AGG should be considered in terms of heterogeneous grain boundary mobility and grain size. AGG occurred only in the microstructures with twocomponent texture under a condition of selective growth of the larger-than-average size grains with highmobility boundaries. As shown in Fig. 2(a), these grains have high mobility grain boundaries that maintain their property during growth. Although this kind of situation is conceivable in simulation, it should be difficult to find frequently in real system because the special grain boundary characteristics will be altered during growth.
Recently the process of texture formation in Fe–Si alloys has been debated with respect to the grain boundary mobility. On the grounds of frequency variation during grain growth, two researchers claimed different grain boundaries as responsible for the Goss texture: special CSL boundaries and the boundaries of particular misorientation (20–45°) by Harase et al. and Hayakawa et al., respectively [3–6]. According to the present result, it is unlikely that the high mobility alone, without the influence of the grain size effect, results in the particular recrystallization texture. In fact Lee and
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Fig. 10. Relationship between the initial grain size and the frequency of survival at t = 3000 MCS. The initial grain size was measured by averages of each ensemble and the frequency data were obtained in each ensemble consisting of five simulation microstructures.
Szpunar [24,25] also raised this question and asserted that there must have been a grain size effect. The present result indicates that AGG requires a very strong texture component and, at the same time, a large grain size in the initial microstructure. The direction of grain boundary movement is predominantly governed by the local curvature effect. Therefore, it must be shown that grains of {110}001 orientation must meet the above conditions to grow into abnormal grains. If this condition is difficult to meet in real materials another mechanism will be sought. Heterogeneity in microstructure, such as clustering, may be considered. According to Inokuti [30], Matsuo [31] and Baudin et al. [32] many {110}001 grains are found as clusters in local areas. This phenomenon provides a condition suitable for coalescence of grains with similar orientations, as suggested by Nielsen [33] in a geometrical model study of recrystallization and grain growth. Selective growth due to the curvature effect follows the coalescence, which, combined with high mobility, results in AGG. Another possibility is the presence of additional driving force as discussed by Dunn and Walter [34]. Grains of particular orientation may selectively grow during annealing heat treatment due to anisotropic surface (solid-vapor) energy. This possibility was also raised by the computer simulation study of Srolovitz et al. [29].
5. Conclusions From the Monte Carlo computer simulation study of microstructure evolution based on the viewpoint of grain boundary character distribution the following conclusions were made:
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1. Microstructure evolves toward minimizing the free energy of the system, which is mainly driven by annihilation of small grains. During this process, local fluctuations such as a transient increase of highly mobile grain boundaries may occur. Therefore high-energy, high-mobility grain boundaries do not necessarily vanish early; they may persist and even increase in number. 2. The most influential parameter that determines the final microstructure is the initial gain size. Initially large grains, having more than average nearest neighbor grains, tend to dominate the final state, as opposed to a claim that the grain boundary mobility alone can account for the anisotropic growth of particular grains. 3. AGG in a two-component texture system can only occur under a local situation of large grains with high-mobility boundaries surrounded by small grains of low mobility. Therefore texture formation may not be simply explained in terms of the disappearance of high mobility grain boundaries and dominance of particular low-energy grain boundaries.
Acknowledgements This work was performed under the auspices of the 1998 Research Fund of POSCO.
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[27] B. Hutchinson, E. Nes, Mater. Sci. Forum 94 – 96 (1992) 385. [28] V. Randle, B. Ralph, N. Hansen, in: Proceedings of The Seventh International RISO Symposium, Metall. Mater. Sci., 1986, pp. 123 – 142. [29] D.J. Srolovitz, G.S. Grest, M.P. Anderson, Acta Metall. 33 (1985) 2233. [30] Y. Inokuti, C. Maeda, Y. Ito, Metall. Trans. A 16A (1985) 1613. [31] M. Matsuo, ISIJ Intl. 29 (1989) 809. [32] T. Baudin, J. Jura, J. Pospiech, R. Penelle, in: Proceedings of The Eleventh International Conference on Textures and Materials (ICOTOM-11), International Academy Publishers, 1996, p. 1319. [33] J.P. Nielsen, Recrystallization, Grain Growth and Textures, ASM, Metals Park, 1966, p. 141. [34] C.G. Dunn, J.L. Walter, Recrystallization, Grain Growth and Textures, ASM, Metals Park, 1966, p. 461.
Monte Carlo simulation of microstructure evolution based on grain boundary character distribution H.N. Lee a, H.S. Ryoo b, S.K. Hwang b,*,1 b
a Agency for Defense De6elopment, Yuseong, 305 -600, South Korea Inha Uni6ersity, Department of Metallurgical Engineering, Incheon, 402 -751, South Korea
Received 14 May 1999; received in revised form 22 October 1999
Abstract A Monte Carlo simulation of microstructure evolution was conducted with emphasis on grain growth anomaly and texture formation arising from grain boundary characteristics. Microstructures consisting of strategically varied grain size, grain boundary mobility and texture components were generated. It was found that the initial grain size was of the most significant parameter that determines the topographical properties of the final microstructure. Contrary to some observations and expectations, high-energy grain boundaries of high mobility did not necessarily vanish first but sometimes persisted and even increased in areal fraction depending on the local distribution of characteristics. In a microstructure with two texture components, an abnormal gain growth occurred when a large grain with high mobility selectively grew consuming surrounding small grains. From the analysis of the variation of grain boundary fraction and associated energy reduction, it was concluded that minimization of the total energy of the system, instead of the local instability, governed the evolution characteristics of the microstructure. Published by Elsevier Science S.A. Keywords: Monte Carlo; Microstructure; Grain boundary; Mobility; Energy; Texture; Grain size
1. Introduction Grain boundary character distribution (GBCD) has been the subject of many recent research efforts concerning texture evolution [1,2]. Attempts have been made to correlate the frequency of particular grain boundaries with the development of Goss texture in Fe-3%Si alloy. For example, Harase et al. [3,4] claimed that the texture development in this alloy was caused by the growth of the grains with the Goss orientation, which were associated with special low CSL (S5 or S9) boundaries during growth. These boundaries, therefore, were concluded to be of high mobility on the ground that they were relatively immune to segregation of solute atoms or secondary phases. Hayakawa et al. [5,6], on the other hand, claimed that the boundaries * Corresponding author. Tel.: +82-328607537; fax: +82328625546. E-mail address: [email protected] (S.K. Hwang) 1 Jointly appointed by the Center for Advanced Aerospace Materials. 0921-5093/00/$ - see front matter Published by Elsevier Science S.A. PII: S 0 9 2 1 - 5 0 9 3 ( 9 9 ) 0 0 7 2 5 - X
with the misorientation angle in the range of 20–45° were of high mobility on the ground that they were of high energy. Precipitate coarsening on these boundaries would reduce the pinning force against abnormal growth of the Goss grains. These apparently conflicting observations were made on the basis of the frequency of particular boundaries, and as such, the disagreement arises from imperfect GBCD information on individual grains. Most of all, there is no a priori reason for the quick disappearance of the mobile boundaries. Evolution of particular boundaries must satisfy the local as well as the system average energy minimization criterion. Therefore, a comprehensive set of information concerning the grain itself as well as the surroundings is necessary to account for in detail the path of microstructural and texture evolution. In the present study we attempted to develop a methodology to define the stages of microstructural evolution from the standpoint of a comprehensive information system. In a simulated microstructure, the orientation and the grain boundary characteristics of each grain were individually assigned and their evolu-
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tion stages were monitored. In the end it will be shown that the boundaries of high mobility do not necessarily disappear quickly and sometimes they persist even longer than low-energy, low mobility boundaries. It will also be shown that the criterion of local energy minimization requires the initial grain size, rather than the grain boundary mobility per se, as the crucial factor to control the microstructure distribution and texture.
2. Monte Carlo simulation method The method to simulate grain growth was according to the one developed by the Exxon group [7,8]. In this method, a system is composed of a triangular lattice, which has a periodic boundary condition at each end. A system contained a total of 40 000 (200× 200) lattice points. Each lattice point is made to have a crystal orientation by assigning a number, Q. A grain is defined as an aggregate of the lattice points with an identical Q number. A grain boundary is defined as the interface between two lattice points of different Q number. Since each grain is assigned with its own Q number ranging from 1 to 988 (Qmax), coalescence of grains with the same orientation is prohibited in this method. Elapse of time is measured as a unit of MCS during which 40,000 flipping attempts of the lattice points occur in a system. Orientation change of lattice points means grain boundary movement. Therefore, the grain boundary mobility can be assigned by adjusting the probability of the orientation change. In the present study, two methods were used to assign mobility to individual grain boundaries. The first method, which was proposed by Rollett et al. [8], defines the grain boundary mobility, M, as a factor modifying the probability of the orientation change. Thus: P=
1 for DE 50 m*
P=
exp(−DE/kBT) for DE \ 0 m*
(1)
where m* is mobility factor (M =1/m*), DE is the energy change associated with the orientation change of a lattice point, kB is the Boltzmann constant and T is absolute temperature. Isotropic mobility corresponds to m*=1. Anisotropy of the mobility can be introduced by assigning each boundary individual value of m* according to the degree of misorientation. This method was used for the study of two-component texture systems. Another method of assigning anisotropy to the grain boundary mobility, newly used in the present work for the multi-component texture systems, was as follows:
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Qi − Qj (2) Qmax − 1 where dQ is the misfit parameter between adjacent grains. In this scheme, orientation switching of a lattice point is allowed only if M is larger than a random number so that the larger the grain boundary misorientation the higher the mobility. In the present work, attempt of a lattice point switching into a different orientation is limited to one nearest neighbor orientation [9]. Therefore, a grain with a new orientation is not allowed to nucleate within an existing grain. M= dQ =
3. Result of Monte Carlo computer simulation
3.1. System with two-component texture The initial microstructure used in the computer simulation was made by the Voronoi mosaic method. The characteristics of this microstructure are explained elsewhere [10]. Distribution of the grain size in the initial microstructure, as determined by the number of lattice points, fits a gamma distribution function as shown in Fig. 1 with the average grain size A( being 40.5. However, topological frequency, as determined by the number of sides of grains, fits a log-normal distribution function. The average number of sides of grains was 6 and the grain size increased linearly with the number of sides. Microstructural evolution was studied in the matrix textured with two components, A and B. Four different types, designated as S1, S2, S3 and S4, of two-component textures were generated. Each type had either different areal proportions of the two texture components or different grain size for a particular texture component. For types S1, S2 and S3, the areal fraction of A-type grains (grains with A component) texture was set as 0.25 and the average grain size was varied. In the S1 type microstructure, all A-type grains were smaller than any B-type grain whereas the case for S2 was opposite. In the S3 type microstructure, the grain size of either texture component was set as random. S4 had the same property as that of S3 except that the areal proportion of A-type and B-type grains was 1:1. These categories of microstructures were designed to study the effect of the initial grain size and grain boundary mobility between the texture components on subsequent growth. Anisotropy of grain boundary mobility was assigned in such a way that the boundaries of grains of the same texture component (either AA or BB) was made slow (M= 0.1) and the others (AB) fast (M= 1.0). This arrangement resulted in mostly fast boundaries around small grains in S1 and large grains in S2. Computer simulation of growth was conducted for three different microstructures for each category at T :0 and ensemble averages are presented.
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Fig. 1. Grain size distribution of the initial microstructure generated by Voronoi method.
Fig. 2. Evolution of microstructure in the systems with two-component texture: (a) S1; (b) S2 and (c) S4. In all microstructures, A-type grains appear as white whereas B-type grains as gray. The black boundary lines and the red lines indicate grain boundaries of low mobility and of high mobility, respectively.
Conditions
S1 S2 S3 S4 a b
AA
0.25 0.25 0.25 0.50
A A:A B (t= 0 MCS)
0.56A0:1.36A0 2.05A0:0.85A0 0.97A0:1.01A0 1.05A0:0.95A0
A A/A0 (t =2000 MCS)
150 10 12 –
tr
1300 MCS 120 MCS 250 MCS –
t =0 MCS
t = 2000 MCS
G H, %
GL AA, %
GL BB, %
G H, %
GL AA, %
GL BB, %
9.02a (33.5)b 6.18 (22.9) 9.77 (36.3) 12.84 (47.7)
4.54 1.91 1.94 6.73
13.37 18.84 15.22 7.37
1.16 (25.8) 0 (0) 0.61 (7.9) 2.73 (38.3)
1.61 8.45 6.99 1.33
1.72 (38.4) 0 (0) 0.02 (0.3) 3.08 (43.1)
(16.8) (7.1) (7.2) (25.0)
(49.7) (70.0) (56.5) (27.3)
(35.8) (100) (91.8) (18.6)
In columns of t= 0 and 2000 MCS, the number is the fraction of lattice sites of grain boundaries between different texture components to total lattice sites in a system. In columns of t= 0 and 2000 MCS, the number within ( ) means the fraction of grain boundaries between different texture components to total grain boundaries in the system.
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Table 1 Analysis result of microstructures before and after grain growth in two-component texture systems
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Table 1 summarizes the result of computer simulation of microstructural evolution up to 2000 MCS. Graphical presentation of grain growth stages of representative microstructures of each category is given in Fig. 2. The one for S3 was omitted since it was very similar to that of S2. Grain growth rates of each texture component and that of the total system are shown in Fig. 3. All the microstructures in the categories S1, S2 and S3 showed abnormal grain growth (AGG), characterized by a bimodal grain size distribution, and a texture reversal. Characteristics of these developments, however, considerably differ depending on the starting size of A-type grains. In the case of S1, where the starting size of A-type grains was small, most of the A-type grains vanished early, before 100 MCS. Only a few of the A-type grains survived, which began to grow abnormally at around 500 MCS, becoming about 150 times the initial size at 2000 MCS. In the case of S2, where the starting size of A-type grains was large, A-type grains grew into abnormal grains in the very early stage of growth, at about 40 MCS, and consumed B-type grains at 500 MCS. Beyond this point there were no more high-mobility grain boundaries, thus normal grain growth set in. During the growth, there appeared a bimodal distribution of grain sizes as shown in Fig. 2(b) and Fig. 3(b). In the case of S3, where the sizes of A-type or B-type grains were random, the growth pattern was very similar to that of the S2 case except for a late texture reversal and slightly larger size of fully grown A-type grains compared to those in case S2. This is because of the increased distance of collisions of the large high-mobility grains that were formed by annihilation of small grains of high mobility. Therefore if the areal fraction of A-type grains were lower than 0.25 then there would be more abundant AGG in S2 than that in S1. In the case of S4, where the two types of texture components were balanced with random size distribution, only normal grain growth was observed. During the normal growth, the time exponent [11], n, of the growth rate was 0.47, excluding the initial transition. Variation of the number fraction of grain boundaries, of high mobility and low mobility, during microstructural evolution was examined as a function of time. The number fraction of grain boundaries was calculated by counting the lattice points at boundaries such as AA-type (G LAA), BB-type (G LBB) or AB-type (G H). The former two are low mobility boundaries and the latter one is the high mobility boundary. Fig. 4 shows the number fraction of grain boundaries varying with time for the four categories of microstructure. Due to the large areal fraction of B-type grains, BB-type grain boundaries are initially predominant in S1, S2 and S3. Most of the AA and AB-type grain boundaries quickly disappear in S1, yielding to BB-type
boundaries, due to the small initial grain size of A-type grains; however, they increased slightly again due to the abnormal growth of several surviving grains. In contrast, AB-type grain boundaries quickly became annihilated along with BB-type boundaries in S2, yielding to AA-type boundaries. This behavior was repeated in S3 except that some AB and BB-type boundaries persisted. In the case of S4, all the grain boundaries showed a monotonic decrease. Overall, it must be emphasized that highly mobile AB-type boundaries do not always vanish first. The energy of the total system, as defined by the following equation decreased with time as shown in Fig. 5: J N nn Et = % % (1−dij ) 2 i j
(3)
In the above equation, Et is the energy of the total system consisting of N lattice sites, J is the unit energy, dij is the Kronecker delta and nn is the number of nearest neighbor sites. As expected from the variation of the fraction of grain boundaries (Fig. 4), the reduction rate of the total energy was most significant in the early stage of growth. System S1 showed a rather peculiar trend in that its energy decreased fast in the beginning; maintained the highest energy level in the intermediate stage; then decreased fast again in the final stage. AGG was responsible for this development.
3.2. Systems of multi-component texture (Random texture) Using identical initial microstructures to those of the two-component texture system, systems of random texture were generated by assigning each grain an individual orientation as well as individual mobilities. According to Eq. (2), then, grain boundaries with a variety of mobility exist in a microstructure. To simplify analysis, grain boundaries were categorized into five different groups: G1 for 0BM50.2, G2 for 0.2B M50.4, . . . and G5 for 0.8 BM5 1.0, with an increasing order of mobility. Two different systems of microstructures were generated; N1, with most grains of similar orientations being neighbored, and N2, with those randomly mixed orientations. In system N1, therefore, low mobility grain boundaries were predominant (some areas as clustering) whereas in N2, grain boundaries of different mobility were evenly distributed. Five simulations were performed for each condition of T:0 and ensemble averages are presented. The results of microstructural evolution for systems N1 and N2 are shown in Fig. 6(a) and (b), respectively. In system N1, a few initially large grains surrounded by high mobility grain boundaries grew abnormally to
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Fig. 3. Grain growth rates of systems with two-component texture. See text for the description of microstructure of (a) S1, (b) S2, (c) S3 and (d) S4.
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eight times the average size at 3000 MCS. In system N2, on the other hand, a normal growth was observed evenly for most of grains.
Variations with time of the fraction of grain boundaries in multi-component texture microstructure are summarized in Table 2 and Fig. 7. The trends of grain boundary annihilation or reinforcement shown in Fig. 7 are considerably different from those shown in Fig. 4. All the grain boundary areas gradually diminished along with grain growth, except for a slight departure of the highest mobility grain boundaries, G5, due to abnormal growth of several grains in local areas in N1 system. Fig. 8 shows the rate of energy reduction of the system with grain growth in microstructures with multicomponent textures. The two systems, N1 and N2, showed a linear (in the log–log plot) decrease of energy with grain growth. Thus: −
dEt R( =k dt R( 0
2a
(4)
where k and a are constants, R( 0 and R( are the average radii of grains at t= 0 and t=t, respectively. From the slopes of the curves in Fig. 8 it was found that a= − 1.25 and n= 0.67 for N1 and a= − 1.39 and n=0.56 for N2. During the initial growth stage the energy reduction rate of system N2 was higher than that of N1; however, in the later stage of growth the two systems showed a similar rate of energy reduction. This phenomenon was due to the grain boundary characteristics. In system N2 there are more high mobility grain boundaries, which efficiently annihilate low mobility grain boundaries in the initial stage. In the later stage, however, the rate of annihilation is reduced due to collisions of high mobility grain boundaries. In the case of N1, there are mostly low mobility grain boundaries; therefore, the annihilation process by a few high-mobility grain boundaries takes more time [see Fig. 7(a)]. Comparison of the n values substantiates this interpretation.
4. Discussion
Fig. 4. Time-dependent variation of the fraction of grain boundaries L in the microstructures with two-component texture. The G L AA, G BB and G H stand for boundaries of A-A type, B-B type and A-B type, respectively. See text for the description of microstructure of (a) S1, (b) S2, (c) S3 and (d) S4.
Whether or not grain boundaries of high mobility always disappear early in the grain growth stage is an important question in the field of grain size control and texture control. Among many studies made so far, the following observations are noteworthy. Martikanen and Lindroos [12] reported an increasing frequency of low-mobility low-energy grain boundaries during growth of low-carbon, low-nitrogen 25wt%Cr-1wt%Mo ferritic stainless steel. In case of Fe-3wt%Si where grain growth was controlled with AlN and MnS particles, increasing frequency of S1 boundaries was found after secondary recrystallization [3,4]. In a similar alloy system, the boundaries with misorientation angle of 20– 45° disappeared quickly [5,6]. A few previous computer
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183
Fig. 5. Variation with time of the system energy in microstructures with two-component texture.
simulation works [13,14] also asserted the quick disappearance of the high-mobility boundaries regardless of texture. These interpretations, however, have not presented the rationale for the lack of the effect of the texture of the entire system. In principle, it must be possible that grains of particular boundary characteristics continually form during growth due to the special orientation distribution of all the gains contained in the system, thus the phenomenon should be texturedependent. The increasing frequency of low energy grain boundaries during grain growth appears to be consistent with the thermodynamical concept of minimization of the system free energy. However, the total energy of a system is not determined solely by the ‘frequency’ of particular boundaries but by the sum of all grain boundary energies, which depends on the distribution of grain boundaries. Therefore, although the total energy of the system gradually decreases with time as shown in Fig. 8, the initial microstructure may evolve into a microstructure apparently consisting of frequent high-energy boundaries. It is possible for some local high-energy boundaries that the rate of decrease in energy is temporarily negative. This is because the main factor governing the system energy reduction is the total grain boundary area instead of preferential disappearance of some high-energy boundaries. During the process of grain growth, there may be a transient energy increase in local areas due to annihilation of small grains with low-energy boundaries by
large grains with high-energy boundaries. This was predicted by a geometrical model study [15] and was also experimentally observed in 316L stainless steel [9], pure Al [16] and Nimonic alloy [17]. Boundaries of large grains in these cases were found to have a random high-energy structure. Many evidences (for example, Figs. 5 and 8) found in the present work support the hypothesis of the total energy criterion for microstructural evolution. There is no a priori reason why high-energy high-mobility grain boundaries should disappear faster than low energy boundaries do. Contradictory results reported in the earlier works, therefore, must be carefully re-analyzed. For example, encounters of the grains of minor texture component were quoted as the reason for the increase of low-mobility grain boundaries in the later stage of growth [14]. However, it is not highly likely that grains of minor texture component should meet frequently, forming low-energy, low-mobility boundaries. In an earlier work [18] the present authors showed that the texture component consisting of small grains (AB 0.6A) vanishes quickly regardless of its volume fraction. This phenomenon is easily understood from the standpoint of the total energy of the system as well as from the von Neumann–Mullins analysis [19,20] that states the predominance of the growth stage by large grains having more than six nearest neighbor grains. Since the energy of a two-dimensional system is proportional to the fraction of the grain boundaries per
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area the system consisting of small grains experiences a high driving force for growth, which requires sacrifice of small grains. In the present work, however, it was shown [Fig. 2(a)] that only several grains of a minor texture component can grow abnormally and become the major
texture of the system. This indicates that although the mobility of grain boundaries is important, the grain size effect is even more important. Compared at the same volume fraction of the minor texture component, small grains vanished whereas large grains survived.
Fig. 6. Evolution of microstructures in the systems composed of multi-component textures with grain boundaries of individually different mobility: (a) N1 and (b) N2.
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Table 2 Analysis result of grain boundary fractions before and after grain growth in multi-component systems Boundary
G1 G2 G3 G4 G5
Mobility
0BM50.2 0.2BM50.4 0.4BM50.6 0.6BM50.8 0.8BM51.0
N1
N2
t =0 MCS,%
t=3000 MCS, %
t=0 MCS, %
t= 3000 MCS, %
10.41a (38.6)b 7.18 (26.7) 5.22 (19.4) 3.07 (11.4) 1.07 (3.9)
3.52 0.81 0.71 0.70 0.86
5.25 5.90 5.23 4.34 6.22
1.56 1.27 1.03 1.09 1.36
(53.3) (12.3) (10.8) (10.6) (13.0)
(19.5) (21.9) (19.4) (16.1) (23.1)
(24.7) (20.1) (16.3) (17.3) (21.6)
a In Columns of N1 and N2, the number is the fraction of lattice sites of grain boundaries between different texture components to total lattice sites in a system. b The number within ( ) means the fraction of grain boundaries between different texture components to total grain boundaries in the system.
The effect of size on grain growth behavior was more clearly visualized by tracing several grains in a special environment. An arrangement was made such that Btype grains surrounded ap small grain of A-type in system S1. Surrounding B-type grains were made to have low-mobility grain boundaries except for the sides in contact with the A-type grain. Growth rates of all these grains are shown in Fig. 9. The A-type grain grew rapidly during the early stage due to the high mobility, reaching about four times the initial size at about 500 MCS. However, this particular grain then started to shrink due to the growth of large neighboring B-type grains. Therefore, the grain size effect is more important than the mobility effect for the microstructure evolution since the direction of grain boundary movement is mainly affected by the local curvature induced from the difference in grain size, i.e. grain topology. Another evidence of the grain size effect was found in the multi-component texture system. Fig. 10 shows the frequency of grains found at t = 3000 MCS in five microstructures of each ensemble as a function of their initial size. Microstructures N1, N2 and N3 were generated with differently assigned grain boundary mobilities, ranging from : 0.001 to 1 (Qmax =988), even though individual grains have the same local size environment. Microstructure N3 was defined to have similar conditions as those of N2 in that the areal fraction of anisotropic grain boundaries was same but grain orientation was differently assigned. Despite the significant difference in the grain boundary mobility, initially large grains definitely tend to survive longer. Most of the surviving grains are larger than the average size of the system. This result is consistent with those experimentally found in Al – Mn alloy [21], commercial purity Al [22], high purity Cu [23], and Fe-3wt.%Si alloy [24,25], where selective growth of large grains governed the final texture [26,27]. AGG is a competitive growth process between a few abnormal grains and the majority of normal grains. To incur AGG, therefore, either an extra driving force should exist for the abnormal grains or a restraining
force for the normal grains. The nature of the extra driving force was suggested [28] to be the anisotropic mobility of grain boundaries due to texture. According to the result of the present work, however, the anisotropic movement alone did not cause AGG. For the case of the restraining force, it was suggested [28] that the source could be either solute atoms or secondary phases. According to Srolovitz’s computer simulation [29], however, the restraining action of second phase
Fig. 7. Variation with time of the fraction of grain boundaries of different mobility during grain growth. Misorientation among grains of the initial microstructure was made small in (a) N1, whereas it was made as random in (b) N2.
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Fig. 8. Rate of system energy reduction due to grain growth in microstructures with multi-component textures.
Fig. 9. Growth behavior of a small A-type grain with high mobility boundaries (black dots) and its neighboring B-type grains in system S1.
particles alone is insufficient to allow AGG. Therefore, we think that AGG should be considered in terms of heterogeneous grain boundary mobility and grain size. AGG occurred only in the microstructures with twocomponent texture under a condition of selective growth of the larger-than-average size grains with highmobility boundaries. As shown in Fig. 2(a), these grains have high mobility grain boundaries that maintain their property during growth. Although this kind of situation is conceivable in simulation, it should be difficult to find frequently in real system because the special grain boundary characteristics will be altered during growth.
Recently the process of texture formation in Fe–Si alloys has been debated with respect to the grain boundary mobility. On the grounds of frequency variation during grain growth, two researchers claimed different grain boundaries as responsible for the Goss texture: special CSL boundaries and the boundaries of particular misorientation (20–45°) by Harase et al. and Hayakawa et al., respectively [3–6]. According to the present result, it is unlikely that the high mobility alone, without the influence of the grain size effect, results in the particular recrystallization texture. In fact Lee and
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Fig. 10. Relationship between the initial grain size and the frequency of survival at t = 3000 MCS. The initial grain size was measured by averages of each ensemble and the frequency data were obtained in each ensemble consisting of five simulation microstructures.
Szpunar [24,25] also raised this question and asserted that there must have been a grain size effect. The present result indicates that AGG requires a very strong texture component and, at the same time, a large grain size in the initial microstructure. The direction of grain boundary movement is predominantly governed by the local curvature effect. Therefore, it must be shown that grains of {110}001 orientation must meet the above conditions to grow into abnormal grains. If this condition is difficult to meet in real materials another mechanism will be sought. Heterogeneity in microstructure, such as clustering, may be considered. According to Inokuti [30], Matsuo [31] and Baudin et al. [32] many {110}001 grains are found as clusters in local areas. This phenomenon provides a condition suitable for coalescence of grains with similar orientations, as suggested by Nielsen [33] in a geometrical model study of recrystallization and grain growth. Selective growth due to the curvature effect follows the coalescence, which, combined with high mobility, results in AGG. Another possibility is the presence of additional driving force as discussed by Dunn and Walter [34]. Grains of particular orientation may selectively grow during annealing heat treatment due to anisotropic surface (solid-vapor) energy. This possibility was also raised by the computer simulation study of Srolovitz et al. [29].
5. Conclusions From the Monte Carlo computer simulation study of microstructure evolution based on the viewpoint of grain boundary character distribution the following conclusions were made:
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1. Microstructure evolves toward minimizing the free energy of the system, which is mainly driven by annihilation of small grains. During this process, local fluctuations such as a transient increase of highly mobile grain boundaries may occur. Therefore high-energy, high-mobility grain boundaries do not necessarily vanish early; they may persist and even increase in number. 2. The most influential parameter that determines the final microstructure is the initial gain size. Initially large grains, having more than average nearest neighbor grains, tend to dominate the final state, as opposed to a claim that the grain boundary mobility alone can account for the anisotropic growth of particular grains. 3. AGG in a two-component texture system can only occur under a local situation of large grains with high-mobility boundaries surrounded by small grains of low mobility. Therefore texture formation may not be simply explained in terms of the disappearance of high mobility grain boundaries and dominance of particular low-energy grain boundaries.
Acknowledgements This work was performed under the auspices of the 1998 Research Fund of POSCO.
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