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Tertiary grain growth driven by surface energy
Scripta Materialia 45 (2001) 267±272
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Tertiary grain growth driven by surface energy Jung-Kyu Junga, Young-Joon Parkb*, Nong-Moon Hwangac, and Young-Chang Jooa a School of Materials Science and Engineering, Seoul National University, Seoul 151-742, South Korea Thin Film Technology Research Center, Korea Institute of Science and Technology, P.O. Box 131, Cheongryang, Seoul 136-791, South Korea c Korea Research Institute of Standards and Science, Taejon 305-600, South Korea
b
Received 31 January 2001; accepted 19 March 2001
Keywords: Grain growth; Texture; Computer simulation
Introduction Grain growth has been an important issue because numerous properties of structures or devices show strong dependence on grain size [1]. Behavior of grain growth can be classi®ed into normal and abnormal one. While the former results in narrow and monomodal grain size distribution, the latter, in which only a few grains are selected to grow and consume the other grains or the matrix grains, develops transient bimodal grain size distribution before impingements among them [1,2]. The reduction of the grain boundary area normally provides a driving force for grain growth of bulk materials, of which behavior can be characterized as normal growth. In materials processing of ®lms or sheets, however, the surface energy provides an additional driving force for grain growth. In this case, grains with the minimum surface energy grow at the expense of other grains with higher surface energy [3±5]. This additional driving force begins to play an active role when the grain size is comparable to the thickness of a thin ®lm. After stagnation of the `primary' grain growth driven by the grain boundary curvature, the second stage of grain growth can be induced by the surface energy. This second stage grain growth is also referred to as secondary grain growth, in which a few grains with the low surface energy can undergo exclusive growth over other grains, replacing grains with the high surface energy by the grains with the low surface energy. Those grains undergoing exclusive growth are usually much larger than the other grains and are called secondary grains. The microstructure is characterized by bimodal grain size distribution. *
Corresponding author. E-mail address: [email protected] (Y.-J. Park).
1359-6462/01/$ - see front matter Ó 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 6 2 ( 0 1 ) 0 1 0 1 7 - X
268
J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
Such exclusive or abnormal growth by secondary grains continues until they impinge each other. After the impingement, the secondary growth becomes stagnated. Since grains with the lowest surface energy might have similar orientations, a strong texture is often accompanied by secondary grain growth. Even after the completion of secondary grain growth, a few grains may undergo further exclusive growth, which leads to the stage of tertiary grain growth. Often, the tertiary grain growth is triggered by the change in the gaseous environment, which is believed to change the surface orientation of the lowest energy changes by the adsorption of gas atoms [2,6,7]. Again, a stronger texture can result from tertiary grain growth. In this paper, we will show that the microstructure evolution similar to tertiary grain growth can be achieved without the change of ambient gases when the grains with the lowest surface energy have a very small initial fraction. We will consider a simple system with three types of grains corresponding to dierent surface energies: high, intermediate, and low. We will show that the initial fraction of each grain type as well as the anisotropic surface energy aects the microstructure evolution. If the initial fraction of the high surface energy is high and the initial fraction of the grains with the low surface energy is much smaller than that with the intermediate surface energy, the grains with the intermediate surface energy would grow abnormally and replace most of the grains with the high surface energy resembling secondary grain growth. Although the grains with the low surface energy may continue to grow at the highest rate, the initial microstructure evolution would be dominated by growth of the grains with the intermediate surface energy because of their much higher initial fraction. After impingements among secondary grains, however, only grains with the low surface energy can continue to grow abnormally over the secondary grains, resulting in tertiary-like grain growth. These aspects of grain growth are examined using Monte Carlo simulation. Description of simulation Two-dimensional Monte Carlo simulation based on the modi®ed Potts model, in which only ®rst-nearest-neighbor lattice points are of concern [3,4], was used to study the grain growth driven by surface energies. The initial triangular lattice structure had 10 000 (100 100) grains of uniform size with random orientations on a 200 200 matrix. 40 000 attempts for atomic jumps correspond to one Monte Carlo step (MCS). We also used a ®xed lattice temperature T for the Boltzmann probability p exp Q= kT to be 0.03, where Q and k are the activation energy of grain boundary migration and the Boltzmann constant, respectively [3,4]. Contrary to others' model [3,4], we introduced three, rather than just two, types of grains according to their surface energies: matrix, type I and type II grains. Matrix grains would have the highest surface energy, and dominate the initial microstructure, while much less type I and II grains would have the intermediate and the lowest surface energies such that cm > cI > cII , where cm , cI and cII were the surface energies per unit area of matrix, type I, and type II grains, respectively.
J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
269
For simplicity, we assumed the equivalency between interface energy and surface energy of any grain with the same orientation and restricted the condition on the relative magnitude of each energy term to cm 1:5cI 3cII . Although surface energies were considered to be anisotropic, the grain boundary energy (cGB ) was assumed to be isotropic and set to cGB cII =3. Abnormal and competitive growth of type I and II grains would necessarily give rise to more complicated phenomena. Of course type II grains should dominate the ®nal grain structure, but the transient structure before completion of all stages of grain growth is likely to be aected by initial fraction of each type. Considering that the rate of impingement is strongly dependent upon initial fractions, relative initial fraction of each grain type may determine which type of grains will be dominant and how fast they will grow. Designating xI and xII as initial fractions of type I and type II grains, respectively, we restricted initial fractions conditions to xI P xII , where the initial fraction of the matrix grains would be 1 xI xII . If the initial fraction of type II grains is higher than that of type I grains (xI < xII ), type II grains would immediately replace the whole microstructure, which is physically trivial. Results and discussion Case A: xII being comparable to xI To examine the initial fraction eect, we performed the simulation with the condition of xI xII at ®rst. We set xI xII 0:05 so that the initial fraction of the matrix grains would be 0.9. The resultant microstructure evolution is illustrated in Fig. 1, in which matrix grains with the high surface energy are in black, type II grains with the low surface energy are in white, and type I grains with the intermediate energy are in gray. In Fig. 1, type II grains with the lowest surface energy (cII ) immediately predominate the microstructure because the initial fraction of type II grains is so high that they can catch up with and overwhelm the kinetics of type I grains. Since type I grains disappear almost simultaneously with matrix grains at early times, type II grains may be regarded as secondary grains. Note that, however, both types of grains exhibit abnormal growth behavior. The results from Fig. 1 can be veri®ed kinetically in Fig. 2. We plotted areal fraction of each type to represent the transition of texture in Fig. 2(a) showing that type I cannot
Fig. 1. Microstructure evolution with the initial fractions of matrix (black), type I (gray) and II (white) grains being 0.9, 0.05 and 0.05, respectively. The unit time elapsed is referred to as 1 MCS.
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J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
Fig. 2. (a) Areal fraction and (b) relative growth velocity for each type of grain with regard to an elapsed time from the data of Fig. 1. `A' and `B' indicate the time at which matrix grains and type I grains become extinct, respectively.
become a component of major texture. It is because the growth kinetics of type II grains easily surpass that of type I grains, i.e. type II grains consume type I grains as well as matrix grains almost simultaneously. The kinetics of grain growth may be also examined using the relative grain growth velocity vr [8] ! o r vr 1 ot r where r is the average radius of grains of a concerned type and r is the average radius of all grains. r and r are determined by assuming that the initial fraction of type I or type II grains is negligible compared to that of matrix grains. The growth behavior is de®ned to be abnormal if vr > 0 [8]. In our simulation, two kinds of vr are introduced: the relative growth velocity of type I grains (vI ), and that of type II grains (vII ). As in Fig. 2(b), vI and vII are high and positive at the beginning, i.e. they show abnormal growth behavior. Then vI and vII decrease rapidly due to impingements among type I or type II grains. The resultant negative values of vI and vII mean that type I and type II grains now exhibit normal growth behavior. However, after about 15 MCS, vI and vII begin to increase because of a mathematical reason: each r would approach to r as there remain less matrix grains. After about 30 MCS (the point `A' in Fig. 2(b)), no matrix grain remains and vI becomes nearly zero. Similarly, no type I grain remains, i.e. there are only type II grains after about 35 MCS (the point `B'). Both results of Fig. 2 shows that matrix and type I grains disappear almost simultaneously. One can deduce from the results of Fig. 2 that, although the growth behavior of both type I and type II grains are much alike as in Fig. 2(b), type II grains having the lowest energy could grow immediately and overwhelmingly to predominate the microstructure by consuming all the other grains. There are an appreciable number of type II grains at the beginning of grain growth, which make type I grains consumed by type II grains almost simultaneously with matrix grains, as in Fig. 2. Thus the microstructure evolution in Fig. 1 can be said to show secondary grain growth, in which type II grains would be secondary ones.
J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
271
Case B. xII being much smaller than xI If the fraction of type II grains is much smaller than that of type I grains, i.e. xI xII , a drastically dierent microstructure is evolved. Type I grains will grow dominantly over matrix grains at ®rst, but eventually they will be replaced by type II grains. The microstructure evolution in which the fractions of type I and II grains were 0.099 and 0.001, respectively, is shown in Fig. 3. In Fig. 3, type I grains are dominant in the transient microstructure since the initial fraction of type II grains is too low to be signi®cant at early times (for example, see 25 MCS of Fig. 3). After impingements and resultant stagnated growth of type I grains, however, only type II grains can grow over type I grains still exhibiting abnormal behavior. In this case, type I and type II grains may be identi®ed with secondary and tertiary grains, respectively. The kinetic aspect of tertiary grain growth is examined in Fig. 4 in a similar way to that of Fig. 2. Fig. 4(a) exhibits that type I or secondary grains become dominant at ®rst, but the ®nal microstructure consists of type II or tertiary ones, i.e. transition of texture occurs following the sequence of matrix ± type I ± type II. Similarly, in Fig. 4(b), comparison between vI and vII shows that secondary growth of type I grains is followed by tertiary growth of type II grains. As in the results of Fig. 2(b), Fig. 4(b) indicates that matrix and type I grains disappear completely about 30 MCS (the point `A') and 200 MCS (the point `B'), respectively.
Fig. 3. Microstructure evolution with the initial fractions of matrix, type I (gray), and II (white) grains being 0.9, 0.099, and 0.001, respectively. The large grain size of type II grains is due to low initial fraction of them.
Fig. 4. (a) Areal fraction and (b) relative growth velocity for each type of with regard to an elapsed time from the data of Fig. 3. `A' and `B' indicate the time at which matrix grains and type I grains become extinct, respectively.
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J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
Figs. 3 and 4 show that the microstructure characteristics of tertiary grain growth can be evolved without any ambient change resulting in the modi®cation of surface energies by a simple assumption on the initial fractions of the type I and II grains. This tertiary-like grain growth can take place if the initial fraction of grains with the lowest surface energy (xII ) is suciently lower than that of grains with intermediate surface energy (xI ). The grains with the intermediate surface energy may predominate the microstructure by consuming the entire matrix grains transiently due to their high initial fraction (secondary grain growth), but their growth would be stagnant after the impingements among them. Finally, the grains with the lowest surface energy will consume all the other grains (tertiary grain growth). For the ®nal microstructure shown in Fig. 3 to be regarded as tertiary grain growth, one should be aware that tertiary grains in our suggestion have existed in the initial microstructure and continued to grow abnormally. That is, when grain growth is mainly driven by surface energy, determination of secondary or tertiary grain growth may be a matter of the initial fraction of the respective grains. Such a kinetic possibility is meaningful from the point of view that the same process of grain structure evolution can explain the characteristics of both secondary and tertiary growth only by the differences in an initial fraction of each component. Conclusion A possibility of tertiary grain growth is suggested. If the fraction of the grains with the lowest surface energy is suciently low and the fraction of the grains with the intermediate surface energy is appreciable, the microstructure initially evolved is dominated by the grains with the intermediate surface energy. But this microstructure is still unstable energetically and the grains with the lowest surface energy will replace all the others. If the initial microstructure evolution is called the secondary grain growth, the later microstructure evolution can be called the tertiary grain growth. It is dierent from the conventional tertiary grain growth in that there is no need to change gaseous environment for selection of tertiary grains among secondary ones that have been stagnated. Although our `tertiary-like' grain growth may not be general but just a special case, it indicates another aspect of abnormal grain growth driven by anisotropic surface energy. References [1] [2] [3] [4] [5] [6] [7] [8]
Humphreys, F. J., & Hatherly, M. (1995). Grain growth and related annealing phenomena. Oxford, UK: Elselvier. Thompson, C. V. (1990). Annu Rev Mater Sci 20, 245. Srolovitz, D. J., Grest, G. S., & Anderson, M. P. (1985). Acta Metall 33, 2233. Rollett, A. D., Srolovitz, D. J., & Anderson, M. P. (1989). Acta Metall 37, 1227. Frost, H. J., Thompson, C. V., & Walton, D. T. (1992). Acta Metall Mater 40, 779. Kramer, J. J., & Foster, K. (1965). Trans Met Soc AIME 233, 1244. McLean, M., & Mykura, H. (1965). Acta Metall 13, 1291. Rios, P. R. (1992). Acta Metall Mater 40, 2765.
www.elsevier.com/locate/scriptamat
Tertiary grain growth driven by surface energy Jung-Kyu Junga, Young-Joon Parkb*, Nong-Moon Hwangac, and Young-Chang Jooa a School of Materials Science and Engineering, Seoul National University, Seoul 151-742, South Korea Thin Film Technology Research Center, Korea Institute of Science and Technology, P.O. Box 131, Cheongryang, Seoul 136-791, South Korea c Korea Research Institute of Standards and Science, Taejon 305-600, South Korea
b
Received 31 January 2001; accepted 19 March 2001
Keywords: Grain growth; Texture; Computer simulation
Introduction Grain growth has been an important issue because numerous properties of structures or devices show strong dependence on grain size [1]. Behavior of grain growth can be classi®ed into normal and abnormal one. While the former results in narrow and monomodal grain size distribution, the latter, in which only a few grains are selected to grow and consume the other grains or the matrix grains, develops transient bimodal grain size distribution before impingements among them [1,2]. The reduction of the grain boundary area normally provides a driving force for grain growth of bulk materials, of which behavior can be characterized as normal growth. In materials processing of ®lms or sheets, however, the surface energy provides an additional driving force for grain growth. In this case, grains with the minimum surface energy grow at the expense of other grains with higher surface energy [3±5]. This additional driving force begins to play an active role when the grain size is comparable to the thickness of a thin ®lm. After stagnation of the `primary' grain growth driven by the grain boundary curvature, the second stage of grain growth can be induced by the surface energy. This second stage grain growth is also referred to as secondary grain growth, in which a few grains with the low surface energy can undergo exclusive growth over other grains, replacing grains with the high surface energy by the grains with the low surface energy. Those grains undergoing exclusive growth are usually much larger than the other grains and are called secondary grains. The microstructure is characterized by bimodal grain size distribution. *
Corresponding author. E-mail address: [email protected] (Y.-J. Park).
1359-6462/01/$ - see front matter Ó 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 6 2 ( 0 1 ) 0 1 0 1 7 - X
268
J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
Such exclusive or abnormal growth by secondary grains continues until they impinge each other. After the impingement, the secondary growth becomes stagnated. Since grains with the lowest surface energy might have similar orientations, a strong texture is often accompanied by secondary grain growth. Even after the completion of secondary grain growth, a few grains may undergo further exclusive growth, which leads to the stage of tertiary grain growth. Often, the tertiary grain growth is triggered by the change in the gaseous environment, which is believed to change the surface orientation of the lowest energy changes by the adsorption of gas atoms [2,6,7]. Again, a stronger texture can result from tertiary grain growth. In this paper, we will show that the microstructure evolution similar to tertiary grain growth can be achieved without the change of ambient gases when the grains with the lowest surface energy have a very small initial fraction. We will consider a simple system with three types of grains corresponding to dierent surface energies: high, intermediate, and low. We will show that the initial fraction of each grain type as well as the anisotropic surface energy aects the microstructure evolution. If the initial fraction of the high surface energy is high and the initial fraction of the grains with the low surface energy is much smaller than that with the intermediate surface energy, the grains with the intermediate surface energy would grow abnormally and replace most of the grains with the high surface energy resembling secondary grain growth. Although the grains with the low surface energy may continue to grow at the highest rate, the initial microstructure evolution would be dominated by growth of the grains with the intermediate surface energy because of their much higher initial fraction. After impingements among secondary grains, however, only grains with the low surface energy can continue to grow abnormally over the secondary grains, resulting in tertiary-like grain growth. These aspects of grain growth are examined using Monte Carlo simulation. Description of simulation Two-dimensional Monte Carlo simulation based on the modi®ed Potts model, in which only ®rst-nearest-neighbor lattice points are of concern [3,4], was used to study the grain growth driven by surface energies. The initial triangular lattice structure had 10 000 (100 100) grains of uniform size with random orientations on a 200 200 matrix. 40 000 attempts for atomic jumps correspond to one Monte Carlo step (MCS). We also used a ®xed lattice temperature T for the Boltzmann probability p exp Q= kT to be 0.03, where Q and k are the activation energy of grain boundary migration and the Boltzmann constant, respectively [3,4]. Contrary to others' model [3,4], we introduced three, rather than just two, types of grains according to their surface energies: matrix, type I and type II grains. Matrix grains would have the highest surface energy, and dominate the initial microstructure, while much less type I and II grains would have the intermediate and the lowest surface energies such that cm > cI > cII , where cm , cI and cII were the surface energies per unit area of matrix, type I, and type II grains, respectively.
J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
269
For simplicity, we assumed the equivalency between interface energy and surface energy of any grain with the same orientation and restricted the condition on the relative magnitude of each energy term to cm 1:5cI 3cII . Although surface energies were considered to be anisotropic, the grain boundary energy (cGB ) was assumed to be isotropic and set to cGB cII =3. Abnormal and competitive growth of type I and II grains would necessarily give rise to more complicated phenomena. Of course type II grains should dominate the ®nal grain structure, but the transient structure before completion of all stages of grain growth is likely to be aected by initial fraction of each type. Considering that the rate of impingement is strongly dependent upon initial fractions, relative initial fraction of each grain type may determine which type of grains will be dominant and how fast they will grow. Designating xI and xII as initial fractions of type I and type II grains, respectively, we restricted initial fractions conditions to xI P xII , where the initial fraction of the matrix grains would be 1 xI xII . If the initial fraction of type II grains is higher than that of type I grains (xI < xII ), type II grains would immediately replace the whole microstructure, which is physically trivial. Results and discussion Case A: xII being comparable to xI To examine the initial fraction eect, we performed the simulation with the condition of xI xII at ®rst. We set xI xII 0:05 so that the initial fraction of the matrix grains would be 0.9. The resultant microstructure evolution is illustrated in Fig. 1, in which matrix grains with the high surface energy are in black, type II grains with the low surface energy are in white, and type I grains with the intermediate energy are in gray. In Fig. 1, type II grains with the lowest surface energy (cII ) immediately predominate the microstructure because the initial fraction of type II grains is so high that they can catch up with and overwhelm the kinetics of type I grains. Since type I grains disappear almost simultaneously with matrix grains at early times, type II grains may be regarded as secondary grains. Note that, however, both types of grains exhibit abnormal growth behavior. The results from Fig. 1 can be veri®ed kinetically in Fig. 2. We plotted areal fraction of each type to represent the transition of texture in Fig. 2(a) showing that type I cannot
Fig. 1. Microstructure evolution with the initial fractions of matrix (black), type I (gray) and II (white) grains being 0.9, 0.05 and 0.05, respectively. The unit time elapsed is referred to as 1 MCS.
270
J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
Fig. 2. (a) Areal fraction and (b) relative growth velocity for each type of grain with regard to an elapsed time from the data of Fig. 1. `A' and `B' indicate the time at which matrix grains and type I grains become extinct, respectively.
become a component of major texture. It is because the growth kinetics of type II grains easily surpass that of type I grains, i.e. type II grains consume type I grains as well as matrix grains almost simultaneously. The kinetics of grain growth may be also examined using the relative grain growth velocity vr [8] ! o r vr 1 ot r where r is the average radius of grains of a concerned type and r is the average radius of all grains. r and r are determined by assuming that the initial fraction of type I or type II grains is negligible compared to that of matrix grains. The growth behavior is de®ned to be abnormal if vr > 0 [8]. In our simulation, two kinds of vr are introduced: the relative growth velocity of type I grains (vI ), and that of type II grains (vII ). As in Fig. 2(b), vI and vII are high and positive at the beginning, i.e. they show abnormal growth behavior. Then vI and vII decrease rapidly due to impingements among type I or type II grains. The resultant negative values of vI and vII mean that type I and type II grains now exhibit normal growth behavior. However, after about 15 MCS, vI and vII begin to increase because of a mathematical reason: each r would approach to r as there remain less matrix grains. After about 30 MCS (the point `A' in Fig. 2(b)), no matrix grain remains and vI becomes nearly zero. Similarly, no type I grain remains, i.e. there are only type II grains after about 35 MCS (the point `B'). Both results of Fig. 2 shows that matrix and type I grains disappear almost simultaneously. One can deduce from the results of Fig. 2 that, although the growth behavior of both type I and type II grains are much alike as in Fig. 2(b), type II grains having the lowest energy could grow immediately and overwhelmingly to predominate the microstructure by consuming all the other grains. There are an appreciable number of type II grains at the beginning of grain growth, which make type I grains consumed by type II grains almost simultaneously with matrix grains, as in Fig. 2. Thus the microstructure evolution in Fig. 1 can be said to show secondary grain growth, in which type II grains would be secondary ones.
J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
271
Case B. xII being much smaller than xI If the fraction of type II grains is much smaller than that of type I grains, i.e. xI xII , a drastically dierent microstructure is evolved. Type I grains will grow dominantly over matrix grains at ®rst, but eventually they will be replaced by type II grains. The microstructure evolution in which the fractions of type I and II grains were 0.099 and 0.001, respectively, is shown in Fig. 3. In Fig. 3, type I grains are dominant in the transient microstructure since the initial fraction of type II grains is too low to be signi®cant at early times (for example, see 25 MCS of Fig. 3). After impingements and resultant stagnated growth of type I grains, however, only type II grains can grow over type I grains still exhibiting abnormal behavior. In this case, type I and type II grains may be identi®ed with secondary and tertiary grains, respectively. The kinetic aspect of tertiary grain growth is examined in Fig. 4 in a similar way to that of Fig. 2. Fig. 4(a) exhibits that type I or secondary grains become dominant at ®rst, but the ®nal microstructure consists of type II or tertiary ones, i.e. transition of texture occurs following the sequence of matrix ± type I ± type II. Similarly, in Fig. 4(b), comparison between vI and vII shows that secondary growth of type I grains is followed by tertiary growth of type II grains. As in the results of Fig. 2(b), Fig. 4(b) indicates that matrix and type I grains disappear completely about 30 MCS (the point `A') and 200 MCS (the point `B'), respectively.
Fig. 3. Microstructure evolution with the initial fractions of matrix, type I (gray), and II (white) grains being 0.9, 0.099, and 0.001, respectively. The large grain size of type II grains is due to low initial fraction of them.
Fig. 4. (a) Areal fraction and (b) relative growth velocity for each type of with regard to an elapsed time from the data of Fig. 3. `A' and `B' indicate the time at which matrix grains and type I grains become extinct, respectively.
272
J.-K. Jung et al./Scripta Materialia 45 (2001) 267±272
Figs. 3 and 4 show that the microstructure characteristics of tertiary grain growth can be evolved without any ambient change resulting in the modi®cation of surface energies by a simple assumption on the initial fractions of the type I and II grains. This tertiary-like grain growth can take place if the initial fraction of grains with the lowest surface energy (xII ) is suciently lower than that of grains with intermediate surface energy (xI ). The grains with the intermediate surface energy may predominate the microstructure by consuming the entire matrix grains transiently due to their high initial fraction (secondary grain growth), but their growth would be stagnant after the impingements among them. Finally, the grains with the lowest surface energy will consume all the other grains (tertiary grain growth). For the ®nal microstructure shown in Fig. 3 to be regarded as tertiary grain growth, one should be aware that tertiary grains in our suggestion have existed in the initial microstructure and continued to grow abnormally. That is, when grain growth is mainly driven by surface energy, determination of secondary or tertiary grain growth may be a matter of the initial fraction of the respective grains. Such a kinetic possibility is meaningful from the point of view that the same process of grain structure evolution can explain the characteristics of both secondary and tertiary growth only by the differences in an initial fraction of each component. Conclusion A possibility of tertiary grain growth is suggested. If the fraction of the grains with the lowest surface energy is suciently low and the fraction of the grains with the intermediate surface energy is appreciable, the microstructure initially evolved is dominated by the grains with the intermediate surface energy. But this microstructure is still unstable energetically and the grains with the lowest surface energy will replace all the others. If the initial microstructure evolution is called the secondary grain growth, the later microstructure evolution can be called the tertiary grain growth. It is dierent from the conventional tertiary grain growth in that there is no need to change gaseous environment for selection of tertiary grains among secondary ones that have been stagnated. Although our `tertiary-like' grain growth may not be general but just a special case, it indicates another aspect of abnormal grain growth driven by anisotropic surface energy. References [1] [2] [3] [4] [5] [6] [7] [8]
Humphreys, F. J., & Hatherly, M. (1995). Grain growth and related annealing phenomena. Oxford, UK: Elselvier. Thompson, C. V. (1990). Annu Rev Mater Sci 20, 245. Srolovitz, D. J., Grest, G. S., & Anderson, M. P. (1985). Acta Metall 33, 2233. Rollett, A. D., Srolovitz, D. J., & Anderson, M. P. (1989). Acta Metall 37, 1227. Frost, H. J., Thompson, C. V., & Walton, D. T. (1992). Acta Metall Mater 40, 779. Kramer, J. J., & Foster, K. (1965). Trans Met Soc AIME 233, 1244. McLean, M., & Mykura, H. (1965). Acta Metall 13, 1291. Rios, P. R. (1992). Acta Metall Mater 40, 2765.