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Microstructure evolution during grain growth in materials with disperse particles
Materials Letters 68 (2012) 413–415
Contents lists available at SciVerse ScienceDirect
Materials Letters journal homepage: www.elsevier.com/locate/matlet
Microstructure evolution during grain growth in materials with disperse particles Vladimir Yu. Novikov 1 Moscow Institute of Steel and Alloys, Moscow, Russian Federation
a r t i c l e
i n f o
Article history: Received 26 October 2011 Accepted 28 October 2011 Available online 12 November 2011 Keywords: Grain growth Disperse particles Limiting grain size Modeling
a b s t r a c t Numerical simulations of grain growth are carried out on micro-grained polycrystals containing different volume fractions f of particles of radius r. It was supposed that particles are distributed randomly and do not change their size with time. Particles promote abnormal grain growth; this process is not completed and results in formation of duplex microstructure. Disposition to abnormal growth at first increases and then reduces with an increase in f, the latter being especially noticeable at f > r/D0 where D0 is the initial grain diameter. Limiting grain size is found to be equal to ~0.4 r/f. © 2011 Elsevier B.V. All rights reserved.
This supplies the “limiting” grain diameter:
1. Introduction It is commonly known that disperse particles inhibit the grain growth process [1]. In many cases, they are intentionally introduced into metallic materials and oxide ceramics in order to control the average grain size because, according to the Hall–Petch equation, it determines mechanical properties. At the same time, presence of secondphase particles can lead to development of abnormal grain growth (AG) [1,2]. The latter often results in formation of microstructures consisting of very large crystallites surrounded by grains of “normal” size, which deteriorates mechanical properties. The drag caused by immobile particles is explained by an increase in the grain boundary (GB) area when, in the course of grain growth, some GBs release from particles located on them [3]. In compliance with Ref. [3], randomly distributed spherical particles of volume fraction f and radius r exert the drag force:
〈D〉 lim ¼ 4r=3f :
ð3Þ
A theoretical analysis of grain growth inhibition carried out by other authors [5–8] leads to similar expression: n
〈D〉 lim ¼ kr=f ;
ð4Þ
The growth process should stop [3] if the driving force becomes equal to Zener's drag force:
where the exponent n is ≤1 and the coefficient k depends on details of the approach used. According to Ref. [9], n = 1 should be observed if particles are distributed randomly. In the majority of experimental studies of grain growth inhibition caused by disperse particles (see e.g. [5–8,10]), the main aim was to find parameters k and n in Eq. (4). Experimental results often supplied n = 1 whereas k values deviated from that predicted by Eq. (3) and revealed a large scatter. It is worth mentioning that these results can be affected by coarsening/dissolution of particles in the course of annealing. Another point is that grain growth is frequently accompanied by AG evolution, which makes it difficult to determine 〈D〉lim experimentally as e.g. in Ref. [10]. At last, to the best of the author's knowledge, experiments are usually fulfilled on materials with certain amount and size of particles, which makes the conclusions not general enough. Our aim is to investigate the specific features of microstructure evolution, in particular, conditions for AG development, in the case that the particle volume fraction systematically varies in a wide range. The study has been fulfilled by means of numerical simulations.
2γ B =〈D〉 lim ¼ 1:5γB f =r:
2. Model description and details of simulation
ΔF drag ¼ γB Z ¼ 1:5γB f =r
ð1Þ
where γB is the GB energy. Driving force for grain growth is directly proportional to the specific GB area and, thus, inversely proportional to the average grain size 〈D〉 [4]: ΔF ¼ 2γ B =〈D〉:
1
E-mail address: [email protected]. Permanent address: Treptower Str. 74 d, D-22147 Hamburg, Germany.
0167-577X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2011.10.101
ð2Þ
The present work was carried out with the aid of a statistical model [11] that supplies evolution of microstructure by tracing temporal changes in the grain size distribution. It was supposed that the
414
V.Y. Novikov / Materials Letters 68 (2012) 413–415
model polycrystals studied contain randomly distributed particles, whose size and volume fraction do not change with time. According to the model, the grain of size Di is able to consume an immediate neighbor of size Dj if the driving force of the former is: ΔF ij ¼ cγB 1=Dj −1=Di −Z > 0
(a)
ð5Þ
where c is a constant close to 1 and Z is given by Eq. (1). In order to trace changes in the grain size distribution, displacements of boundaries between each pair of adjacent grains Δsij ¼ M B ΔF ij Δt (MB is the GB mobility and Δt the time step) are calculated for various i and j and used for deriving size increments of growing crystallites as well as size decrements of grains consumed by them. These size alterations, in turn, are used for calculating the corresponding changes in the grain size distribution under the condition of volume preservation. The influence of disperse particles on microstructure evolution was investigated on two model polycrystals containing randomly distributed spherical particles of radius 5 nm. The initial volume fraction of particles was varied from 0.002 to ~0.10%. Both the size and volume fraction of particles were independent of time. The initial mean grain diameter 〈D〉0 in the polycrystals was 3.453 and 3.640 μm while the ratio Dmax /〈D〉0 was 3.8 and 4.7 (they are named further P1 and P2, respectively); the initial size distributions were almost log-normal. Magnitudes of γB = 0.5∙10 −4 J cm −2 and MB = 2∙10 −4 cm 4 J −1 s −1, taken from Ref. [12] for Al at 673 K, were identical for all GBs and independent of time. Grain growth kinetics was described by the time dependence of the number-averaged, three-dimensional grain diameter 〈D〉. The microstructure was characterized, besides 〈D〉, by the Dmax/〈D〉 ratio; both were found from the grain size distribution. The type of the growth process was judged by the scatter of grain sizes characterized by the Dmax/〈D〉 magnitude. Namely, normal grain growth was assumed to proceed at Dmax/〈D〉 ≤ 5 and AG — at Dmax/〈D〉 > 5. The time of AG start was defined as the point where Dmax/〈D〉 becomes >5 and steadily increases further. The maximum grain size reached by that time was taken as the minimum size of abnormal grains. This made it possible to find their volume fraction VA. 3. Results and discussion As can be seen in Fig. 1a, an increase in the volume fraction of particles leads to an expected inhibition of the grain growth process. In the time interval studied, all kinetic curves possess a parabolic shape characteristic of normal grain growth. At the same time, Fig. 1b shows that the growth process is accompanied by an increase in the Dmax/〈D〉 ratio strongly depending on f and being the largest at high f. Thus, under the influence of disperse particles, not normal grain growth but AG evolves in a wide range of f. The results presented in Fig. 1a differ from the data [10] where changes in the shape of kinetic curves were observed. This difference is obviously connected with a decreased contribution of abnormally large grain into 〈D〉 in our simulations, which can be explained as follows. First of all, magnitude of the contribution depends on the averaging procedure. If the linear intercept method that supplies the volumeaveraged value is used as in Ref. [10], the contribution is larger than in the case of the number-averaged value used in the present work. Besides, the number of abnormal grains in our simulations was apparently smaller than in Ref. [10]. Fig. 2 shows that the volume fraction of abnormal grains VA increases with time, at first rather rapidly but, after some period of time, more slowly. This is consistent with changes in Dmax/〈D〉, which is most evident from comparison of data for f = 0.07% in Figs. 1b and 2. An initial increase in Dmax/〈D〉 can be connected with
(b)
Fig. 1. (a) Grain growth kinetics and (b) time dependence of microstructure nonhomogeneity in polycrystal P1 with f = 0.01% (squares), 0.02% (circles) and 0.07% (triangles).
emergence and rapid growth of large crystallites, which is typical of AG. However later on, their size and number increase more slowly, which reduces dVA/dt and eventually leads to AG cessation and, thus, to formation of duplex structure. Apparently, this is a result of the fact that the drag force remains the entire time constant. Owing to this, abnormal grains at first have time to emerge and consume those matrix grains whose size obeys the condition Eq. (5). However afterwards, their growth as well as emergence of new abnormal grains becomes impeded because there remain much less matrix grains, in comparison to the initial state, that could be consumed. That is why thermally stable particles do not permit the AG process to come to completion. If, on the contrary, particles were unstable and the particle drag reduced with time, the size of the matrix grains consumed as well as their number would increase (because the grain size distribution has a peaked shape) and thereby promote further AG development. It follows from Fig. 2 that the AG evolution depends on the f/r magnitude. Fig. 3a shows that the volume fraction of abnormal grains reached by the time t = 120 min has a maximum at f/r ≈ 0.06 μm −1. It
Fig. 2. AG evolution in polycrystal P1 with f = 0.02% (circles) and 0.07% (triangles).
V.Y. Novikov / Materials Letters 68 (2012) 413–415
415
(a)
(b) Fig. 4. Dependence of 〈D〉lim on r/f for polycrystals P1 (open symbols) and P2 (solid symbols).
Fig. 3. Impact of f/r and Z〈D〉0 in polycrystal P1 on VA at t = 120 min (a) and (b) time elapsed to reach VA ≈ 0.01.
is seen in Fig. 3b that the time t0.01, necessary to reach VA ≈ 0.01, has a minimum at the same f/r. So, at both low and large f/r magnitudes, AG does not evolve, however, owing to different reasons. At low f/r, AG is not observed because the particle drag is not high enough to suppress GB movement in such a way that some grains lose the ability to grow (see Eq. (1)). On the contrary at large f/r, AG does not evolve because the majority of crystallites cannot grow at all except those whose immediate neighbors are the smallest grains. Obviously, just this range of f/r magnitudes is the most promising for practical applications. It should be noted that a slight increase in 〈D〉 and a decrease in the total number of grains take place even at very high f/r corresponding to Z > 1/〈D〉, which contradicts the Zener approach. This contradiction can be solved by taking into consideration that our results are obtained under the supposition of interaction of grains of different sizes whereas Zener's approach is based on a one-grain approximation oversimplifying the situation. Our studies of grain growth kinetics gave the opportunity to estimate 〈D〉lim at different f/r and to find the coefficient k in Eq. (4). Since the kinetic curves obtained had always some slope, the magnitude of 〈D〉lim was defined as 〈D〉 at VA ≈ 0.01. The results are presented in
Fig. 4 and lead to several conclusions. Firstly, comparison of data for P1 and P2 shows that the initial grain size non-homogeneity affects the simulation results rather weakly. Secondly, the bulk of data for P1 in Fig. 4 can be approximated by a straight line with a slope of 0.43, which is ~3 times lower than that predicted by Eq. (3). This can be explained by the essential difference in the descriptions of grain growth in our model and in Ref. [3] outlined in the preceding paragraph. Thirdly, the magnitude of k can vary in a rather wide range if only a part of the data is considered. This explains why different k values have been observed in different experimental studies. References [1] Novikov V. Grain growth and control of microstructure and texture in polycrystalline materials. Boca Raton (USA): CRC Press; 1996. [2] Beck PA, Holzworth ML, Sperry PR. Effect of dispersed phase on grain growth in Al–Mn alloys. Trans AIME 1949;180:163–92. [3] Zener C, quoted by Smith CS. Introduction to grains, phases, and interfaces—an interpretation of microstructure. Trans AIME 1948;175:15–51. [4] Burke JE. Some factors affecting the rate of grain growth in metals. Trans AIME 1949;180:73–91. [5] Gladman T. On the theory of the effect of precipitate particles on grain growth in metals. Proc R Soc London A 1966;294:298–309. [6] Haroun NA, Budworth DW. Modification to the Zener formula for limitation of grain size. J Mater Sci 1968;3:326–8. [7] Hellman P, Hillert M. On the effect of second phase particles on grain growth. Scand J Metall 1975;4:211–9. [8] Anand L, Gourland J. The relationship between the size of zementite particles and the subgraín size in quenched and tempered steel. Metall Trans 1975;6A:928–31. [9] Liu Y, Patterson BR. Stereological analysis of Zener pinning. Acta Mater 1996;44: 4327–35. [10] Gao N, Baker TN. Austenite grain growth behaviour of microalloyed Al–V–N and Al–V–Ti–N steels. ISIJ Int 1998;38:744–51. [11] Novikov VYu. Computer simulation of normal grain growth. Acta Metall 1978;26: 1739–44. [12] Gottstein G, Shvindlerman LS. Grain boundary migration in metals: thermodynamics, kinetics, applications. Boca Raton (USA): CRC Press; 2010.
Contents lists available at SciVerse ScienceDirect
Materials Letters journal homepage: www.elsevier.com/locate/matlet
Microstructure evolution during grain growth in materials with disperse particles Vladimir Yu. Novikov 1 Moscow Institute of Steel and Alloys, Moscow, Russian Federation
a r t i c l e
i n f o
Article history: Received 26 October 2011 Accepted 28 October 2011 Available online 12 November 2011 Keywords: Grain growth Disperse particles Limiting grain size Modeling
a b s t r a c t Numerical simulations of grain growth are carried out on micro-grained polycrystals containing different volume fractions f of particles of radius r. It was supposed that particles are distributed randomly and do not change their size with time. Particles promote abnormal grain growth; this process is not completed and results in formation of duplex microstructure. Disposition to abnormal growth at first increases and then reduces with an increase in f, the latter being especially noticeable at f > r/D0 where D0 is the initial grain diameter. Limiting grain size is found to be equal to ~0.4 r/f. © 2011 Elsevier B.V. All rights reserved.
This supplies the “limiting” grain diameter:
1. Introduction It is commonly known that disperse particles inhibit the grain growth process [1]. In many cases, they are intentionally introduced into metallic materials and oxide ceramics in order to control the average grain size because, according to the Hall–Petch equation, it determines mechanical properties. At the same time, presence of secondphase particles can lead to development of abnormal grain growth (AG) [1,2]. The latter often results in formation of microstructures consisting of very large crystallites surrounded by grains of “normal” size, which deteriorates mechanical properties. The drag caused by immobile particles is explained by an increase in the grain boundary (GB) area when, in the course of grain growth, some GBs release from particles located on them [3]. In compliance with Ref. [3], randomly distributed spherical particles of volume fraction f and radius r exert the drag force:
〈D〉 lim ¼ 4r=3f :
ð3Þ
A theoretical analysis of grain growth inhibition carried out by other authors [5–8] leads to similar expression: n
〈D〉 lim ¼ kr=f ;
ð4Þ
The growth process should stop [3] if the driving force becomes equal to Zener's drag force:
where the exponent n is ≤1 and the coefficient k depends on details of the approach used. According to Ref. [9], n = 1 should be observed if particles are distributed randomly. In the majority of experimental studies of grain growth inhibition caused by disperse particles (see e.g. [5–8,10]), the main aim was to find parameters k and n in Eq. (4). Experimental results often supplied n = 1 whereas k values deviated from that predicted by Eq. (3) and revealed a large scatter. It is worth mentioning that these results can be affected by coarsening/dissolution of particles in the course of annealing. Another point is that grain growth is frequently accompanied by AG evolution, which makes it difficult to determine 〈D〉lim experimentally as e.g. in Ref. [10]. At last, to the best of the author's knowledge, experiments are usually fulfilled on materials with certain amount and size of particles, which makes the conclusions not general enough. Our aim is to investigate the specific features of microstructure evolution, in particular, conditions for AG development, in the case that the particle volume fraction systematically varies in a wide range. The study has been fulfilled by means of numerical simulations.
2γ B =〈D〉 lim ¼ 1:5γB f =r:
2. Model description and details of simulation
ΔF drag ¼ γB Z ¼ 1:5γB f =r
ð1Þ
where γB is the GB energy. Driving force for grain growth is directly proportional to the specific GB area and, thus, inversely proportional to the average grain size 〈D〉 [4]: ΔF ¼ 2γ B =〈D〉:
1
E-mail address: [email protected]. Permanent address: Treptower Str. 74 d, D-22147 Hamburg, Germany.
0167-577X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2011.10.101
ð2Þ
The present work was carried out with the aid of a statistical model [11] that supplies evolution of microstructure by tracing temporal changes in the grain size distribution. It was supposed that the
414
V.Y. Novikov / Materials Letters 68 (2012) 413–415
model polycrystals studied contain randomly distributed particles, whose size and volume fraction do not change with time. According to the model, the grain of size Di is able to consume an immediate neighbor of size Dj if the driving force of the former is: ΔF ij ¼ cγB 1=Dj −1=Di −Z > 0
(a)
ð5Þ
where c is a constant close to 1 and Z is given by Eq. (1). In order to trace changes in the grain size distribution, displacements of boundaries between each pair of adjacent grains Δsij ¼ M B ΔF ij Δt (MB is the GB mobility and Δt the time step) are calculated for various i and j and used for deriving size increments of growing crystallites as well as size decrements of grains consumed by them. These size alterations, in turn, are used for calculating the corresponding changes in the grain size distribution under the condition of volume preservation. The influence of disperse particles on microstructure evolution was investigated on two model polycrystals containing randomly distributed spherical particles of radius 5 nm. The initial volume fraction of particles was varied from 0.002 to ~0.10%. Both the size and volume fraction of particles were independent of time. The initial mean grain diameter 〈D〉0 in the polycrystals was 3.453 and 3.640 μm while the ratio Dmax /〈D〉0 was 3.8 and 4.7 (they are named further P1 and P2, respectively); the initial size distributions were almost log-normal. Magnitudes of γB = 0.5∙10 −4 J cm −2 and MB = 2∙10 −4 cm 4 J −1 s −1, taken from Ref. [12] for Al at 673 K, were identical for all GBs and independent of time. Grain growth kinetics was described by the time dependence of the number-averaged, three-dimensional grain diameter 〈D〉. The microstructure was characterized, besides 〈D〉, by the Dmax/〈D〉 ratio; both were found from the grain size distribution. The type of the growth process was judged by the scatter of grain sizes characterized by the Dmax/〈D〉 magnitude. Namely, normal grain growth was assumed to proceed at Dmax/〈D〉 ≤ 5 and AG — at Dmax/〈D〉 > 5. The time of AG start was defined as the point where Dmax/〈D〉 becomes >5 and steadily increases further. The maximum grain size reached by that time was taken as the minimum size of abnormal grains. This made it possible to find their volume fraction VA. 3. Results and discussion As can be seen in Fig. 1a, an increase in the volume fraction of particles leads to an expected inhibition of the grain growth process. In the time interval studied, all kinetic curves possess a parabolic shape characteristic of normal grain growth. At the same time, Fig. 1b shows that the growth process is accompanied by an increase in the Dmax/〈D〉 ratio strongly depending on f and being the largest at high f. Thus, under the influence of disperse particles, not normal grain growth but AG evolves in a wide range of f. The results presented in Fig. 1a differ from the data [10] where changes in the shape of kinetic curves were observed. This difference is obviously connected with a decreased contribution of abnormally large grain into 〈D〉 in our simulations, which can be explained as follows. First of all, magnitude of the contribution depends on the averaging procedure. If the linear intercept method that supplies the volumeaveraged value is used as in Ref. [10], the contribution is larger than in the case of the number-averaged value used in the present work. Besides, the number of abnormal grains in our simulations was apparently smaller than in Ref. [10]. Fig. 2 shows that the volume fraction of abnormal grains VA increases with time, at first rather rapidly but, after some period of time, more slowly. This is consistent with changes in Dmax/〈D〉, which is most evident from comparison of data for f = 0.07% in Figs. 1b and 2. An initial increase in Dmax/〈D〉 can be connected with
(b)
Fig. 1. (a) Grain growth kinetics and (b) time dependence of microstructure nonhomogeneity in polycrystal P1 with f = 0.01% (squares), 0.02% (circles) and 0.07% (triangles).
emergence and rapid growth of large crystallites, which is typical of AG. However later on, their size and number increase more slowly, which reduces dVA/dt and eventually leads to AG cessation and, thus, to formation of duplex structure. Apparently, this is a result of the fact that the drag force remains the entire time constant. Owing to this, abnormal grains at first have time to emerge and consume those matrix grains whose size obeys the condition Eq. (5). However afterwards, their growth as well as emergence of new abnormal grains becomes impeded because there remain much less matrix grains, in comparison to the initial state, that could be consumed. That is why thermally stable particles do not permit the AG process to come to completion. If, on the contrary, particles were unstable and the particle drag reduced with time, the size of the matrix grains consumed as well as their number would increase (because the grain size distribution has a peaked shape) and thereby promote further AG development. It follows from Fig. 2 that the AG evolution depends on the f/r magnitude. Fig. 3a shows that the volume fraction of abnormal grains reached by the time t = 120 min has a maximum at f/r ≈ 0.06 μm −1. It
Fig. 2. AG evolution in polycrystal P1 with f = 0.02% (circles) and 0.07% (triangles).
V.Y. Novikov / Materials Letters 68 (2012) 413–415
415
(a)
(b) Fig. 4. Dependence of 〈D〉lim on r/f for polycrystals P1 (open symbols) and P2 (solid symbols).
Fig. 3. Impact of f/r and Z〈D〉0 in polycrystal P1 on VA at t = 120 min (a) and (b) time elapsed to reach VA ≈ 0.01.
is seen in Fig. 3b that the time t0.01, necessary to reach VA ≈ 0.01, has a minimum at the same f/r. So, at both low and large f/r magnitudes, AG does not evolve, however, owing to different reasons. At low f/r, AG is not observed because the particle drag is not high enough to suppress GB movement in such a way that some grains lose the ability to grow (see Eq. (1)). On the contrary at large f/r, AG does not evolve because the majority of crystallites cannot grow at all except those whose immediate neighbors are the smallest grains. Obviously, just this range of f/r magnitudes is the most promising for practical applications. It should be noted that a slight increase in 〈D〉 and a decrease in the total number of grains take place even at very high f/r corresponding to Z > 1/〈D〉, which contradicts the Zener approach. This contradiction can be solved by taking into consideration that our results are obtained under the supposition of interaction of grains of different sizes whereas Zener's approach is based on a one-grain approximation oversimplifying the situation. Our studies of grain growth kinetics gave the opportunity to estimate 〈D〉lim at different f/r and to find the coefficient k in Eq. (4). Since the kinetic curves obtained had always some slope, the magnitude of 〈D〉lim was defined as 〈D〉 at VA ≈ 0.01. The results are presented in
Fig. 4 and lead to several conclusions. Firstly, comparison of data for P1 and P2 shows that the initial grain size non-homogeneity affects the simulation results rather weakly. Secondly, the bulk of data for P1 in Fig. 4 can be approximated by a straight line with a slope of 0.43, which is ~3 times lower than that predicted by Eq. (3). This can be explained by the essential difference in the descriptions of grain growth in our model and in Ref. [3] outlined in the preceding paragraph. Thirdly, the magnitude of k can vary in a rather wide range if only a part of the data is considered. This explains why different k values have been observed in different experimental studies. References [1] Novikov V. Grain growth and control of microstructure and texture in polycrystalline materials. Boca Raton (USA): CRC Press; 1996. [2] Beck PA, Holzworth ML, Sperry PR. Effect of dispersed phase on grain growth in Al–Mn alloys. Trans AIME 1949;180:163–92. [3] Zener C, quoted by Smith CS. Introduction to grains, phases, and interfaces—an interpretation of microstructure. Trans AIME 1948;175:15–51. [4] Burke JE. Some factors affecting the rate of grain growth in metals. Trans AIME 1949;180:73–91. [5] Gladman T. On the theory of the effect of precipitate particles on grain growth in metals. Proc R Soc London A 1966;294:298–309. [6] Haroun NA, Budworth DW. Modification to the Zener formula for limitation of grain size. J Mater Sci 1968;3:326–8. [7] Hellman P, Hillert M. On the effect of second phase particles on grain growth. Scand J Metall 1975;4:211–9. [8] Anand L, Gourland J. The relationship between the size of zementite particles and the subgraín size in quenched and tempered steel. Metall Trans 1975;6A:928–31. [9] Liu Y, Patterson BR. Stereological analysis of Zener pinning. Acta Mater 1996;44: 4327–35. [10] Gao N, Baker TN. Austenite grain growth behaviour of microalloyed Al–V–N and Al–V–Ti–N steels. ISIJ Int 1998;38:744–51. [11] Novikov VYu. Computer simulation of normal grain growth. Acta Metall 1978;26: 1739–44. [12] Gottstein G, Shvindlerman LS. Grain boundary migration in metals: thermodynamics, kinetics, applications. Boca Raton (USA): CRC Press; 2010.