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On grain growth in the presence of mobile particles
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Acta Materialia 58 (2010) 3326–3331 www.elsevier.com/locate/actamat
On grain growth in the presence of mobile particles V.Yu. Novikov * Moscow Institute of Steel and Alloys, Moscow, Russia Received 16 January 2010; received in revised form 4 February 2010; accepted 5 February 2010 Available online 16 March 2010
Abstract The ability of second phase particles to migrate along with grain boundaries is shown to be determined not only by the particle mobility but also by the migration rate of the grain boundary where they locate. This leads to a duality in the mobile particle behaviour: they behave as either movable or immovable depending on the boundary migration rate. In the first case, they reduce the boundary mobility; in the second one they decrease the driving force for boundary migration. It is demonstrated by numerical modeling that mobile particles with low mobility can suppress grain growth even in nanocrystalline material, the limiting grains size being several times smaller than in the case of randomly distributed immobile particles. It is also shown that the Zener solution to the problem of the grain growth retardation by disperse particles is a specific case of the proposed approach. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain growth suppression; Grain boundary migration; Disperse particles; Nanocrystalline materials
1. Introduction The universally accepted approach to explaining the impact of disperse particles on grain growth was proposed by C. Zener (quoted in Ref. [1]). According to this approach, randomly distributed spherical particles of volume fraction f and radius r exert the drag force on grain boundaries (GBs): DF drag ¼ cB Z ¼ 1:5cB f =r
ð1Þ
where cB is the GB energy. The term Z = 1.5f/r is further named a Zener drag. This drag is caused by an increase in the GB area owing to release of GBs from particles located on them. So, disperse particles are considered in Ref. [1] to be immobile although they can be of relatively small size and so could possess rather high mobility [2]. As has been shown recently [3,4] particles can migrate along with GBs provided the migration rate of the latter is not higher than m ¼ pcB M P r *
Present address: Treptower Str. 74d, Hamburg 22147, Germany. E-mail address: [email protected].
ð2Þ
where MP is the particle mobility. Lately it was proposed [5] that the ability or inability of mobile particles to move along with migrating GBs not only depends on the properties of particles but also is a result of the interaction of particles with certain GBs. Namely, the same mobile particles either move along with GBs (further, movable particles) or detach from them (immovable particles), depending on the migration rate, vm, of the latter: if mm < v*, particles behave as movable, in the opposite case as immovable. Obviously, the Zener approach relates to the second case, which can be confirmed by the following consideration. If the grain growth process can be represented by the behaviour of an “average” grain (i.e. if a one-grain approximation is used), the grain growth rate is described after Ref. [6] as: mm ¼
3cB M B 2D
ð3Þ
where D is the three-dimensional average grain diameter and MB the GB mobility; both cB and MB are supposed to be identical for all GBs and independent of time. Then in the presence of movable particles located on migrating GBs, the growth rate is [4]:
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.02.006
V.Yu. Novikov / Acta Materialia 58 (2010) 3326–3331
mm ¼ 1:5M eff cB =D In Eq. (4), M
eff
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ð4Þ is the “effective GB mobility”:
M eff ¼ M B =ð1 þ nA M B =M P Þ
ð5Þ
nA being the number of particles per unit GB area. If nAMB/MP 1, then mm ffi
1:5cB M P nA D
ð6Þ
Since for randomly distributed particles nA ¼
3 f =r2 2p
ð7Þ
it can be found from the condition mm ¼ m that GBs detach from the particles if 1=Dlim ¼ f =r
ð8Þ
where Dlim is the “limiting average grain size” after Ref. [1]. This is in a good agreement with the term Z in Eq. (1)1 and leads to the conclusion that the Zener approach can be considered as a specific case of interaction between mobile particles and GBs. It is clear, however, that the one-grain approximation ignores the fact that any microstructure is formed by grains of different sizes and, thus, the grain of size Ri adjacent to neighbours of different sizes Rj can consume an immediate neighbour of size Rj < Ri. Migration rate of the boundary between the grain and its neighbour in the presence of movable particles is: ¼ cM eff cB ð1=Rj 1=Ri Þ mmov ij
ð9Þ
where c is a constant close in value to 1 and Meff is the effective GB mobility. It follows from Eq. (9) that movable particles do not affect the driving force for grain growth. Fig. 1a displays the growth rate of a grain of radius Ri vs. the size of consumed grains according to Eq. (9). Two magnitudes of Ri, namely, Ri = R and 3R, were used, where the former is the average grain size and the latter approximately corresponds to the maximum grain size in a typical single-phase annealed material. It is seen in Fig. 1a that, in the presence of movable particles, grains of size Ri are able to consume all their smaller neighbours and that their GBs migrate at rates increasing with a decrease in the size of consumed grains Rj. Since v* in Eq. (2) is independent of Rj, it should be displayed by a horizontal line in the coordinates used, which leads to the appearance of two ranges on the Rj/R scale. In the right-hand range where mm < m , GBs can migrate along with particles and so the
1 If particles are located at GBs only, it can be derived in the same way that the condition for the particle detachment is:
cB =D2lim ¼ acB f =r2 which corresponds to the dependence [7]:
Dlim =r / f 1=2
eff Fig. 1. (a) Reduced GB migration rate mmov ij =ðcM cB =RÞ according to Eq. (9) vs. size of consumed grains Rj/R. The size of the growing grain is R (solid line) and 3R (dot line) R being the average grain radius. (b) Normalized log-normal grain size distribution.
latter behave as movable. In the left-hand one, i.e. at small Rj/R, the GB migration rate is larger than m*. Thus, boundaries of the corresponding grains would detach from the particles owing to which the latter behave as immovable although they have the same radius and mobility as the movable ones. Owing to this, they are further referred to as movable/immovable particles. All particles behave as immovable at a very low m*, e.g. at MPi ? 0, whereas at very high m* all particles behave as movable. Obviously, a decrease in m* should lead to an increase in the fraction of consumed grains whose GBs migrate at rates mm P m (compare Fig. 1a and b). This, in turn, should increase the fraction of immovable particles. The impact of immovable particles on the GB migration rate can be presented as follows: mim ij ¼ cM B cB ð1=Rj 1=Ri ZÞ
ð10Þ
where 1/Rj 1/Ri Z > 0 and Z is the Zener drag. As follows from Eq. (10), immovable particles, in contrast to movable ones, decrease the driving force for grain growth. As a result, there should appear a gap between the size of a growing grain and the maximum size of the consumed . Fig. 2a illustrates the situation for the case that grain Rmax j the grain of the average size R is the growing one. Such a gap should also appear between the minimum size of the and the size of the consumed grain. growing grain Rmin i The corresponding illustration is given in Fig. 2b for the case that the size of the consumed grain is R. It is seen in decreases and Rmin increases with an inFig. 2 that Rmax j i crease in the product ZR. Thus, if disperse particles behave
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V.Yu. Novikov / Acta Materialia 58 (2010) 3326–3331
GB migration rate is Pv*, particles pin the GB; in the opposite case they move along with the GB. As soon as this becomes obvious, we can attempt to control the particle behaviour and thus the microstructure evolution in the course of grain growth. This paper is aimed at the study of the impact on grain growth of movable/immovable particles at different magnitudes of v*. 2. Model description and details of simulation
Fig. 2. Effect of ZR on (a) relative size of grains consumed by immediate neighbours of the mean size R and (b) relative size of crystallites able to grow at the expense of grains of size R.
as immovable, this results in inability of certain grains to grow at the expense of some of their smaller neighbours and to be consumed by some of their larger neighbours. By contrast with studies of Zener pinning, to the best of our knowledge there are only a few investigations on the effect of mobile particles. The impact of mobile particles on grain growth kinetics has been studied in Ref. [8] where the particle behaviour was supposed to be analogous to that of mobile impurities. According to Ref. [9], in the case that the driving force on the GB is low or the impurity mobility is high, impurity atoms diffuse along with the GB and the GB migration rate is small. In the opposite case (the driving force on the GB is high or the impurity mobility is low) the GB becomes free from impurities and migrates at a much higher rate, leaving impurity atoms behind. In the model polycrystal studied, particles were supposed to be distributed randomly and their mobility constant, whereas the particle volume fraction was varied. The Monte Carlo simulations [8] have shown that when the mean grain size becomes approximately equal to the average inter-particle distance, the grain growth process is temporarily arrested. This was explained by the inhibiting effect of “static” particles. A further increase in the growth duration led to an increase in both the mean grain size and the growth rate, which was explained by the GB detachment from static particles and its movement along with “mobile” ones. In the theoretical analysis [10] of the drag effect of second phase particles located at GB and growing in the course of grain growth, it was also shown that the dual behaviour of such particles can be revealed. Thus, the mechanism of grain growth inhibition by mobile second phase particles is established in Ref. [8]: particles, able to diffuse along with GBs, decrease the mobility of loaded GBs, whereas the others “pin” GBs in the manner described by Zener. However, it remains unclear what is responsible for the particle behaviour and whether the relative number of the first or second ones could be influenced. We believe that the answer to these questions is found: the particle behaviour is determined by the migration rate of GBs interacting with them. Namely, if the
This work was carried out by numerical simulation with the aid of a statistical model [11,12] successfully used to investigate various aspects of grain growth (see e.g. Refs. [13–16]). It describes the evolution of both the grain size distribution and its parameters with time. The influence of disperse particles on grain growth was investigated on a model nanocrystalline polycrystal. Its initial microstructure was characterized by an almost log-normal grain size distribution (see Fig. 1b) with the mean three-dimensional grain diameter D = 73 nm. Initial spatial distribution of particles was supposed to be random. Particle size was chosen in such a way that their radius r = 3 nm was about one order of magnitude smaller than the mean grain size; their volume fraction was 0.002. Grain growth kinetics was characterized by the time dependence of the number averaged, three-dimensional grain diameter D. The impact of v* on both the growth kinetics and microstructure evolution was investigated. Microstructure changes were studied in a polycrystal with identical cB and MB, both being supposed to be independent of time. It was also supposed that (i) disperse particles are initially distributed randomly but in the course of grain growth they redistribute between GBs and grain interior owing to both the sweeping-out of movable particles from the grain interior and the detachment of immovable particle from GBs, (ii) they have an identical size and identical mobility, (iii) the total number of particles remains unchanged, (iv) the instantaneous mean number of particles per unit boundary area, nA, is identical for all GBs, and (v) the instantaneous mean particle density in the grain interior is identical for all grains. In order to trace changes in the grain size distribution, the displacements, Ddij, of the boundary between each pair of adjacent grains should be found. If particles at ¼ mmov the GB behave as movable, Dd mov ij Dt, where Dt is ij mob the time step. The product of Dd ij and the contact area between corresponding grains delivers the magnitude of the volume swept by the GB migrating at the rate
V.Yu. Novikov / Acta Materialia 58 (2010) 3326–3331
migrating GBs. Then the magnitude of f in Eq. (1) should be derived from Eq. (7) so that ð11Þ
Z ¼ pnA r Dd mob ij
Dd im ij ;
or for various The GB displacements, either i and j were used to derive the size increments of growing grains of radius Ri as well as the size decrements of grains consumed by them. This necessitates the consideration of the mean contact probabilities of growing and consumed grains of different sizes. The size alterations, in turn, were used to derive the corresponding changes in the grain size distribution under the condition of volume preservation. and Values of cB = 0.5 104 J cm2 4 4 1 1 MB = 2 10 cm J s were taken from the data [17] for Al at T = 673 K. The particle mobility is supposed to be controlled by the interface diffusion [2]: M P ¼ ðdDS X=kT Þ=pr4
ð12Þ
where d is the “thickness” of the particle/matrix interface, DS the diffusion coefficient along the interface and X the atomic volume. Numerical values of d, DS, X and T used in this work were 107 cm, 109 cm2 s1, 10 cm3 mol1 and 673 K, respectively; the corresponding MP magnitude is further designated as M # P. 3. Results and discussion The kinetics of grain growth influenced by movable/ immovable particles with M P ¼ M # P (according to the concept described above) and that affected by immobile and randomly distributed particles (according to Ref. [1]) are compared in Fig. 3. As could be expected, in both cases the growth process is retarded. At the same time, the kinetic curves for grain growth affected by movable/ immovable particles and by immobile particles are princi-
Fig. 3. Grain growth kinetics simulated taking into account movable/ immovable particles with M P ¼ M # P (squares), under the supposition that all particles are immobile and randomly distributed [1] (triangles), and in the absence of particles (dash line).
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pally different. Under the influence of movable/immovable particles the mean grain size D steadily increases with time. This leads to the conclusion that the majority of particles in this case behave as movable. In fact, if all particles were movable, grains of any size would remain able to grow at the expense of any smaller grains and to be consumed by even slightly larger grains (see Fig. 1a). Owing to this, the mean grain size should continuously increase. To the contrary, in the case of immobile particles, D first increases and after some period of time becomes almost constant, reaching a “limiting grain size” [1]. In our opinion, this reflects the fact that, under the influence of immobile particles, some grains become unable to grow at the expense of their slightly smaller neighbours and at the same time cannot be consumed by their slightly larger neighbours, as has been shown in Section 1. Obviously, this apparent restriction of grain growth by immobile particles takes place at some “critical” magnitude of the product ZR of the particle drag Z and the mean grain radius R. On the basis of Fig. 2 it can be proposed that this magnitude is not smaller than 0.6. As follows from Fig. 2a, at Z P 0.6/R, grains of the mean size are able to consume neighbours of size 60.5R but this only slightly can contribute to an increase in R because the number of such neighbours is relatively small. Simultaneously, at this Z, the size of crystallites, able to consume the grains of the mean size, is P 5R (see Fig. 2b). Since the relative number of these crystallites is very small, this hardly contributes to a decrease in R. As a result, the mean grain size alteration with time cannot be noticeable because of compensation of these two contributions. It is worth mentioning that the numerical simulation of grain growth kinetics with immobile particles (see Fig. 3) yields Dlim ffi 1:2r=f ; in a good accordance with Eq. (8). It follows from the comparison of Fig. 1a with Fig. 1b that the fraction of consumed grains, whose GBs migrate at rates Pm*, should increase with a decrease in v*. This, in turn, should be accompanied by an increase in the relative number of immovable particles. Since in our simulations the magnitudes of cB and r are supposed to be constant, a reduction in v* was achieved, according to Eq. (2), by a decrease in MP through decreasing DS. Fig. 4 shows the growth kinetic curves for polycrystals with # 3 particles of different MP, from M # P to 2 10 M P . It is seen # that at M P ¼ 0:1M P ; the shape of the kinetic curve is similar to that at M P ¼ M # P : This allows the assumption that the effect of movable particles is predominant not only at # MP ¼ M# P but also at M P ¼ 0:1M P . A much more significant reduction in particle mobility to M P ¼ 0:002M # P leads to pronounced changes in the growth kinetics (see Fig. 4). The kinetic curve becomes similar to that predicted according to Ref. [1] (compare with Fig. 3) and so at this MP the effect of immovable particles becomes the strongest. Difference in the microstructure evolution at various MP becomes much more obvious if the grain size non-homogeneities are compared (see Fig. 5). The latter are characterized by the ratio Dmax/D of the maximum grain size to the
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V.Yu. Novikov / Acta Materialia 58 (2010) 3326–3331
# Fig. 4. Grain growth kinetics at MP = M # P (squares), 0:1M P (circles) and # 3 2 10 M P (triangles).
increases and becomes very strong, namely, Dmax/D attains the magnitude of 20–25, characteristic of abnormal grain growth. Thus, the vast majority of grains do not increase their size, which can only be a result of the impact of immovable particles (see the first paragraph in this section). It is interesting that in the case M P ¼ 0:002M # P ; Dlim is 5 times smaller than that obtained according to Ref. [1] for immobile particles of the same size and same volume fraction (compare with Fig. 3). It can be proposed that the grain size stabilization under the influence of movable/immovable particles is also a result of reaching a “critical” ZR value, as discussed in the preceding paragraph. However, in the case considered the magnitude of Z is determined not by the volume fraction of particles but by the magnitude of particle density at GBs, nA, in accordance with Eq. (11). Since it increases owing to sweeping out of movable particles, the critical ZR value is attained at much smaller R than in the case of immobile and randomly distributed particles. 4. Concluding remarks and summary
Fig. 5. Evolution of grain size non-homogeneity in the course of grain growth in the presence of particles with different mobility: M P ¼ M # P # (squares), M P ¼ 0:01M # P (solid circles) and M P ¼ 0:002M P (triangles).
average grain size. In the case M P ¼ M # P ; this ratio first increases from the initial magnitude of 4.7 to 6 and then decreases to 3, typical of normal grain growth. In the case M P ¼ 0:01M # P ; Dmax =D first increases to 11 and then continuously decreases to 5. An increase in Dmax/D means that at the initial stages, the growth of the majority of grains is strongly inhibited and abnormal grain growth evolves. However, afterwards the particle inhibition decays, which results in the evolution of normal grain growth. These results are in a qualitative agreement with the data [8] where grain growth was observed to be strongly inhibited at relatively low D, apparently by immovable particles, but at larger D the inhibition became weaker owing to the impact of movable particles. At last in the case M P ¼ 0:002M # P ; the grain size non-homogeneity steadily
The retardation of GB migration by disperse particles, manifesting in particular in the slowing down of the grain growth process, has been known for more than six decades. The vast majority of results on the grain growth retardation by particles have been analysed in the belief that they are immobile and so unable to move along with GBs. However, since the particles are usually rather small, it should be taken into account that they can possess a finite, nonzero mobility. So, the question arises whether mobile particles could behave as immovable. A positive answer to this question has been given in Ref. [8]. However, it remained unclear why it is possible. An answer to the latter question has been given in the present paper. It takes into account not only the particle mobility but also the GB migration rate. Namely, in any polycrystal there is a wide range of GB migration rates, vm, and, besides, there exists certain GB rate, v*, at which particles located at GBs should detach from the latter. Usually, v* is within the above-mentioned range. Owing to this, particles possessing certain mobility are immovable if the GB where they locate migrates at the rate mm P m . (In the case that the minimum GB migration rate equals v*, all particles should behave as immovable and so the well-known Zener approach is a specific case of interaction between GBs and mobile particles.) At the same time, particles located at all other GBs behave as movable. So a duality in the behaviour of mobile particles must be observed in any polycrystal, and thus the label “mobile” does not mean that the particle can move along with any GB. The impact of movable/immovable particles on grain growth was studied in the present paper on nanocrystalline polycrystal where the driving force for grain growth is especially high. This was done to widen the range of GB migration rates. In contrast to Ref. [8], the volume fraction of mobile particles was supposed to be constant whereas the magnitude of MP was varied. This was used as a means to
V.Yu. Novikov / Acta Materialia 58 (2010) 3326–3331
control the particle behaviour because it affects, in accordance with Eq. (2), the magnitude of v*, causing the GB detachment from particles (see above). It has been found that a reduction in MP and thus in v* leads not only to a decrease in the rate of grain growth, which could be expected, but simultaneously, it increases the fraction of immovable particles, which manifests in the evolution of abnormal grain growth at the initial stages of the growth process. It has been shown that even in such a material the growth process can be completely suppressed provided the particle mobility is low but some particles behave as movable. It should be emphasized that the latter condition is crucial: the movable particles accumulate at the GBs migrating at mm < m ; which increases the particle density at GBs and eventually leads to an increase in the drag force when those GBs detach from the particles. It has also been shown that the limiting grain size achieved under the influence of mobile particles can be much smaller than that found in accordance with the supposition that all particles are immobile and distributed randomly. Acknowledgement Fruitful discussions with Prof. L. Shvindlerman are gratefully acknowledged.
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References [1] Smith CS. Trans AIME 1948;175:15. [2] Shewmon PG. Trans AIME 1964;230:1134. [3] Ashby MF. In: Hansen N, Jones AR, Leffers T, editors. Recrystallization and grain growth of multi-phase and particle containing materials. Roskilde (Denmark): Risø; 1980. p. 325. [4] Gottstein G, Shvindlerman LS. Acta Metall 1993;41:3267. [5] Novikov VYu. Int J Mater Res 2007;98:1031. [6] Beck PA, Kremer JC, Demer LJ, Holzworth ML. Trans AIME 1948;175:372. [7] Haroun NA, Budworth DW. J Mater Sci 1968;3:326. [8] Hassold GN, Srolovitz DJ. Scripta Metal Mater 1995;32:1541. [9] Luecke K, Detert K. Acta Metall 1957;5:628. [10] Rabkin E. Scripta Mater 2000;42:1199. [11] Novikov VYu. Acta Metall 1978;26:1739. [12] Novikov VYu. Z Metallkd 1989;98:767. [13] Novikov VYu. Mater Lett 2008;62:3748. [14] Novikov VYu. Scripta Mater 1999;47:1935. [15] Novikov VYu. Scripta Mater 1999;47:4507. [16] Novikov VYu. Mater Lett 2008;62:2071. [17] Gottstein G, Shvindlerman LS. Grain boundary migration in metals: thermodynamics, kinetics, applications. Boca Raton (FL): CRC Press; 1999.
Acta Materialia 58 (2010) 3326–3331 www.elsevier.com/locate/actamat
On grain growth in the presence of mobile particles V.Yu. Novikov * Moscow Institute of Steel and Alloys, Moscow, Russia Received 16 January 2010; received in revised form 4 February 2010; accepted 5 February 2010 Available online 16 March 2010
Abstract The ability of second phase particles to migrate along with grain boundaries is shown to be determined not only by the particle mobility but also by the migration rate of the grain boundary where they locate. This leads to a duality in the mobile particle behaviour: they behave as either movable or immovable depending on the boundary migration rate. In the first case, they reduce the boundary mobility; in the second one they decrease the driving force for boundary migration. It is demonstrated by numerical modeling that mobile particles with low mobility can suppress grain growth even in nanocrystalline material, the limiting grains size being several times smaller than in the case of randomly distributed immobile particles. It is also shown that the Zener solution to the problem of the grain growth retardation by disperse particles is a specific case of the proposed approach. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain growth suppression; Grain boundary migration; Disperse particles; Nanocrystalline materials
1. Introduction The universally accepted approach to explaining the impact of disperse particles on grain growth was proposed by C. Zener (quoted in Ref. [1]). According to this approach, randomly distributed spherical particles of volume fraction f and radius r exert the drag force on grain boundaries (GBs): DF drag ¼ cB Z ¼ 1:5cB f =r
ð1Þ
where cB is the GB energy. The term Z = 1.5f/r is further named a Zener drag. This drag is caused by an increase in the GB area owing to release of GBs from particles located on them. So, disperse particles are considered in Ref. [1] to be immobile although they can be of relatively small size and so could possess rather high mobility [2]. As has been shown recently [3,4] particles can migrate along with GBs provided the migration rate of the latter is not higher than m ¼ pcB M P r *
Present address: Treptower Str. 74d, Hamburg 22147, Germany. E-mail address: [email protected].
ð2Þ
where MP is the particle mobility. Lately it was proposed [5] that the ability or inability of mobile particles to move along with migrating GBs not only depends on the properties of particles but also is a result of the interaction of particles with certain GBs. Namely, the same mobile particles either move along with GBs (further, movable particles) or detach from them (immovable particles), depending on the migration rate, vm, of the latter: if mm < v*, particles behave as movable, in the opposite case as immovable. Obviously, the Zener approach relates to the second case, which can be confirmed by the following consideration. If the grain growth process can be represented by the behaviour of an “average” grain (i.e. if a one-grain approximation is used), the grain growth rate is described after Ref. [6] as: mm ¼
3cB M B 2D
ð3Þ
where D is the three-dimensional average grain diameter and MB the GB mobility; both cB and MB are supposed to be identical for all GBs and independent of time. Then in the presence of movable particles located on migrating GBs, the growth rate is [4]:
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.02.006
V.Yu. Novikov / Acta Materialia 58 (2010) 3326–3331
mm ¼ 1:5M eff cB =D In Eq. (4), M
eff
3327
ð4Þ is the “effective GB mobility”:
M eff ¼ M B =ð1 þ nA M B =M P Þ
ð5Þ
nA being the number of particles per unit GB area. If nAMB/MP 1, then mm ffi
1:5cB M P nA D
ð6Þ
Since for randomly distributed particles nA ¼
3 f =r2 2p
ð7Þ
it can be found from the condition mm ¼ m that GBs detach from the particles if 1=Dlim ¼ f =r
ð8Þ
where Dlim is the “limiting average grain size” after Ref. [1]. This is in a good agreement with the term Z in Eq. (1)1 and leads to the conclusion that the Zener approach can be considered as a specific case of interaction between mobile particles and GBs. It is clear, however, that the one-grain approximation ignores the fact that any microstructure is formed by grains of different sizes and, thus, the grain of size Ri adjacent to neighbours of different sizes Rj can consume an immediate neighbour of size Rj < Ri. Migration rate of the boundary between the grain and its neighbour in the presence of movable particles is: ¼ cM eff cB ð1=Rj 1=Ri Þ mmov ij
ð9Þ
where c is a constant close in value to 1 and Meff is the effective GB mobility. It follows from Eq. (9) that movable particles do not affect the driving force for grain growth. Fig. 1a displays the growth rate of a grain of radius Ri vs. the size of consumed grains according to Eq. (9). Two magnitudes of Ri, namely, Ri = R and 3R, were used, where the former is the average grain size and the latter approximately corresponds to the maximum grain size in a typical single-phase annealed material. It is seen in Fig. 1a that, in the presence of movable particles, grains of size Ri are able to consume all their smaller neighbours and that their GBs migrate at rates increasing with a decrease in the size of consumed grains Rj. Since v* in Eq. (2) is independent of Rj, it should be displayed by a horizontal line in the coordinates used, which leads to the appearance of two ranges on the Rj/R scale. In the right-hand range where mm < m , GBs can migrate along with particles and so the
1 If particles are located at GBs only, it can be derived in the same way that the condition for the particle detachment is:
cB =D2lim ¼ acB f =r2 which corresponds to the dependence [7]:
Dlim =r / f 1=2
eff Fig. 1. (a) Reduced GB migration rate mmov ij =ðcM cB =RÞ according to Eq. (9) vs. size of consumed grains Rj/R. The size of the growing grain is R (solid line) and 3R (dot line) R being the average grain radius. (b) Normalized log-normal grain size distribution.
latter behave as movable. In the left-hand one, i.e. at small Rj/R, the GB migration rate is larger than m*. Thus, boundaries of the corresponding grains would detach from the particles owing to which the latter behave as immovable although they have the same radius and mobility as the movable ones. Owing to this, they are further referred to as movable/immovable particles. All particles behave as immovable at a very low m*, e.g. at MPi ? 0, whereas at very high m* all particles behave as movable. Obviously, a decrease in m* should lead to an increase in the fraction of consumed grains whose GBs migrate at rates mm P m (compare Fig. 1a and b). This, in turn, should increase the fraction of immovable particles. The impact of immovable particles on the GB migration rate can be presented as follows: mim ij ¼ cM B cB ð1=Rj 1=Ri ZÞ
ð10Þ
where 1/Rj 1/Ri Z > 0 and Z is the Zener drag. As follows from Eq. (10), immovable particles, in contrast to movable ones, decrease the driving force for grain growth. As a result, there should appear a gap between the size of a growing grain and the maximum size of the consumed . Fig. 2a illustrates the situation for the case that grain Rmax j the grain of the average size R is the growing one. Such a gap should also appear between the minimum size of the and the size of the consumed grain. growing grain Rmin i The corresponding illustration is given in Fig. 2b for the case that the size of the consumed grain is R. It is seen in decreases and Rmin increases with an inFig. 2 that Rmax j i crease in the product ZR. Thus, if disperse particles behave
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V.Yu. Novikov / Acta Materialia 58 (2010) 3326–3331
GB migration rate is Pv*, particles pin the GB; in the opposite case they move along with the GB. As soon as this becomes obvious, we can attempt to control the particle behaviour and thus the microstructure evolution in the course of grain growth. This paper is aimed at the study of the impact on grain growth of movable/immovable particles at different magnitudes of v*. 2. Model description and details of simulation
Fig. 2. Effect of ZR on (a) relative size of grains consumed by immediate neighbours of the mean size R and (b) relative size of crystallites able to grow at the expense of grains of size R.
as immovable, this results in inability of certain grains to grow at the expense of some of their smaller neighbours and to be consumed by some of their larger neighbours. By contrast with studies of Zener pinning, to the best of our knowledge there are only a few investigations on the effect of mobile particles. The impact of mobile particles on grain growth kinetics has been studied in Ref. [8] where the particle behaviour was supposed to be analogous to that of mobile impurities. According to Ref. [9], in the case that the driving force on the GB is low or the impurity mobility is high, impurity atoms diffuse along with the GB and the GB migration rate is small. In the opposite case (the driving force on the GB is high or the impurity mobility is low) the GB becomes free from impurities and migrates at a much higher rate, leaving impurity atoms behind. In the model polycrystal studied, particles were supposed to be distributed randomly and their mobility constant, whereas the particle volume fraction was varied. The Monte Carlo simulations [8] have shown that when the mean grain size becomes approximately equal to the average inter-particle distance, the grain growth process is temporarily arrested. This was explained by the inhibiting effect of “static” particles. A further increase in the growth duration led to an increase in both the mean grain size and the growth rate, which was explained by the GB detachment from static particles and its movement along with “mobile” ones. In the theoretical analysis [10] of the drag effect of second phase particles located at GB and growing in the course of grain growth, it was also shown that the dual behaviour of such particles can be revealed. Thus, the mechanism of grain growth inhibition by mobile second phase particles is established in Ref. [8]: particles, able to diffuse along with GBs, decrease the mobility of loaded GBs, whereas the others “pin” GBs in the manner described by Zener. However, it remains unclear what is responsible for the particle behaviour and whether the relative number of the first or second ones could be influenced. We believe that the answer to these questions is found: the particle behaviour is determined by the migration rate of GBs interacting with them. Namely, if the
This work was carried out by numerical simulation with the aid of a statistical model [11,12] successfully used to investigate various aspects of grain growth (see e.g. Refs. [13–16]). It describes the evolution of both the grain size distribution and its parameters with time. The influence of disperse particles on grain growth was investigated on a model nanocrystalline polycrystal. Its initial microstructure was characterized by an almost log-normal grain size distribution (see Fig. 1b) with the mean three-dimensional grain diameter D = 73 nm. Initial spatial distribution of particles was supposed to be random. Particle size was chosen in such a way that their radius r = 3 nm was about one order of magnitude smaller than the mean grain size; their volume fraction was 0.002. Grain growth kinetics was characterized by the time dependence of the number averaged, three-dimensional grain diameter D. The impact of v* on both the growth kinetics and microstructure evolution was investigated. Microstructure changes were studied in a polycrystal with identical cB and MB, both being supposed to be independent of time. It was also supposed that (i) disperse particles are initially distributed randomly but in the course of grain growth they redistribute between GBs and grain interior owing to both the sweeping-out of movable particles from the grain interior and the detachment of immovable particle from GBs, (ii) they have an identical size and identical mobility, (iii) the total number of particles remains unchanged, (iv) the instantaneous mean number of particles per unit boundary area, nA, is identical for all GBs, and (v) the instantaneous mean particle density in the grain interior is identical for all grains. In order to trace changes in the grain size distribution, the displacements, Ddij, of the boundary between each pair of adjacent grains should be found. If particles at ¼ mmov the GB behave as movable, Dd mov ij Dt, where Dt is ij mob the time step. The product of Dd ij and the contact area between corresponding grains delivers the magnitude of the volume swept by the GB migrating at the rate
V.Yu. Novikov / Acta Materialia 58 (2010) 3326–3331
migrating GBs. Then the magnitude of f in Eq. (1) should be derived from Eq. (7) so that ð11Þ
Z ¼ pnA r Dd mob ij
Dd im ij ;
or for various The GB displacements, either i and j were used to derive the size increments of growing grains of radius Ri as well as the size decrements of grains consumed by them. This necessitates the consideration of the mean contact probabilities of growing and consumed grains of different sizes. The size alterations, in turn, were used to derive the corresponding changes in the grain size distribution under the condition of volume preservation. and Values of cB = 0.5 104 J cm2 4 4 1 1 MB = 2 10 cm J s were taken from the data [17] for Al at T = 673 K. The particle mobility is supposed to be controlled by the interface diffusion [2]: M P ¼ ðdDS X=kT Þ=pr4
ð12Þ
where d is the “thickness” of the particle/matrix interface, DS the diffusion coefficient along the interface and X the atomic volume. Numerical values of d, DS, X and T used in this work were 107 cm, 109 cm2 s1, 10 cm3 mol1 and 673 K, respectively; the corresponding MP magnitude is further designated as M # P. 3. Results and discussion The kinetics of grain growth influenced by movable/ immovable particles with M P ¼ M # P (according to the concept described above) and that affected by immobile and randomly distributed particles (according to Ref. [1]) are compared in Fig. 3. As could be expected, in both cases the growth process is retarded. At the same time, the kinetic curves for grain growth affected by movable/ immovable particles and by immobile particles are princi-
Fig. 3. Grain growth kinetics simulated taking into account movable/ immovable particles with M P ¼ M # P (squares), under the supposition that all particles are immobile and randomly distributed [1] (triangles), and in the absence of particles (dash line).
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pally different. Under the influence of movable/immovable particles the mean grain size D steadily increases with time. This leads to the conclusion that the majority of particles in this case behave as movable. In fact, if all particles were movable, grains of any size would remain able to grow at the expense of any smaller grains and to be consumed by even slightly larger grains (see Fig. 1a). Owing to this, the mean grain size should continuously increase. To the contrary, in the case of immobile particles, D first increases and after some period of time becomes almost constant, reaching a “limiting grain size” [1]. In our opinion, this reflects the fact that, under the influence of immobile particles, some grains become unable to grow at the expense of their slightly smaller neighbours and at the same time cannot be consumed by their slightly larger neighbours, as has been shown in Section 1. Obviously, this apparent restriction of grain growth by immobile particles takes place at some “critical” magnitude of the product ZR of the particle drag Z and the mean grain radius R. On the basis of Fig. 2 it can be proposed that this magnitude is not smaller than 0.6. As follows from Fig. 2a, at Z P 0.6/R, grains of the mean size are able to consume neighbours of size 60.5R but this only slightly can contribute to an increase in R because the number of such neighbours is relatively small. Simultaneously, at this Z, the size of crystallites, able to consume the grains of the mean size, is P 5R (see Fig. 2b). Since the relative number of these crystallites is very small, this hardly contributes to a decrease in R. As a result, the mean grain size alteration with time cannot be noticeable because of compensation of these two contributions. It is worth mentioning that the numerical simulation of grain growth kinetics with immobile particles (see Fig. 3) yields Dlim ffi 1:2r=f ; in a good accordance with Eq. (8). It follows from the comparison of Fig. 1a with Fig. 1b that the fraction of consumed grains, whose GBs migrate at rates Pm*, should increase with a decrease in v*. This, in turn, should be accompanied by an increase in the relative number of immovable particles. Since in our simulations the magnitudes of cB and r are supposed to be constant, a reduction in v* was achieved, according to Eq. (2), by a decrease in MP through decreasing DS. Fig. 4 shows the growth kinetic curves for polycrystals with # 3 particles of different MP, from M # P to 2 10 M P . It is seen # that at M P ¼ 0:1M P ; the shape of the kinetic curve is similar to that at M P ¼ M # P : This allows the assumption that the effect of movable particles is predominant not only at # MP ¼ M# P but also at M P ¼ 0:1M P . A much more significant reduction in particle mobility to M P ¼ 0:002M # P leads to pronounced changes in the growth kinetics (see Fig. 4). The kinetic curve becomes similar to that predicted according to Ref. [1] (compare with Fig. 3) and so at this MP the effect of immovable particles becomes the strongest. Difference in the microstructure evolution at various MP becomes much more obvious if the grain size non-homogeneities are compared (see Fig. 5). The latter are characterized by the ratio Dmax/D of the maximum grain size to the
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# Fig. 4. Grain growth kinetics at MP = M # P (squares), 0:1M P (circles) and # 3 2 10 M P (triangles).
increases and becomes very strong, namely, Dmax/D attains the magnitude of 20–25, characteristic of abnormal grain growth. Thus, the vast majority of grains do not increase their size, which can only be a result of the impact of immovable particles (see the first paragraph in this section). It is interesting that in the case M P ¼ 0:002M # P ; Dlim is 5 times smaller than that obtained according to Ref. [1] for immobile particles of the same size and same volume fraction (compare with Fig. 3). It can be proposed that the grain size stabilization under the influence of movable/immovable particles is also a result of reaching a “critical” ZR value, as discussed in the preceding paragraph. However, in the case considered the magnitude of Z is determined not by the volume fraction of particles but by the magnitude of particle density at GBs, nA, in accordance with Eq. (11). Since it increases owing to sweeping out of movable particles, the critical ZR value is attained at much smaller R than in the case of immobile and randomly distributed particles. 4. Concluding remarks and summary
Fig. 5. Evolution of grain size non-homogeneity in the course of grain growth in the presence of particles with different mobility: M P ¼ M # P # (squares), M P ¼ 0:01M # P (solid circles) and M P ¼ 0:002M P (triangles).
average grain size. In the case M P ¼ M # P ; this ratio first increases from the initial magnitude of 4.7 to 6 and then decreases to 3, typical of normal grain growth. In the case M P ¼ 0:01M # P ; Dmax =D first increases to 11 and then continuously decreases to 5. An increase in Dmax/D means that at the initial stages, the growth of the majority of grains is strongly inhibited and abnormal grain growth evolves. However, afterwards the particle inhibition decays, which results in the evolution of normal grain growth. These results are in a qualitative agreement with the data [8] where grain growth was observed to be strongly inhibited at relatively low D, apparently by immovable particles, but at larger D the inhibition became weaker owing to the impact of movable particles. At last in the case M P ¼ 0:002M # P ; the grain size non-homogeneity steadily
The retardation of GB migration by disperse particles, manifesting in particular in the slowing down of the grain growth process, has been known for more than six decades. The vast majority of results on the grain growth retardation by particles have been analysed in the belief that they are immobile and so unable to move along with GBs. However, since the particles are usually rather small, it should be taken into account that they can possess a finite, nonzero mobility. So, the question arises whether mobile particles could behave as immovable. A positive answer to this question has been given in Ref. [8]. However, it remained unclear why it is possible. An answer to the latter question has been given in the present paper. It takes into account not only the particle mobility but also the GB migration rate. Namely, in any polycrystal there is a wide range of GB migration rates, vm, and, besides, there exists certain GB rate, v*, at which particles located at GBs should detach from the latter. Usually, v* is within the above-mentioned range. Owing to this, particles possessing certain mobility are immovable if the GB where they locate migrates at the rate mm P m . (In the case that the minimum GB migration rate equals v*, all particles should behave as immovable and so the well-known Zener approach is a specific case of interaction between GBs and mobile particles.) At the same time, particles located at all other GBs behave as movable. So a duality in the behaviour of mobile particles must be observed in any polycrystal, and thus the label “mobile” does not mean that the particle can move along with any GB. The impact of movable/immovable particles on grain growth was studied in the present paper on nanocrystalline polycrystal where the driving force for grain growth is especially high. This was done to widen the range of GB migration rates. In contrast to Ref. [8], the volume fraction of mobile particles was supposed to be constant whereas the magnitude of MP was varied. This was used as a means to
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control the particle behaviour because it affects, in accordance with Eq. (2), the magnitude of v*, causing the GB detachment from particles (see above). It has been found that a reduction in MP and thus in v* leads not only to a decrease in the rate of grain growth, which could be expected, but simultaneously, it increases the fraction of immovable particles, which manifests in the evolution of abnormal grain growth at the initial stages of the growth process. It has been shown that even in such a material the growth process can be completely suppressed provided the particle mobility is low but some particles behave as movable. It should be emphasized that the latter condition is crucial: the movable particles accumulate at the GBs migrating at mm < m ; which increases the particle density at GBs and eventually leads to an increase in the drag force when those GBs detach from the particles. It has also been shown that the limiting grain size achieved under the influence of mobile particles can be much smaller than that found in accordance with the supposition that all particles are immobile and distributed randomly. Acknowledgement Fruitful discussions with Prof. L. Shvindlerman are gratefully acknowledged.
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