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On the influence of triple junctions on grain growth kinetics and microstructure evolution in 2D polycrystals
Scripta Materialia 52 (2005) 857–861 www.actamat-journals.com
On the influence of triple junctions on grain growth kinetics and microstructure evolution in 2D polycrystals Vladimir Yu Novikov
*
Moscow Institute of Steel and Alloys, Leninski pr. 4, 119049 Moscow, Russian Federation Received 20 May 2004; received in revised form 19 December 2004; accepted 10 January 2005
Abstract Grain growth in 2D microstructure is modelled under the supposition that triple junctions possess a limited mobility. The impact of the triple junctions on the grain growth kinetics is shown to depend on a dimensionless parameter K equal to the product of the triple junction mobility and the average grain size divided by the grain boundary mobility; it was varied in the range from 0.03 to 100. With low initial K, the growth kinetics is at first linear and becomes parabolic at the later stages only. The value of the triple junction mobility estimated from the available experimental data is used to prove that an initial K as low as 0.005 can be observed in nano-crystalline materials. Thus, the triple junction drag could explain some features of the growth kinetics in these materials. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain growth; Triple junction; Grain boundary; Theory and modeling
1. Introduction The grain growth process is known to proceed through migration of grain boundaries to the centers of their curvature, and triple junctions (TJ) connecting the boundaries in a continuous network move together with them. It is usually supposed that TJ do not affect the rate of the process because they possess very high mobility, owing to which the boundaries meet at TJ always at equilibrium angles. However, it has been shown experimentally that in 2D tri-crystals with the consumed grain of a half-loop shape, the angles between the boundaries at TJ deviate from equilibrium values and, thus, the mobility of TJ is limited [1–3]. Simulations [4] with the same geometries as the consumed 2D grains confirm these experimental data. On the basis of these results the von Neumann–Mullins relation, describing
*
Tel.: +49 40 647 2845. E-mail address: [email protected]
the growth of a 2D individual grain, was reconsidered [5]. According to Ref. [5] the effect of limited TJ mobility should reveal itself in zero growth rate not only for 6sided grains, as follows from the von Neumann–Mullins relation, but also for grains whose number of sides is either greater or smaller than 6. This has just recently been confirmed in the experiments on Al–Mg alloy with 2D microstructure [6]. It appears that, in fact, some grains with 3–8 sides possess a growth rate close to zero. Thus, the validity of the approach [5] seems proved. In terms of boundary curvature, the influence of TJ on the migration rate v of the boundary of an individual grain is described in Ref. [5] as follows: v¼
cB M B K ; 1 þ 1=K
ð1Þ
where cB, MB and K are the energy, mobility and curvature of the boundary, respectively, and K is a TJ drag parameter: K¼
aM TJ ; MB
1359-6462/$ - see front matter Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2005.01.013
ð1aÞ
858
V.Y. Novikov / Scripta Materialia 52 (2005) 857–861
where a is the spacing between the adjacent TJ assumed the same for all the neighboring grains, and MTJ is the TJ mobility. If K is high, the growth rate of an individual grain is controlled by the boundary mobility, whereas with small K it should be slower and even depend on time linearly [5]. In other words, according to Ref. [5] the boundary migration rate is determined not only by the boundary energy and curvature but also by its ‘‘effective’’ mobility depending on the TJ spacing and mobility: M ¼
MB : 1 þ 1=K
ð2Þ
It is obvious that both Eq. (1) and the predicted deviation from the von Neumann–Mullins condition do not describe the grain growth process in a polycrystal but relate to individual grains. It is common knowledge that a description of the growth of an individual grain does not provide information on the polycrystal behavior, especially in the cases where some factors inhibiting the boundary migration are active (see e.g. Ref. [7]). So, it seems very interesting to study the TJ effect on the growth kinetics in polycrystals, which to the best of our knowledge has not been done so far. The present study addresses the possible influence of limited TJ mobility on the grain growth process in 2D microstructure.
2. Theoretical and experimental details The impact of TJ on the grain growth process in 2D polycrystals was investigated by means of computer simulation. A statistical model [8,9] whose reliability has been proved on various aspects of normal and abnormal grain growth [9–12] was used. The model was modified in such a way that the migration rate of the boundary between a growing grain of diameter Di and the adjacent grain of diameter Dj (Dj < Di) is described as follows: Dd ij ccB M B ð1=Dj 1=Di Þ ¼ ; 1 þ 1=Kj Dt where Ddij is the displacement of the boundary of the growing grain, Dt the time step, c a constant, and, by analogy with Eq. (1a): Kj ¼
Dj M TJ ; 2M B
provided a = Dj/2. At a given MTJ/MB, Kj depends on the neighborÕs size Dj and varies in a wide range determined by the width of the grain size distribution. Taking into account the current amount of different grains and their mean contact probability, an increase in Di and a decrease in Dj are derived from Ddij under the condition of total volume preservation. This procedure is repeated for all Di and Dj at every time step, which makes it pos-
sible to find the overall size increments (due to the growth of grains of diameter Di) and decrements (owing to the consumption of grains of diameter Dj) for grains of different sizes and thus to derive the corresponding changes in the grain size distribution as well as the time dependence of the average grain diameter D. In order to analyze further the TJ-affected grain growth kinetics in a polycrystal, it is necessary to modify Eq. (1). A common way of describing the growth process is to consider the behavior of an ‘‘average’’ grain of size D surrounded by the grains of the same size. In the absence of TJ drag, the mean growth rate is: dD cB M B ¼ ; dt D
ð3Þ
on the condition that MB and cB do not change with time. Since the TJ spacing normally varies along the boundary of any grain, owing to different sizes of its immediate neighbors, the same approach allows us to introduce an average TJ spacing a ¼ D=2 and an average drag parameter: K¼
DM TJ : 2M B
ð4Þ
(It is worth noting that, in contrast to the experimental studies [1–3] and simulations [4] as well as to the theoretical treatment [5], K does not relate to a single grain but is averaged over all the grains in the polycrystal considered.) Then the impact of TJ on the growth process in a polycrystal can be described by the equation obtained from Eq. (1) through substitution of 1=D for K and K for K: dD cB M B ¼ ; dt Dð1 þ 1=KÞ
ð5Þ
provided all the grain boundaries are characterized by identical MB and cB and all TJ possess the same mobility MTJ. (Moreover, it is necessary to suppose that all the boundaries and TJ newly appeared because of neighbor switching have the same properties as their counterparts in the initial structure.) It is seen in Eq. (5) that, besides the driving force cB =D, the growth rate dD=dt is determined by the average effective boundary mobility:
M ¼
MB ; 1 þ 1=K
ð6Þ
that should be time-dependent owing to an increase in D (see Eq. (4)). In contrast to Eq. (1), Eq. (5) describes changes in the average grain size D with time and thus the evolution of microstructure in the course of the growth process. In the present study, grain growth in 2D polycrystals P1 and P2 with the initial average grain diameter D0 ¼ 53 and 106.1 arbitrary length units (lu), respectively, was investigated. In both the polycrystals, the initial grain size distribution was of the same shape with
V.Y. Novikov / Scripta Materialia 52 (2005) 857–861
Dmax/Dmod ffi 3.6, where Dmax and Dmod are the maximum and the most probable grain diameter, respectively. The values of cB and MB for all the grain boundaries were assumed identical. Since MTJ was supposed the same for all TJ, this results in a constant MTJ/ MB In different simulation runs, MTJ/MB was from 0.001 to 2 lu1 owing to which the initial average drag parameter K was in the range from 0.03 to 100. The magnitude of MTJ/MB was chosen in such a way that the initial K was in the range of the experimentally observed K for tri-crystals [2] and individual grains in 2D microstructure [6]. Time is expressed in units (tu) equal to 200 Dt.
3. Results and discussion The effect of TJ on the growth kinetics in polycrystal P1 with different initial K is shown in Fig. 1. As could be expected, the growth rate under the influence of TJ is reduced in comparison to the case where grain growth is controlled by the grain boundary mobility alone, and the retarding effect increases with a reduction of the initial K. However as follows from the simulations, the TJ effect in P1 and P2 with the initial K P 26:5 is very small
so that the difference in D with respect to the boundarycontrolled kinetics does not exceed 1–2%. In polycrystal P1, the impact of TJ is noticeable if the initial K is not greater than 5.3, and with small initial K the kinetic curve is at first linear but the linear part grades further into a curvilinear one (see Fig. 1). The kinetics of TJ-affected grain growth can be found from Eq. (5) by the substitution of K (from Eq. (4)) and subsequent integration: 2
Dt þ
4Dt ffi 2cB M B t; M TJ =M B
ð7Þ
on the condition that Dt D0 , where Dt and D0 are the current and the initial average grain diameter, respectively. The first term in the left-hand side of Eq. (7) describes the well-known kinetics controlled by grain boundary mobility whereas the second one relates to the effect of TJ. It is obvious that the second term can be comparable with the first one if (i) MTJ/MB is relatively low and (ii) Dt is small, either owing to low D0 or to slow growth rate. In fact, as can be seen in Fig. 1, kinetic curves for microstructure with the initial K 6 0:1 are at first almost linear and their slopes are mutually related as the corresponding K, namely for the initial K ¼ 0:0265, 0.053, and 0.106 the slopes relate approximately as 1:2:4. Simultaneously, the duration of the linear stage depends on the initial K and thus on the growth rate, decreasing with its increase. An increase in Dt makes the first left-hand term in Eq. (7) larger and larger, which leads to transition from the linear TJ-controlled kinetics to a curvilinear one and eventually to the parabolic, boundary-controlled kinetics. The following analysis confirms the validity of this supposition. The boundary-controlled kinetics is described by the well-known parabolic law: 2
2
Dt D0 ¼ kt;
Fig. 1. Grain growth kinetics in polycrystal P1 not affected by TJ (curve 1) and subjected to the TJ effect with initial K ¼ 5:3 (2), 0.53 (3), 0.159 (4), 0.106 (5), 0.053 (6) and 0.027 (7). Straight lines show the slope of the corresponding curves.
859
ð8Þ
derived from Eq. (3) on the condition that the kinetic constant k = cBMB is independent of time. In the case of TJ-affected grain growth, kcB M , where M is determined by Eq. (6), and should vary with time owing to an increase in K with D increasing (see Eq. (4)). As can be seen in Fig. 2 (curves 3 and 4) in polycrystal P1 with the initial K < 5:3, the time dependence of 2 2 (Dt D0 ) becomes linear at the later stages of the growth process, which supports the above supposition. Fig. 2 shows that an increase in the initial K leads to an increase in the duration of the linear regime and with K P 5:3 it is observed from the beginning of the growth process. In the strict sense, the influence of TJ should al2 2 ways lead to deviation of the (Dt D0 ) dependence from linear one because M changes with time. Then, the ob2 2 served linear time dependence of (Dt D0 ) is an approximation, which obviously results from insignificant changes in the current magnitude of K and thus from
860
V.Y. Novikov / Scripta Materialia 52 (2005) 857–861 5
4x10
4.5
2 1
3.5
4
3.0
3
5
2x10
1
2.5
2
2
D - D0 , lu
2
3x10
Dmax / D mod
4.0
3
5
4
0
2 10
20
30
40
50
Time , tu
5
1x10
Fig. 3. Temporal evolution of Dmax/Dmod in polycrystal P1 not affected by TJ drag (curve 1) and under TJ influence with initial K ¼ 0:53 (2), 0.106 (3) and 0.053 (4).
0 0
10 2
20
30 40 50 Time , tu
60
70
2
Fig. 2. Dependence of (Dt D0 ) on time in polycrystal P1 not affected by TJ drag (curve 1) and subjected to the TJ effect with initial K ¼ 5:3 (2), 0.53 (3) and 0.159 (4). Straight lines correspond to the parabolic law, Eq. (8).
almost constant M (this was confirmed by a detailed examination of the simulation data). As can be seen from the comparison of curves 2–4 with curve 1 in Fig. 2, at the stage of nearly linear behavior of 2 2 (Dt D0 ) the slopes of curves 2–4 describing the TJ-controlled growth are smaller than in the case of the boundary-controlled growth (curve 1). Obviously, this results from the influence of MTJ/MB on the magnitude of the effective boundary mobility M . Thus, while the growth kinetics is almost linear if the initial and the current K are small (60.1), it becomes further curvilinear, and eventually parabolic with K P 5, the kinetic constant at the latter stage decreasing with reduction of K. Obviously, all this results from changes in M with the evolution of the growth process. Fig. 3 shows the evolution in the course of grain growth of grain size non-homogeneity characterized by the ratio Dmax/Dmod. In contrast to continuous reduction of the ratio in the absence of the TJ effect (curve 1), at the stage where the TJ-affected growth kinetics is nearly linear it increases and reaches a maximum (compare Figs. 3 and 1). The initial increase and the subsequent decrease in Dmax/Dmod are typical of abnormal grain growth although in the latter case the duration and the amplitude of these changes are much greater. So, it can be supposed that the behavior of Dmax/Dmod shown in Fig. 3 stems from the action of some factor retarding normal grain growth. Obviously, in our case this factor is a decreased M at the initial stage of the
process (see above), which is supported by the fact that both the height and the width of the maximum on the Dmax/Dmod dependence are the greatest if the initial K is the lowest (compare curves 2–4 in Fig. 3). It should however be noted that such a time dependence of Dmax/Dmod cannot be considered as an inherent characteristic of TJ-affected grain growth because the same behavior can also be observed in the case where abnormal grain growth goes over into normal one [11]. In summary, the impact of TJ on the grain growth evolution in 2D microstructure with all TJ of limited mobility can reveal itself in (i) retardation of the process, (ii) occurrence of linear kinetics at the initial stage of the process, and (iii) temporal increase of the grain size nonhomogeneity at this stage. Since all these effects are more pronounced in polycrystals with low initial K and thus with small initial grain size, they could apparently be found on fine-grained or nano-crystalline materials. To prove this supposition we estimated the ratio MTJ/ MB from the experimental data [2,6]; its maximum value appeared to be 50–100 mm1. If we further assume that D0 ¼ 100 nm then, according to Eq. (4), the initial K ¼ 0:0025–0:005. This value is at the lower limit of K studied in the present work, which confirms that the above-mentioned features of TJ-controlled grain growth could be observed on nano-crystalline materials. In fact, both an inhibited grain growth and linear growth kinetics have already been detected on such materials [13,14] and were explained by pinning action of nano-pores [13] or by intrinsic drag of excess vacancies occurring because of disappearance of grain boundaries [14]. However, as has been shown in this work, these features could also be ascribed to the effect of TJ. Thus, the influence of limited TJ mobility on the grain growth process should be taken into account as an alternative explanation of experimental observations.
V.Y. Novikov / Scripta Materialia 52 (2005) 857–861
Acknowledgment The author is grateful to Prof. L.S. Shvindlerman for stimulating and fruitful discussions. References [1] Czubayko U, Sursaeva VG, Gottstein G, Shvindlerman LS. Acta Mater 1998;46:5863. [2] Protasova SG, Gottstein G, Molodov DA, Sursaeva VG, Shvindlerman LS. Acta Mater 2001;49:2519. [3] Gottstein G, Sursaeva VG, Shvindlerman LS. Interface Sci 1999;7:273.
861
[4] Upmanyu M, Srolovitz DJ, Shvindlerman LS, Gottstein G. Acta Mater 2002;50:1405. [5] Gottstein G, Shvindlerman LS. Acta Mater 2002;50:703. [6] Matissen D, Kirch D, Molodov DA, Shvindlerman LS, Gottstein G. Mater Sci Forum 2004;467–470:777. [7] Gladman T. Proc Roy Soc A 1966;294:298. [8] Novikov VYu. Acta Metall 1978;26:1739. [9] Novikov VYu. Scripta Mater 1999;40:91. [10] Novikov VYu. Acta Metall 1979;27:1461. [11] Novikov VYu. Scripta Mater 1999;47:1935. [12] Novikov VYu. Scripta Mater 1999;47:4507. [13] Gleiter H. Acta Mater 2000;48:1. [14] Krill CE, Helfen L, Michels D, Natter H, Fitch A, Masson O, et al. Phys Rev Lett 2001;86:842.
On the influence of triple junctions on grain growth kinetics and microstructure evolution in 2D polycrystals Vladimir Yu Novikov
*
Moscow Institute of Steel and Alloys, Leninski pr. 4, 119049 Moscow, Russian Federation Received 20 May 2004; received in revised form 19 December 2004; accepted 10 January 2005
Abstract Grain growth in 2D microstructure is modelled under the supposition that triple junctions possess a limited mobility. The impact of the triple junctions on the grain growth kinetics is shown to depend on a dimensionless parameter K equal to the product of the triple junction mobility and the average grain size divided by the grain boundary mobility; it was varied in the range from 0.03 to 100. With low initial K, the growth kinetics is at first linear and becomes parabolic at the later stages only. The value of the triple junction mobility estimated from the available experimental data is used to prove that an initial K as low as 0.005 can be observed in nano-crystalline materials. Thus, the triple junction drag could explain some features of the growth kinetics in these materials. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain growth; Triple junction; Grain boundary; Theory and modeling
1. Introduction The grain growth process is known to proceed through migration of grain boundaries to the centers of their curvature, and triple junctions (TJ) connecting the boundaries in a continuous network move together with them. It is usually supposed that TJ do not affect the rate of the process because they possess very high mobility, owing to which the boundaries meet at TJ always at equilibrium angles. However, it has been shown experimentally that in 2D tri-crystals with the consumed grain of a half-loop shape, the angles between the boundaries at TJ deviate from equilibrium values and, thus, the mobility of TJ is limited [1–3]. Simulations [4] with the same geometries as the consumed 2D grains confirm these experimental data. On the basis of these results the von Neumann–Mullins relation, describing
*
Tel.: +49 40 647 2845. E-mail address: [email protected]
the growth of a 2D individual grain, was reconsidered [5]. According to Ref. [5] the effect of limited TJ mobility should reveal itself in zero growth rate not only for 6sided grains, as follows from the von Neumann–Mullins relation, but also for grains whose number of sides is either greater or smaller than 6. This has just recently been confirmed in the experiments on Al–Mg alloy with 2D microstructure [6]. It appears that, in fact, some grains with 3–8 sides possess a growth rate close to zero. Thus, the validity of the approach [5] seems proved. In terms of boundary curvature, the influence of TJ on the migration rate v of the boundary of an individual grain is described in Ref. [5] as follows: v¼
cB M B K ; 1 þ 1=K
ð1Þ
where cB, MB and K are the energy, mobility and curvature of the boundary, respectively, and K is a TJ drag parameter: K¼
aM TJ ; MB
1359-6462/$ - see front matter Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2005.01.013
ð1aÞ
858
V.Y. Novikov / Scripta Materialia 52 (2005) 857–861
where a is the spacing between the adjacent TJ assumed the same for all the neighboring grains, and MTJ is the TJ mobility. If K is high, the growth rate of an individual grain is controlled by the boundary mobility, whereas with small K it should be slower and even depend on time linearly [5]. In other words, according to Ref. [5] the boundary migration rate is determined not only by the boundary energy and curvature but also by its ‘‘effective’’ mobility depending on the TJ spacing and mobility: M ¼
MB : 1 þ 1=K
ð2Þ
It is obvious that both Eq. (1) and the predicted deviation from the von Neumann–Mullins condition do not describe the grain growth process in a polycrystal but relate to individual grains. It is common knowledge that a description of the growth of an individual grain does not provide information on the polycrystal behavior, especially in the cases where some factors inhibiting the boundary migration are active (see e.g. Ref. [7]). So, it seems very interesting to study the TJ effect on the growth kinetics in polycrystals, which to the best of our knowledge has not been done so far. The present study addresses the possible influence of limited TJ mobility on the grain growth process in 2D microstructure.
2. Theoretical and experimental details The impact of TJ on the grain growth process in 2D polycrystals was investigated by means of computer simulation. A statistical model [8,9] whose reliability has been proved on various aspects of normal and abnormal grain growth [9–12] was used. The model was modified in such a way that the migration rate of the boundary between a growing grain of diameter Di and the adjacent grain of diameter Dj (Dj < Di) is described as follows: Dd ij ccB M B ð1=Dj 1=Di Þ ¼ ; 1 þ 1=Kj Dt where Ddij is the displacement of the boundary of the growing grain, Dt the time step, c a constant, and, by analogy with Eq. (1a): Kj ¼
Dj M TJ ; 2M B
provided a = Dj/2. At a given MTJ/MB, Kj depends on the neighborÕs size Dj and varies in a wide range determined by the width of the grain size distribution. Taking into account the current amount of different grains and their mean contact probability, an increase in Di and a decrease in Dj are derived from Ddij under the condition of total volume preservation. This procedure is repeated for all Di and Dj at every time step, which makes it pos-
sible to find the overall size increments (due to the growth of grains of diameter Di) and decrements (owing to the consumption of grains of diameter Dj) for grains of different sizes and thus to derive the corresponding changes in the grain size distribution as well as the time dependence of the average grain diameter D. In order to analyze further the TJ-affected grain growth kinetics in a polycrystal, it is necessary to modify Eq. (1). A common way of describing the growth process is to consider the behavior of an ‘‘average’’ grain of size D surrounded by the grains of the same size. In the absence of TJ drag, the mean growth rate is: dD cB M B ¼ ; dt D
ð3Þ
on the condition that MB and cB do not change with time. Since the TJ spacing normally varies along the boundary of any grain, owing to different sizes of its immediate neighbors, the same approach allows us to introduce an average TJ spacing a ¼ D=2 and an average drag parameter: K¼
DM TJ : 2M B
ð4Þ
(It is worth noting that, in contrast to the experimental studies [1–3] and simulations [4] as well as to the theoretical treatment [5], K does not relate to a single grain but is averaged over all the grains in the polycrystal considered.) Then the impact of TJ on the growth process in a polycrystal can be described by the equation obtained from Eq. (1) through substitution of 1=D for K and K for K: dD cB M B ¼ ; dt Dð1 þ 1=KÞ
ð5Þ
provided all the grain boundaries are characterized by identical MB and cB and all TJ possess the same mobility MTJ. (Moreover, it is necessary to suppose that all the boundaries and TJ newly appeared because of neighbor switching have the same properties as their counterparts in the initial structure.) It is seen in Eq. (5) that, besides the driving force cB =D, the growth rate dD=dt is determined by the average effective boundary mobility:
M ¼
MB ; 1 þ 1=K
ð6Þ
that should be time-dependent owing to an increase in D (see Eq. (4)). In contrast to Eq. (1), Eq. (5) describes changes in the average grain size D with time and thus the evolution of microstructure in the course of the growth process. In the present study, grain growth in 2D polycrystals P1 and P2 with the initial average grain diameter D0 ¼ 53 and 106.1 arbitrary length units (lu), respectively, was investigated. In both the polycrystals, the initial grain size distribution was of the same shape with
V.Y. Novikov / Scripta Materialia 52 (2005) 857–861
Dmax/Dmod ffi 3.6, where Dmax and Dmod are the maximum and the most probable grain diameter, respectively. The values of cB and MB for all the grain boundaries were assumed identical. Since MTJ was supposed the same for all TJ, this results in a constant MTJ/ MB In different simulation runs, MTJ/MB was from 0.001 to 2 lu1 owing to which the initial average drag parameter K was in the range from 0.03 to 100. The magnitude of MTJ/MB was chosen in such a way that the initial K was in the range of the experimentally observed K for tri-crystals [2] and individual grains in 2D microstructure [6]. Time is expressed in units (tu) equal to 200 Dt.
3. Results and discussion The effect of TJ on the growth kinetics in polycrystal P1 with different initial K is shown in Fig. 1. As could be expected, the growth rate under the influence of TJ is reduced in comparison to the case where grain growth is controlled by the grain boundary mobility alone, and the retarding effect increases with a reduction of the initial K. However as follows from the simulations, the TJ effect in P1 and P2 with the initial K P 26:5 is very small
so that the difference in D with respect to the boundarycontrolled kinetics does not exceed 1–2%. In polycrystal P1, the impact of TJ is noticeable if the initial K is not greater than 5.3, and with small initial K the kinetic curve is at first linear but the linear part grades further into a curvilinear one (see Fig. 1). The kinetics of TJ-affected grain growth can be found from Eq. (5) by the substitution of K (from Eq. (4)) and subsequent integration: 2
Dt þ
4Dt ffi 2cB M B t; M TJ =M B
ð7Þ
on the condition that Dt D0 , where Dt and D0 are the current and the initial average grain diameter, respectively. The first term in the left-hand side of Eq. (7) describes the well-known kinetics controlled by grain boundary mobility whereas the second one relates to the effect of TJ. It is obvious that the second term can be comparable with the first one if (i) MTJ/MB is relatively low and (ii) Dt is small, either owing to low D0 or to slow growth rate. In fact, as can be seen in Fig. 1, kinetic curves for microstructure with the initial K 6 0:1 are at first almost linear and their slopes are mutually related as the corresponding K, namely for the initial K ¼ 0:0265, 0.053, and 0.106 the slopes relate approximately as 1:2:4. Simultaneously, the duration of the linear stage depends on the initial K and thus on the growth rate, decreasing with its increase. An increase in Dt makes the first left-hand term in Eq. (7) larger and larger, which leads to transition from the linear TJ-controlled kinetics to a curvilinear one and eventually to the parabolic, boundary-controlled kinetics. The following analysis confirms the validity of this supposition. The boundary-controlled kinetics is described by the well-known parabolic law: 2
2
Dt D0 ¼ kt;
Fig. 1. Grain growth kinetics in polycrystal P1 not affected by TJ (curve 1) and subjected to the TJ effect with initial K ¼ 5:3 (2), 0.53 (3), 0.159 (4), 0.106 (5), 0.053 (6) and 0.027 (7). Straight lines show the slope of the corresponding curves.
859
ð8Þ
derived from Eq. (3) on the condition that the kinetic constant k = cBMB is independent of time. In the case of TJ-affected grain growth, kcB M , where M is determined by Eq. (6), and should vary with time owing to an increase in K with D increasing (see Eq. (4)). As can be seen in Fig. 2 (curves 3 and 4) in polycrystal P1 with the initial K < 5:3, the time dependence of 2 2 (Dt D0 ) becomes linear at the later stages of the growth process, which supports the above supposition. Fig. 2 shows that an increase in the initial K leads to an increase in the duration of the linear regime and with K P 5:3 it is observed from the beginning of the growth process. In the strict sense, the influence of TJ should al2 2 ways lead to deviation of the (Dt D0 ) dependence from linear one because M changes with time. Then, the ob2 2 served linear time dependence of (Dt D0 ) is an approximation, which obviously results from insignificant changes in the current magnitude of K and thus from
860
V.Y. Novikov / Scripta Materialia 52 (2005) 857–861 5
4x10
4.5
2 1
3.5
4
3.0
3
5
2x10
1
2.5
2
2
D - D0 , lu
2
3x10
Dmax / D mod
4.0
3
5
4
0
2 10
20
30
40
50
Time , tu
5
1x10
Fig. 3. Temporal evolution of Dmax/Dmod in polycrystal P1 not affected by TJ drag (curve 1) and under TJ influence with initial K ¼ 0:53 (2), 0.106 (3) and 0.053 (4).
0 0
10 2
20
30 40 50 Time , tu
60
70
2
Fig. 2. Dependence of (Dt D0 ) on time in polycrystal P1 not affected by TJ drag (curve 1) and subjected to the TJ effect with initial K ¼ 5:3 (2), 0.53 (3) and 0.159 (4). Straight lines correspond to the parabolic law, Eq. (8).
almost constant M (this was confirmed by a detailed examination of the simulation data). As can be seen from the comparison of curves 2–4 with curve 1 in Fig. 2, at the stage of nearly linear behavior of 2 2 (Dt D0 ) the slopes of curves 2–4 describing the TJ-controlled growth are smaller than in the case of the boundary-controlled growth (curve 1). Obviously, this results from the influence of MTJ/MB on the magnitude of the effective boundary mobility M . Thus, while the growth kinetics is almost linear if the initial and the current K are small (60.1), it becomes further curvilinear, and eventually parabolic with K P 5, the kinetic constant at the latter stage decreasing with reduction of K. Obviously, all this results from changes in M with the evolution of the growth process. Fig. 3 shows the evolution in the course of grain growth of grain size non-homogeneity characterized by the ratio Dmax/Dmod. In contrast to continuous reduction of the ratio in the absence of the TJ effect (curve 1), at the stage where the TJ-affected growth kinetics is nearly linear it increases and reaches a maximum (compare Figs. 3 and 1). The initial increase and the subsequent decrease in Dmax/Dmod are typical of abnormal grain growth although in the latter case the duration and the amplitude of these changes are much greater. So, it can be supposed that the behavior of Dmax/Dmod shown in Fig. 3 stems from the action of some factor retarding normal grain growth. Obviously, in our case this factor is a decreased M at the initial stage of the
process (see above), which is supported by the fact that both the height and the width of the maximum on the Dmax/Dmod dependence are the greatest if the initial K is the lowest (compare curves 2–4 in Fig. 3). It should however be noted that such a time dependence of Dmax/Dmod cannot be considered as an inherent characteristic of TJ-affected grain growth because the same behavior can also be observed in the case where abnormal grain growth goes over into normal one [11]. In summary, the impact of TJ on the grain growth evolution in 2D microstructure with all TJ of limited mobility can reveal itself in (i) retardation of the process, (ii) occurrence of linear kinetics at the initial stage of the process, and (iii) temporal increase of the grain size nonhomogeneity at this stage. Since all these effects are more pronounced in polycrystals with low initial K and thus with small initial grain size, they could apparently be found on fine-grained or nano-crystalline materials. To prove this supposition we estimated the ratio MTJ/ MB from the experimental data [2,6]; its maximum value appeared to be 50–100 mm1. If we further assume that D0 ¼ 100 nm then, according to Eq. (4), the initial K ¼ 0:0025–0:005. This value is at the lower limit of K studied in the present work, which confirms that the above-mentioned features of TJ-controlled grain growth could be observed on nano-crystalline materials. In fact, both an inhibited grain growth and linear growth kinetics have already been detected on such materials [13,14] and were explained by pinning action of nano-pores [13] or by intrinsic drag of excess vacancies occurring because of disappearance of grain boundaries [14]. However, as has been shown in this work, these features could also be ascribed to the effect of TJ. Thus, the influence of limited TJ mobility on the grain growth process should be taken into account as an alternative explanation of experimental observations.
V.Y. Novikov / Scripta Materialia 52 (2005) 857–861
Acknowledgment The author is grateful to Prof. L.S. Shvindlerman for stimulating and fruitful discussions. References [1] Czubayko U, Sursaeva VG, Gottstein G, Shvindlerman LS. Acta Mater 1998;46:5863. [2] Protasova SG, Gottstein G, Molodov DA, Sursaeva VG, Shvindlerman LS. Acta Mater 2001;49:2519. [3] Gottstein G, Sursaeva VG, Shvindlerman LS. Interface Sci 1999;7:273.
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