- Email: [email protected]
Formation of dendritic lamella in eutectic alloys
November 2002
Materials Letters 56 (2002) 921 – 926 www.elsevier.com/locate/matlet
Formation of dendritic lamella in eutectic alloys W.M. Wang a,b,*, J.-M. Liu a, J.F. Webb a, X.F. Bian b, G.L. Yuan a, Z.G. Liu a b
a Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China Key Lab of Liquid Structure and Heredity of Materials, Shandong University, Jinan 250061, China
Received 23 February 2002; accepted 4 March 2002
Abstract The dendritic lamella in eutectic growth is simulated with a Monte Carlo (MC) model. It shows that the dendritic lamella (h phase) forms at a relatively large supercooling degree DT, with a higher concentration gradient ahead of the solid/liquid (s/l) interface, and that the formation of dendritic lamella is associated with the concentration perturbation. The microstructures of eutectic Al – Si accord with the simulated results. D 2002 Elsevier Science B.V. All rights reserved. PACS: 61.72-y; 61.42.k Keywords: Eutectic; MC simulation
1. Introduction Eutectic alloys are the basis of many casting alloys. This has led to extensive theoretical and experimental studies on the relationship between microstructure and solidification conditions [1,2]. One of the most significant theoretical studies is Jackson and Hunt’s (JH) [3] analysis of the regular lamellar eutectic structure. Besides a lot of experimental studies, computational studies including numerical calculation and kinetic simulation on eutectic growth are increasing. Numerical calculation is centered on the solution of composition field ahead of the growing interface [1,3 – 5], which mainly deals with only one set of lamellae (a + h) without giving a concrete pattern of eutectic growth. In many practical cases, the eutectic lamella shape is associated not only with the concentration
field directly ahead of the solid/liquid (s/l) interface, but also with the shape of the neighboring lamellae. Kinetic simulations like the Monte Carlo (MC) method are applied to simulate the motion of atoms during eutectic growth [6,7], by such a method we can get the eutectic pattern. However, to our knowledge, there is no MC simulation work on the formation of dendritic lamella in the eutectic growth of binary alloys. In this paper, we apply a MC method to simulate the formation of dendritic lamella, and analyze the concentration field ahead of the s/l interface; we also present the experimental results for the dendritic lamella (Si phase) in Al –Si alloy at different cooling rates.
2. Simulation and experimental procedure 2.1. Model and simulation procedure
*
Corresponding author. Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China. E-mail address: [email protected] (W.M. Wang).
Like the previous simulation work [8,9], the simulation was performed using a rectangular l0 l0
0167-577X/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 5 7 7 X ( 0 2 ) 0 0 6 3 8 - 9
922
W.M. Wang et al. / Materials Letters 56 (2002) 921–926
lattice with a periodic boundary condition parallel to the s/l interface, where the lattice parameter is b (shown in Fig. 1). A thermodynamic framework of the present approach is quite similar to the work of Xiao et al. [7]. The eutectic growth procedure includes two reversible events (shown in Fig. 1): (1) species A (in the given concentration c0) and B emit from the source line and random walk, then stick on the s/l interface; (2) the species depart from the s/l interface and diffuse on the s/l interface, then escape from the regime after possible random walk. The possibility for the system to choose whether departure event or emission event is Dl Pe ¼ 1= 1 þ exp ð1Þ kB T where kB is the Boltzmann constant and T is the temperature, and Dl is a chemical potential difference between the solid and liquid with a unit of kBT. If the interfacial concentration difference between the melt and solid concentration is omitted, we have [10] Dl ¼
Lp DT , where DT ¼ T Te Te
ð2Þ
where Lp and Te are the latent heat and temperature for the equilibrium melt – solid transformation, respectively.
The possibility of sticking on the s/l interface for the species reached there is expðDl=kB T Þ exp½DEbond GL DHÞ=kB T 1 þ expðDl=kB T Þ exp½DEbond GL DHÞ=kB T ð3Þ where GL is the temperature gradient ahead of the s/l interface, DH is the distance of sticking location (x, z) surpassing the average interface lav (show in Fig. 1), and the change of bond energy DEbond for the stick procedure is
Ps ¼
1 2 1 2 DEbond ¼ ðNAA þ f2 NAA Þ/AA þ ðNAB þ f2 NAB Þ/AB ,
ð4aÞ
for A species
1 2 1 2 DEbond ¼ ðNBA þ f2 NBA Þ/BA þ ðNBB þ f2 NBB Þ/BB ,
ð4bÞ
for B species
here / is the bonding energy between two species, and N is the coordination number of the species; the subscripts A and B denote the species A and B, the superscripts 1 and 2 mean the first and second nearest neighboring of two species; f2 is the ratio of the binding energy for the second neighboring species to the first neighboring species. And the interface diffusion possibility is Pi!j ¼ expðDEi!j =kB T Þ=
m X
expðDEi!j =kB T Þ
j¼1
ð5Þ where DEi ! j is the total energy difference before and after the assumed transferring of species from site i to site j, m is the number of the unoccupied nearestneighbors of site i, including site i itself. The local concentration ahead of the s/l interface is dependent on the number of times NvA(x, z) or NvB (x,z) that the spot (x,z) is visited by species A or B: NvA ðx,zÞ ð6Þ CA ðx,zÞ ¼ NvA ðx,zÞ þ NvB ðx,zÞ
Fig. 1. The domain of the MC simulation model.
The parameters for the present simulation are shown in Table 1, where /BB is much greater than /AA, as /Si – Si is bigger than /Al – Al in Al– Si alloys.
W.M. Wang et al. / Materials Letters 56 (2002) 921–926
923
Table 1 Parameters for the present MC simulation l0
/AA
/BB
/AB
GL
f2
c0
lav
Dl
120 – 200b
3 kBT
12 kBT
12 kBT
5 kB T/b
0.15
0.7
50b
5.3 – 2.1 kBT
2.2. Experimental procedure The samples of Al– Si alloys for the experiments were prepared with electrical heating singlecrystal silicon (purity 99.99%) and high purity aluminium (purity 99.995%) in SiC crucible. The
samples we investigated are concentrated on 12.5 wt.% Si. All the samples were pre-treated in a vacuum chamber for several cycles of heating to high temperature and cooling down, in order to eliminate the hydrogen, which normally contaminates the samples.
Fig. 2. Simulated patterns with various chemical potential difference Dl (a: in ‘‘ ’’; h: in ‘‘ + ’’).
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W.M. Wang et al. / Materials Letters 56 (2002) 921–926
3. Results and discussion The simulated patterns with various chemical difference Dl are shown in Fig. 2. When Dl is large enough (Dl = 5.3 kBT), the simulated eutectic morphology appears in dendritic shape, which occurs occasionally, not on every h phase. According to Eq. (2), the larger Dl implies the larger DT (supercooling degree), hence, we get that h phase changes from flake-like to dendritic at larger DT. The average lamellar spacing examined from the simulated patterns is shown in Fig. 3. The average lamellar spacing is approximately linear with the Dl 1, namely k~Dl 1. When the total lattice size increases from 120b to 200b, this relation does not change. In the early 1960s, Hunt and Chilton [11] got the relation between DT and growth rate R as 1
DT ¼ KR 2
ð7Þ
where K is constant. Combining Eqs. (2) and (7) and Fig. 3, we can get 1
k~R 2
ð8Þ
This obeys well the well-known argument of JH theory [3]. The deviation of the lamellar spacing
Fig. 4. Distribution of the concentrations for A species CA(x, z) ahead of the s/l interface with z = F(x) + 1 (a: Dl = 5.3kBT; b: Dl = 4.5kBT; c: Dl = 3.7kBT; d: Dl = 2.9kBT; e: Dl = 2.1kBT).
from the average line (dot line in Fig. 3) originates mainly from the limitation of the size of the simulated lattice. The concentrations of A species CA(x, z) ahead of the s/l interface at z = F(x) + 1 are shown in Fig. 4, here z = F(x) is the s/l interface curve. It shows that the concentration fluctuation is related to the location of the a phase and h phase. The amplitude with larger Dl is greater than that with smaller Dl. For the steady growth condition, the solute conservation at the curved interface (z = F(x)) is given by [2,6,12] DC Rn ¼ ðCE Cas Þ, for phase ð9aÞ Dn z¼FðxÞ D Fig. 3. Variation of the average lamellar spacing of the simulated patterns with Dl (5: l0 = 200, .: l0 = 120).
DC Dn
¼ z¼FðxÞ
Rn ðCE Chs Þ, for phase D
ð9bÞ
W.M. Wang et al. / Materials Letters 56 (2002) 921–926
where n is the direction normal to the s/l interface, and Rn is the growth rate in n direction, CE is the eutectic concentration, Cs is the maximum concentration for one monophase, and D is the diffusion coefficient. From Eqs. (9a) and (9b), Rn increases with increasing ADC/DnA, which accords well with
925
the simulated concentration distribution ahead of the s/l interface (shown in Fig. 4). However, at relatively larger DT (Dl = 5.3 kBT; Fig. 4), we find the perturbation of concentration distribution occurs where the dendritic h phase is located, which is obviously deviating from the predicted concentration. Consequently, the formation of dendritic lamella (h phase) is associated with the concentration perturbation. The microstructures of Al– 12.5%Si alloy at different cooling rates are shown in Fig. 5. At the lower cooling rate (15 jC/min), the eutectic Si phase is in a commonly found flake-like shape (see Fig. 5(a)); at higher cooling rate (30 jC/min), there exists several high dendritic Si phase between the flake-like eutectic lamella (see Fig. 5(b)). Examining the distribution of Si flakes in both upper cases, the average lamellar spacing at lower cooling rate is a little larger than that at higher cooling rate. According to Eq. (7), the Si phases have smaller spacing and tend to be dendritic in shape at higher DT, which accords with the upper simulated results. Finally, we must point out again that the range of DT in our simulation is limited. For Fe – C alloy, if the supercooling degree is large enough, the flake-like structure of the graphite may change to a degenerate flake-like structure (in a random worm shape) [13]. On the other hand, the eutectic sample prepared by directional growth is mostly in three dimensions, and the microstructure in the transverse section is often different from the longitudinal section. The web-like Si phase in the transverse section in Ref. [14], which is very interesting, is rather difficult to simulate in our present model.
4. Conclusions
Fig. 5. Experimental microstrucutures of Al – 12.5%Si alloy with various cooling rates (a: 15 jC/min, b: 30 jC/min).
We present the dendritic lamella in eutectic growth by MC simulation method. It shows that a dendritic lamella (h phase) forms at relatively larger DT, which accords with the formation of dendritic Si phase in eutectic Al –Si alloy by the experimental method. The concentration gradient ahead of the s/l interface with relatively larger DT is greater than the lower DT case. The formation of dendritic lamella is associated with the perturbation of the concentration field ahead of the s/l interface.
926
W.M. Wang et al. / Materials Letters 56 (2002) 921–926
Acknowledgements The authors want to acknowledge the financial support of key and normal projects of NSFC and the post-doctor project of Laboratory of Solid State Microstructures.
References [1] R. Elliott, Eutectic Solidification Processing, Butterworths, London, 1983. [2] J. Liu, R. Elliott, Acta Metall. Mater. 43 (1995) 3301 – 3311. [3] K.A. Jackson, J.D. Hunt, Trans. Metall. Soc. AIME 236 (1966) 1129.
[4] W. Zhang, H. Fu, Y. Yang, Z. Hu, J. Cryst. Growth 194 (1998) 263 – 271. [5] D. Ma, W.Q. Jie, Y. Li, S.C. Ng, Acta Mater. 46 (1998) 3203 – 3210. [6] A. Karma, Phys. Rev. Lett. 59 (1987) 71 – 74. [7] R.-F. Xiao, J.I.D. Alexander, F. Rosenberger, Mater. Sci. Eng., A 178 (1994) 23 – 238. [8] J.M. Liu, Z.G. Liu, Mater. Lett. 36 (1998) 17 – 23. [9] J.M. Liu, Mater. Lett. 34 (1998) 60 – 66. [10] R.-F. Xiao, J.I.D. Alexander, F. Rosenberger, Phys. Rev., A 45 (1992) R571 – R575. [11] J.D. Hunt, J.R. Chilton, J. Inst. Met. 92 (1963) 21. [12] K. Kassner, C. Misbah, Phys. Rev., A 44 (1991) 6533 – 6543. [13] H. Jones, W. Kurz, Z. Metallkd. 72 (1981) 792 – 797. [14] P. Magnin, J.T. Mason, R. Trivedi, Acta Metall. Mater. 39 (1991) 469 – 480.
Materials Letters 56 (2002) 921 – 926 www.elsevier.com/locate/matlet
Formation of dendritic lamella in eutectic alloys W.M. Wang a,b,*, J.-M. Liu a, J.F. Webb a, X.F. Bian b, G.L. Yuan a, Z.G. Liu a b
a Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China Key Lab of Liquid Structure and Heredity of Materials, Shandong University, Jinan 250061, China
Received 23 February 2002; accepted 4 March 2002
Abstract The dendritic lamella in eutectic growth is simulated with a Monte Carlo (MC) model. It shows that the dendritic lamella (h phase) forms at a relatively large supercooling degree DT, with a higher concentration gradient ahead of the solid/liquid (s/l) interface, and that the formation of dendritic lamella is associated with the concentration perturbation. The microstructures of eutectic Al – Si accord with the simulated results. D 2002 Elsevier Science B.V. All rights reserved. PACS: 61.72-y; 61.42.k Keywords: Eutectic; MC simulation
1. Introduction Eutectic alloys are the basis of many casting alloys. This has led to extensive theoretical and experimental studies on the relationship between microstructure and solidification conditions [1,2]. One of the most significant theoretical studies is Jackson and Hunt’s (JH) [3] analysis of the regular lamellar eutectic structure. Besides a lot of experimental studies, computational studies including numerical calculation and kinetic simulation on eutectic growth are increasing. Numerical calculation is centered on the solution of composition field ahead of the growing interface [1,3 – 5], which mainly deals with only one set of lamellae (a + h) without giving a concrete pattern of eutectic growth. In many practical cases, the eutectic lamella shape is associated not only with the concentration
field directly ahead of the solid/liquid (s/l) interface, but also with the shape of the neighboring lamellae. Kinetic simulations like the Monte Carlo (MC) method are applied to simulate the motion of atoms during eutectic growth [6,7], by such a method we can get the eutectic pattern. However, to our knowledge, there is no MC simulation work on the formation of dendritic lamella in the eutectic growth of binary alloys. In this paper, we apply a MC method to simulate the formation of dendritic lamella, and analyze the concentration field ahead of the s/l interface; we also present the experimental results for the dendritic lamella (Si phase) in Al –Si alloy at different cooling rates.
2. Simulation and experimental procedure 2.1. Model and simulation procedure
*
Corresponding author. Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China. E-mail address: [email protected] (W.M. Wang).
Like the previous simulation work [8,9], the simulation was performed using a rectangular l0 l0
0167-577X/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 5 7 7 X ( 0 2 ) 0 0 6 3 8 - 9
922
W.M. Wang et al. / Materials Letters 56 (2002) 921–926
lattice with a periodic boundary condition parallel to the s/l interface, where the lattice parameter is b (shown in Fig. 1). A thermodynamic framework of the present approach is quite similar to the work of Xiao et al. [7]. The eutectic growth procedure includes two reversible events (shown in Fig. 1): (1) species A (in the given concentration c0) and B emit from the source line and random walk, then stick on the s/l interface; (2) the species depart from the s/l interface and diffuse on the s/l interface, then escape from the regime after possible random walk. The possibility for the system to choose whether departure event or emission event is Dl Pe ¼ 1= 1 þ exp ð1Þ kB T where kB is the Boltzmann constant and T is the temperature, and Dl is a chemical potential difference between the solid and liquid with a unit of kBT. If the interfacial concentration difference between the melt and solid concentration is omitted, we have [10] Dl ¼
Lp DT , where DT ¼ T Te Te
ð2Þ
where Lp and Te are the latent heat and temperature for the equilibrium melt – solid transformation, respectively.
The possibility of sticking on the s/l interface for the species reached there is expðDl=kB T Þ exp½DEbond GL DHÞ=kB T 1 þ expðDl=kB T Þ exp½DEbond GL DHÞ=kB T ð3Þ where GL is the temperature gradient ahead of the s/l interface, DH is the distance of sticking location (x, z) surpassing the average interface lav (show in Fig. 1), and the change of bond energy DEbond for the stick procedure is
Ps ¼
1 2 1 2 DEbond ¼ ðNAA þ f2 NAA Þ/AA þ ðNAB þ f2 NAB Þ/AB ,
ð4aÞ
for A species
1 2 1 2 DEbond ¼ ðNBA þ f2 NBA Þ/BA þ ðNBB þ f2 NBB Þ/BB ,
ð4bÞ
for B species
here / is the bonding energy between two species, and N is the coordination number of the species; the subscripts A and B denote the species A and B, the superscripts 1 and 2 mean the first and second nearest neighboring of two species; f2 is the ratio of the binding energy for the second neighboring species to the first neighboring species. And the interface diffusion possibility is Pi!j ¼ expðDEi!j =kB T Þ=
m X
expðDEi!j =kB T Þ
j¼1
ð5Þ where DEi ! j is the total energy difference before and after the assumed transferring of species from site i to site j, m is the number of the unoccupied nearestneighbors of site i, including site i itself. The local concentration ahead of the s/l interface is dependent on the number of times NvA(x, z) or NvB (x,z) that the spot (x,z) is visited by species A or B: NvA ðx,zÞ ð6Þ CA ðx,zÞ ¼ NvA ðx,zÞ þ NvB ðx,zÞ
Fig. 1. The domain of the MC simulation model.
The parameters for the present simulation are shown in Table 1, where /BB is much greater than /AA, as /Si – Si is bigger than /Al – Al in Al– Si alloys.
W.M. Wang et al. / Materials Letters 56 (2002) 921–926
923
Table 1 Parameters for the present MC simulation l0
/AA
/BB
/AB
GL
f2
c0
lav
Dl
120 – 200b
3 kBT
12 kBT
12 kBT
5 kB T/b
0.15
0.7
50b
5.3 – 2.1 kBT
2.2. Experimental procedure The samples of Al– Si alloys for the experiments were prepared with electrical heating singlecrystal silicon (purity 99.99%) and high purity aluminium (purity 99.995%) in SiC crucible. The
samples we investigated are concentrated on 12.5 wt.% Si. All the samples were pre-treated in a vacuum chamber for several cycles of heating to high temperature and cooling down, in order to eliminate the hydrogen, which normally contaminates the samples.
Fig. 2. Simulated patterns with various chemical potential difference Dl (a: in ‘‘ ’’; h: in ‘‘ + ’’).
924
W.M. Wang et al. / Materials Letters 56 (2002) 921–926
3. Results and discussion The simulated patterns with various chemical difference Dl are shown in Fig. 2. When Dl is large enough (Dl = 5.3 kBT), the simulated eutectic morphology appears in dendritic shape, which occurs occasionally, not on every h phase. According to Eq. (2), the larger Dl implies the larger DT (supercooling degree), hence, we get that h phase changes from flake-like to dendritic at larger DT. The average lamellar spacing examined from the simulated patterns is shown in Fig. 3. The average lamellar spacing is approximately linear with the Dl 1, namely k~Dl 1. When the total lattice size increases from 120b to 200b, this relation does not change. In the early 1960s, Hunt and Chilton [11] got the relation between DT and growth rate R as 1
DT ¼ KR 2
ð7Þ
where K is constant. Combining Eqs. (2) and (7) and Fig. 3, we can get 1
k~R 2
ð8Þ
This obeys well the well-known argument of JH theory [3]. The deviation of the lamellar spacing
Fig. 4. Distribution of the concentrations for A species CA(x, z) ahead of the s/l interface with z = F(x) + 1 (a: Dl = 5.3kBT; b: Dl = 4.5kBT; c: Dl = 3.7kBT; d: Dl = 2.9kBT; e: Dl = 2.1kBT).
from the average line (dot line in Fig. 3) originates mainly from the limitation of the size of the simulated lattice. The concentrations of A species CA(x, z) ahead of the s/l interface at z = F(x) + 1 are shown in Fig. 4, here z = F(x) is the s/l interface curve. It shows that the concentration fluctuation is related to the location of the a phase and h phase. The amplitude with larger Dl is greater than that with smaller Dl. For the steady growth condition, the solute conservation at the curved interface (z = F(x)) is given by [2,6,12] DC Rn ¼ ðCE Cas Þ, for phase ð9aÞ Dn z¼FðxÞ D Fig. 3. Variation of the average lamellar spacing of the simulated patterns with Dl (5: l0 = 200, .: l0 = 120).
DC Dn
¼ z¼FðxÞ
Rn ðCE Chs Þ, for phase D
ð9bÞ
W.M. Wang et al. / Materials Letters 56 (2002) 921–926
where n is the direction normal to the s/l interface, and Rn is the growth rate in n direction, CE is the eutectic concentration, Cs is the maximum concentration for one monophase, and D is the diffusion coefficient. From Eqs. (9a) and (9b), Rn increases with increasing ADC/DnA, which accords well with
925
the simulated concentration distribution ahead of the s/l interface (shown in Fig. 4). However, at relatively larger DT (Dl = 5.3 kBT; Fig. 4), we find the perturbation of concentration distribution occurs where the dendritic h phase is located, which is obviously deviating from the predicted concentration. Consequently, the formation of dendritic lamella (h phase) is associated with the concentration perturbation. The microstructures of Al– 12.5%Si alloy at different cooling rates are shown in Fig. 5. At the lower cooling rate (15 jC/min), the eutectic Si phase is in a commonly found flake-like shape (see Fig. 5(a)); at higher cooling rate (30 jC/min), there exists several high dendritic Si phase between the flake-like eutectic lamella (see Fig. 5(b)). Examining the distribution of Si flakes in both upper cases, the average lamellar spacing at lower cooling rate is a little larger than that at higher cooling rate. According to Eq. (7), the Si phases have smaller spacing and tend to be dendritic in shape at higher DT, which accords with the upper simulated results. Finally, we must point out again that the range of DT in our simulation is limited. For Fe – C alloy, if the supercooling degree is large enough, the flake-like structure of the graphite may change to a degenerate flake-like structure (in a random worm shape) [13]. On the other hand, the eutectic sample prepared by directional growth is mostly in three dimensions, and the microstructure in the transverse section is often different from the longitudinal section. The web-like Si phase in the transverse section in Ref. [14], which is very interesting, is rather difficult to simulate in our present model.
4. Conclusions
Fig. 5. Experimental microstrucutures of Al – 12.5%Si alloy with various cooling rates (a: 15 jC/min, b: 30 jC/min).
We present the dendritic lamella in eutectic growth by MC simulation method. It shows that a dendritic lamella (h phase) forms at relatively larger DT, which accords with the formation of dendritic Si phase in eutectic Al –Si alloy by the experimental method. The concentration gradient ahead of the s/l interface with relatively larger DT is greater than the lower DT case. The formation of dendritic lamella is associated with the perturbation of the concentration field ahead of the s/l interface.
926
W.M. Wang et al. / Materials Letters 56 (2002) 921–926
Acknowledgements The authors want to acknowledge the financial support of key and normal projects of NSFC and the post-doctor project of Laboratory of Solid State Microstructures.
References [1] R. Elliott, Eutectic Solidification Processing, Butterworths, London, 1983. [2] J. Liu, R. Elliott, Acta Metall. Mater. 43 (1995) 3301 – 3311. [3] K.A. Jackson, J.D. Hunt, Trans. Metall. Soc. AIME 236 (1966) 1129.
[4] W. Zhang, H. Fu, Y. Yang, Z. Hu, J. Cryst. Growth 194 (1998) 263 – 271. [5] D. Ma, W.Q. Jie, Y. Li, S.C. Ng, Acta Mater. 46 (1998) 3203 – 3210. [6] A. Karma, Phys. Rev. Lett. 59 (1987) 71 – 74. [7] R.-F. Xiao, J.I.D. Alexander, F. Rosenberger, Mater. Sci. Eng., A 178 (1994) 23 – 238. [8] J.M. Liu, Z.G. Liu, Mater. Lett. 36 (1998) 17 – 23. [9] J.M. Liu, Mater. Lett. 34 (1998) 60 – 66. [10] R.-F. Xiao, J.I.D. Alexander, F. Rosenberger, Phys. Rev., A 45 (1992) R571 – R575. [11] J.D. Hunt, J.R. Chilton, J. Inst. Met. 92 (1963) 21. [12] K. Kassner, C. Misbah, Phys. Rev., A 44 (1991) 6533 – 6543. [13] H. Jones, W. Kurz, Z. Metallkd. 72 (1981) 792 – 797. [14] P. Magnin, J.T. Mason, R. Trivedi, Acta Metall. Mater. 39 (1991) 469 – 480.