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Grain growth simulation starting from experimental data
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Pergamon
Materialia. Vol. 36., No. 7.,.. DD.789-794. 1997 Elsevier Science Ltd Copyright 0 1997 Acta Metallurgica Inc. Printed in the USA. All rights reserved 1359~6462/97 $17.00 + .OO
PII 81359-6462(96)00451-4
GRAIN GROWTH SIMULATION STARTING FROM EXPERIMENTAL DATA T. Baudin’, P. Paillard
and R. Penelle’
‘Universite de Paris Sud, Laboratoire de Metallurgic Structurale, URA CNRS 1107, Batiment 4 13,9 1405 Orsay Cedex, France ‘Universite de Lille 1, Laboratoire de Metalhtrgie Physique, URA CNRS 234, Batiment C6,2Bme &age, 59655 Villeneuve d’Ascq, France (Received October 2, 1996) (Accepted October 9, 1996) Introduction
For many years, normal or abnormal grain growth phenomena have been modelled using several ways. The most common techniques (Monte Carlo, Vertex, ...) are described in (1); the Cellular Automata method is discussed in a recent paper by Lepinoux (2). These methods give statistical results in agreement with the experimental measurements (comparison of the grain growth exponent, average number of neighbours, ...). One should note however that the initial microstructure is always an arbitrary one and that the comparisons are always performed at the global level and in an average sense. In order to validate these numerical models at the local level of the grains and the grain boundaries, the microstructure must be precisely represented. First, experimental measurements of the initial microstntcture have to be transformed into sets of data to define the starting grain parameters (size, shape, relative positions) for the simulations. These initial experimental dalta can be characterized by Orientation Imaging Microscopy (OIM) (3): the orientations are automatically measured by Electron Back Scattered Diffraction (EBSD) (4) on each point of a grid and the microstructure is then reconstructed from the orientation measurements. The main objective of this work consists of comparing grain by grain and at any time step the experimental and simulated results and adjusting numerical parameters such as the energy and the mobility of grain boundaries (5). However, and before performing these comparisons, it is necessary to show the feasibility of this new approach. As a simulation tool, the Monte Carlo simulation is chosen. Material and Experimental
Techniques
The OIM is performed on a Fe 3% Si grade HiB samples at the primary recrystallized state (sample 1) and after a 970°C - Smin annealing (sample 2). The normal growth is inhibited by the presence of AlN and MnS precipitates (6) and only abnormal growth of near-Goss grains occurs. Two areas (450 x 450 pm*, Ax = Ay =: 2.0 pm) of sample 1 were characterized at 60.000 points of a hexagonal grid. One area of sample 2 was also measured at 45.000 points (500 x 500 pm*, Ax = Ay = 2.5 pm).
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Numerical Treatment of Experimental Data The introduction of an initial experimental microstructure into a simulation involves the development of new numerical treatments and of introducing minor modifications in the simulation procedures. In a Monte Carlo simulation, each site is characterized by its crystallographic orientation. This is not always possible for example when OIM is used : the electron beam may strike a grain boundary and a fast correction consists in affecting the undefined points to adjacent grains. Moreover, because of the sample preparation, of the precision of the measurements .... the orientations measured inside a same grain can have close although different values. If the grain boundaries (GB) are estimated from a misorientation calculation between adjacent points, the minimum misorientation angle ( A02,) must be stated : if A0:” is high, the number of grains tends to one and if A02, tends to zero each site becomes a grain. The digitalized microstructure obtained by OIM is visualized using a gray level code defined from the quality of the Kikuchi patterns (7) : white for a clear pattern and black for a blurred one obtained for example when the electron beam is diffracted by “a grain boundary” (Figure la). In Figure lb, only the grain boundaries are drawn adjusting A(!):,, (here Aez,, = 3’) to have a good representation of grains when Figures la and lb are superimposed (cf. Figure lc). The measurements are carried out on a hexagonal grid that defines a square or a rectangular area and not a diamond-shaped area as in the simulation (the softwares used are those described in (8)). Simple modifications must be done to take this shape change into account. Moreover, because of the lack of data on grain boundary energy and mobility, simulations are performed assuming crystallographic orientation classes (5,8). A calculation of misorientation must then be performed to affect each orientation to an orientation class. The classes and the energies and mobilities between them are chosen as in (8).
Figure 1. Experimentalmicrostructure(a) gray level code (b) grainboundarydefinition (c) superimpositionof (a) and (b).
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Figure 2. Evolution of microstructure (450 x 450 pm’) during normal grain growth (0,250, 500 MCS).
Results Even if the normal grain growth does not occur in the reality in the studied material, a simulation can still be performed. Figure 2 shows the microstructure evolution for several Monte Carlo steps (MCS). For the abnormal grain growth simulation, one first needs to define the misorientation range to accepting each orientation (0) into an orientation class. Here, this angle is adjusted to identify nearGoss grains in the experimental microstructure (area 1) ( Ae$, = 6”). For the { 11 1}<112> and { 100)<012> orientation classes, AO%i, is arbitrarily taken equal to 15 ’ and the other orientations are assumed to belong to the random class. In these conditions, the simulation starting with experimentally measured microstructure predicts only two small near-Goss grains that grow on Figure 3. Simulations are typically performed using a global description of the initial microstructure and the orientations of the individual grains are chosen according to a homogeneous distribution. However, this hypothesis iisnot always satisfied in the real material as shown on Figure 4a (area 2) where clusters of grains appear when Aezn is increased to 15”. Another kind of representation consists in chasing a grain inside the cluster and finding the adjacent grains with a misorientation angle (de’)
lower than
Figure 3. Evolution of microstructure (area 1) (450 x 450 pm’) during abnormal grain growth (0, 50, 100, .... 250 MCS) - (The black grains are the near-Goss grains).
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Figure 4. Experimental microstructure (area 2) (450 x 450 pm2) (a)
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Aez” = IY, (b) At3:” = 3”. (c) {11l}, (200) and (220)
pole figures of the black near-Goss grain cluster defined on Figure 4b.
15’ (Figure 4b : for At32” equal to 3”). In fact, different kinds of orientation clusters appear but here a large near-Goss grain cluster is found. Figure 4c shows the pole figures that correspond to the cluster defined on Figure 4b. The initial near-Goss grain percentage before secondary recrystallization (9) (and the percentage of the other texture components) has a large effect on the predicted abnormal grain growth simulation. This can be observed by comparing Figures 3 and 5 which were obtained for identical simulation conditions. In the simulation conditions described in (S), when the initial area percentage of near-Goss grains is low (0.04 % for area l), only two near-Goss grains remain at the end of the simulation; when the initial percentage is higher as in area 2 (0.65%) several grains remain. Figure 6 allows the authors to compare the microstructures of areas 1 and 2 after 250 MCS since the grain boundaries are erased inside the black domains of near-Goss grains (cf. Figures 3 and 5). Figure 7 shows the evolution of the area percentage of Goss grains as a function of the MCS number for the two studied microstructures (cf. Figures 3 and 5). This figure shows the large influence of texture inhomogeneity since after 250 MCS, the area percentage is equal to about 10% for area 1 and to about 30% for area 2.
Figure 5. Evolution of microstructure (area2) (450 x 450 pm’) during abnormal grain growth (0, 50, 100, .... 250 MCS) - (The black grains are the near-Goss grains).
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Figure 6. Comparison of microstructures obtained after 250 MCS (a) area 1 and (b) area 2.
Similar differences have been found between the abnormal grain growth of near-Goss grains of this material, and that of near-cube grains in Fe 3% Si ribbons cast directly (10). Indeed, the initial surface fraction of these near-Goss and near-cube grains at the primary recrystallized state are respectively equal to about 0.1% and 1%. The presence of near-Goss grain cluster tends to modify the simulation results since the probability to have near-Goss grains (defmed with A0:, = 6”) is higher. The effects of the textural inhomogeneities on the abnormal growth of near-Goss grains should be studied experimentally. Up to now, the simulation has been performed from experimental microstructures of a sample at the primary recrystallized state. However, it is not easy to compare experimental and simulated results, especially from a local point of view. Indeed, to have a good resolution for drawing the microstructure, the step size must be quite low and the total area remains small if the number of orientations is less than about 60.000 (this value is reasonable for the measurement time and the file size). It is then difftcult to fmd a Goss grain that will grow in such area because the volume traction of Goss grains is very low (6). So, it becomes easier to compare experimental and simulated results obtained when the Goss grain is already large knowing that the growth mechanisms are probably different at the beginning and during the abnormal growth of Goss grains. This kind of experimental study where the grain boundary migration is followed during the annealing time has been already performed by Harase et al. (11). Figure 8 shows an example of the microstructure evolution during the simulation from an experimental microstructure measured by OIM (sample 2). This allows the authors to follow the grain boundary migration and to compare how it evolves experimentally. Conclusion
A new approach is introduced to study grain growth : the measured experimental microstructure is used as a description of a starting microstructure for a simulation. However, the lack of data on grain
Figure 7. Evolution of the area percentage of near-Goss grains as a function of the MCS number.
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36,No. 7
Figure 8. Evolution of microstructure (500 x 500 pm*) during abnormal grain growth (0,10,20 MCS).
boundary energies and mobilities data makes such studies difficult, currently only orientation classes with corresponding adjusted parameters can be used. It becomes thus necessary to introduce an iterative procedure to adjust these parameters by comparing numerical and experimental results. In parallel, numerical and experimental results for growth of bicrystals or tricrystals could also be used to determine the grain boundary energy and mobility. But, if a space scale is introduced with the experimental microstructure, it is also necessary to have a time scale that does not exist in a classical Monte Carlo simulation (12, 13) and that could be adjusted simultaneously with the other parameters defined above. In this study the Monte Carlo method is used, but any other simulation technique could be used. If for instance the Vertex method is prefered, the vertices of the experimental microstructure that are the input data of the simulation can easily be located (14). Acknowledgments The authors thank A. Amiri who prepared the sample at the secondary recrystallization state for the OIM analysis that has permitted them to complete this work performed in the frame of the GDR 1165 “Modelisations et simulations mdsoscopiques en m&allurgie”. They thank also Y. Chaste1 and L.P. Kubin for a critical reading of the manuscript. References 1. Proc. of Grain Growth in Polyctystalline Materials II, Japan, 17-20 May 1995, ed. H. Yoshinaga et al., Materials Science Forum, 204-205. Transtec Publications (1996). 2. J. Lepinoux, Proc. of the NATO Advanced Study Institute on Computer Simulation in Materials Science, Ile dOleron, France, June 1995, ed. H.O.K. Kirchner et al., 247-258.247, Kluwer Academic Publishers (1996). 3. B.L. Adams, S.I. Wright and K. Kunze, Met. Trans. A, 24& 819 (1993). 4. D. Diiley, Proc. of the 8th ICOTOM, Santa Fe, Sept 1987, ed. J.S. Kallend and G. Gottstein, 189, TMS, Warrendale, PA (1988). 5. G. Abbruzzese, A. Campopiano and S. Fortunati, Proc. of the 9th ICOTOM, Avignon, France, Sept 1990, ed. H.J. Bunge, Textures and Microstructures, &I&, 775 (1991). 6. N. Rouag and R. Penelle, Textures and Microstructures, J_L 203 (1989). I. S.I. Wright, I. of Computer Assisted Microscopy, 5, N03, 207 (1993). 8. P. Paillard, T. Baudin and R. Penelle, Proc. of the 10th ICOTOM, Clausthal, Sept 1993, ed. H.J. Bunge, Materials Science Forum, 157_I62. 1027, Trans Tech Publications (1994). 9. T. Baudii, J. Jura, R. Penelle and J. Pospiech, J. Appl. Cryst., a 582 (1995). 10. P. Paillard, These, Universite de Paris Sud Orsay (1994). 11. J. Harase, R. Shimizu and T. Watanabe, Proc. of the 7th Riso International Symposium on Metallurgy and Materials Science, Roskilde, Denmark, Sept 1986, ed. N. Hansen et al., 343 (1986). 12. Y. Breehet and L. Kubin, Proc. of the NATO Advanced Study Institute on Computer Simulation in Materials Science, Ile d’Oleron, France., June 1995, ed. H.O.K. Kirchner et al., 247-258.3, Kluwer Academic Publishers (1996). 13. R. Penelle, P. Paillard, Y. Liu and T. Baudin, Proc. of the 1lth ICOTOM, Xi’an, China (in press). 14. J.C. Louin, DEA de Physique et Genie des Materiaux, ENSMPAJNSA (19%).
Pergamon
Materialia. Vol. 36., No. 7.,.. DD.789-794. 1997 Elsevier Science Ltd Copyright 0 1997 Acta Metallurgica Inc. Printed in the USA. All rights reserved 1359~6462/97 $17.00 + .OO
PII 81359-6462(96)00451-4
GRAIN GROWTH SIMULATION STARTING FROM EXPERIMENTAL DATA T. Baudin’, P. Paillard
and R. Penelle’
‘Universite de Paris Sud, Laboratoire de Metallurgic Structurale, URA CNRS 1107, Batiment 4 13,9 1405 Orsay Cedex, France ‘Universite de Lille 1, Laboratoire de Metalhtrgie Physique, URA CNRS 234, Batiment C6,2Bme &age, 59655 Villeneuve d’Ascq, France (Received October 2, 1996) (Accepted October 9, 1996) Introduction
For many years, normal or abnormal grain growth phenomena have been modelled using several ways. The most common techniques (Monte Carlo, Vertex, ...) are described in (1); the Cellular Automata method is discussed in a recent paper by Lepinoux (2). These methods give statistical results in agreement with the experimental measurements (comparison of the grain growth exponent, average number of neighbours, ...). One should note however that the initial microstructure is always an arbitrary one and that the comparisons are always performed at the global level and in an average sense. In order to validate these numerical models at the local level of the grains and the grain boundaries, the microstructure must be precisely represented. First, experimental measurements of the initial microstntcture have to be transformed into sets of data to define the starting grain parameters (size, shape, relative positions) for the simulations. These initial experimental dalta can be characterized by Orientation Imaging Microscopy (OIM) (3): the orientations are automatically measured by Electron Back Scattered Diffraction (EBSD) (4) on each point of a grid and the microstructure is then reconstructed from the orientation measurements. The main objective of this work consists of comparing grain by grain and at any time step the experimental and simulated results and adjusting numerical parameters such as the energy and the mobility of grain boundaries (5). However, and before performing these comparisons, it is necessary to show the feasibility of this new approach. As a simulation tool, the Monte Carlo simulation is chosen. Material and Experimental
Techniques
The OIM is performed on a Fe 3% Si grade HiB samples at the primary recrystallized state (sample 1) and after a 970°C - Smin annealing (sample 2). The normal growth is inhibited by the presence of AlN and MnS precipitates (6) and only abnormal growth of near-Goss grains occurs. Two areas (450 x 450 pm*, Ax = Ay =: 2.0 pm) of sample 1 were characterized at 60.000 points of a hexagonal grid. One area of sample 2 was also measured at 45.000 points (500 x 500 pm*, Ax = Ay = 2.5 pm).
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Vol. 36. No. 7
Numerical Treatment of Experimental Data The introduction of an initial experimental microstructure into a simulation involves the development of new numerical treatments and of introducing minor modifications in the simulation procedures. In a Monte Carlo simulation, each site is characterized by its crystallographic orientation. This is not always possible for example when OIM is used : the electron beam may strike a grain boundary and a fast correction consists in affecting the undefined points to adjacent grains. Moreover, because of the sample preparation, of the precision of the measurements .... the orientations measured inside a same grain can have close although different values. If the grain boundaries (GB) are estimated from a misorientation calculation between adjacent points, the minimum misorientation angle ( A02,) must be stated : if A0:” is high, the number of grains tends to one and if A02, tends to zero each site becomes a grain. The digitalized microstructure obtained by OIM is visualized using a gray level code defined from the quality of the Kikuchi patterns (7) : white for a clear pattern and black for a blurred one obtained for example when the electron beam is diffracted by “a grain boundary” (Figure la). In Figure lb, only the grain boundaries are drawn adjusting A(!):,, (here Aez,, = 3’) to have a good representation of grains when Figures la and lb are superimposed (cf. Figure lc). The measurements are carried out on a hexagonal grid that defines a square or a rectangular area and not a diamond-shaped area as in the simulation (the softwares used are those described in (8)). Simple modifications must be done to take this shape change into account. Moreover, because of the lack of data on grain boundary energy and mobility, simulations are performed assuming crystallographic orientation classes (5,8). A calculation of misorientation must then be performed to affect each orientation to an orientation class. The classes and the energies and mobilities between them are chosen as in (8).
Figure 1. Experimentalmicrostructure(a) gray level code (b) grainboundarydefinition (c) superimpositionof (a) and (b).
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Figure 2. Evolution of microstructure (450 x 450 pm’) during normal grain growth (0,250, 500 MCS).
Results Even if the normal grain growth does not occur in the reality in the studied material, a simulation can still be performed. Figure 2 shows the microstructure evolution for several Monte Carlo steps (MCS). For the abnormal grain growth simulation, one first needs to define the misorientation range to accepting each orientation (0) into an orientation class. Here, this angle is adjusted to identify nearGoss grains in the experimental microstructure (area 1) ( Ae$, = 6”). For the { 11 1}<112> and { 100)<012> orientation classes, AO%i, is arbitrarily taken equal to 15 ’ and the other orientations are assumed to belong to the random class. In these conditions, the simulation starting with experimentally measured microstructure predicts only two small near-Goss grains that grow on Figure 3. Simulations are typically performed using a global description of the initial microstructure and the orientations of the individual grains are chosen according to a homogeneous distribution. However, this hypothesis iisnot always satisfied in the real material as shown on Figure 4a (area 2) where clusters of grains appear when Aezn is increased to 15”. Another kind of representation consists in chasing a grain inside the cluster and finding the adjacent grains with a misorientation angle (de’)
lower than
Figure 3. Evolution of microstructure (area 1) (450 x 450 pm’) during abnormal grain growth (0, 50, 100, .... 250 MCS) - (The black grains are the near-Goss grains).
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Figure 4. Experimental microstructure (area 2) (450 x 450 pm2) (a)
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Aez” = IY, (b) At3:” = 3”. (c) {11l}, (200) and (220)
pole figures of the black near-Goss grain cluster defined on Figure 4b.
15’ (Figure 4b : for At32” equal to 3”). In fact, different kinds of orientation clusters appear but here a large near-Goss grain cluster is found. Figure 4c shows the pole figures that correspond to the cluster defined on Figure 4b. The initial near-Goss grain percentage before secondary recrystallization (9) (and the percentage of the other texture components) has a large effect on the predicted abnormal grain growth simulation. This can be observed by comparing Figures 3 and 5 which were obtained for identical simulation conditions. In the simulation conditions described in (S), when the initial area percentage of near-Goss grains is low (0.04 % for area l), only two near-Goss grains remain at the end of the simulation; when the initial percentage is higher as in area 2 (0.65%) several grains remain. Figure 6 allows the authors to compare the microstructures of areas 1 and 2 after 250 MCS since the grain boundaries are erased inside the black domains of near-Goss grains (cf. Figures 3 and 5). Figure 7 shows the evolution of the area percentage of Goss grains as a function of the MCS number for the two studied microstructures (cf. Figures 3 and 5). This figure shows the large influence of texture inhomogeneity since after 250 MCS, the area percentage is equal to about 10% for area 1 and to about 30% for area 2.
Figure 5. Evolution of microstructure (area2) (450 x 450 pm’) during abnormal grain growth (0, 50, 100, .... 250 MCS) - (The black grains are the near-Goss grains).
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Figure 6. Comparison of microstructures obtained after 250 MCS (a) area 1 and (b) area 2.
Similar differences have been found between the abnormal grain growth of near-Goss grains of this material, and that of near-cube grains in Fe 3% Si ribbons cast directly (10). Indeed, the initial surface fraction of these near-Goss and near-cube grains at the primary recrystallized state are respectively equal to about 0.1% and 1%. The presence of near-Goss grain cluster tends to modify the simulation results since the probability to have near-Goss grains (defmed with A0:, = 6”) is higher. The effects of the textural inhomogeneities on the abnormal growth of near-Goss grains should be studied experimentally. Up to now, the simulation has been performed from experimental microstructures of a sample at the primary recrystallized state. However, it is not easy to compare experimental and simulated results, especially from a local point of view. Indeed, to have a good resolution for drawing the microstructure, the step size must be quite low and the total area remains small if the number of orientations is less than about 60.000 (this value is reasonable for the measurement time and the file size). It is then difftcult to fmd a Goss grain that will grow in such area because the volume traction of Goss grains is very low (6). So, it becomes easier to compare experimental and simulated results obtained when the Goss grain is already large knowing that the growth mechanisms are probably different at the beginning and during the abnormal growth of Goss grains. This kind of experimental study where the grain boundary migration is followed during the annealing time has been already performed by Harase et al. (11). Figure 8 shows an example of the microstructure evolution during the simulation from an experimental microstructure measured by OIM (sample 2). This allows the authors to follow the grain boundary migration and to compare how it evolves experimentally. Conclusion
A new approach is introduced to study grain growth : the measured experimental microstructure is used as a description of a starting microstructure for a simulation. However, the lack of data on grain
Figure 7. Evolution of the area percentage of near-Goss grains as a function of the MCS number.
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GRAINGROWTHSIMULATION
Vol.
36,No. 7
Figure 8. Evolution of microstructure (500 x 500 pm*) during abnormal grain growth (0,10,20 MCS).
boundary energies and mobilities data makes such studies difficult, currently only orientation classes with corresponding adjusted parameters can be used. It becomes thus necessary to introduce an iterative procedure to adjust these parameters by comparing numerical and experimental results. In parallel, numerical and experimental results for growth of bicrystals or tricrystals could also be used to determine the grain boundary energy and mobility. But, if a space scale is introduced with the experimental microstructure, it is also necessary to have a time scale that does not exist in a classical Monte Carlo simulation (12, 13) and that could be adjusted simultaneously with the other parameters defined above. In this study the Monte Carlo method is used, but any other simulation technique could be used. If for instance the Vertex method is prefered, the vertices of the experimental microstructure that are the input data of the simulation can easily be located (14). Acknowledgments The authors thank A. Amiri who prepared the sample at the secondary recrystallization state for the OIM analysis that has permitted them to complete this work performed in the frame of the GDR 1165 “Modelisations et simulations mdsoscopiques en m&allurgie”. They thank also Y. Chaste1 and L.P. Kubin for a critical reading of the manuscript. References 1. Proc. of Grain Growth in Polyctystalline Materials II, Japan, 17-20 May 1995, ed. H. Yoshinaga et al., Materials Science Forum, 204-205. Transtec Publications (1996). 2. J. Lepinoux, Proc. of the NATO Advanced Study Institute on Computer Simulation in Materials Science, Ile dOleron, France, June 1995, ed. H.O.K. Kirchner et al., 247-258.247, Kluwer Academic Publishers (1996). 3. B.L. Adams, S.I. Wright and K. Kunze, Met. Trans. A, 24& 819 (1993). 4. D. Diiley, Proc. of the 8th ICOTOM, Santa Fe, Sept 1987, ed. J.S. Kallend and G. Gottstein, 189, TMS, Warrendale, PA (1988). 5. G. Abbruzzese, A. Campopiano and S. Fortunati, Proc. of the 9th ICOTOM, Avignon, France, Sept 1990, ed. H.J. Bunge, Textures and Microstructures, &I&, 775 (1991). 6. N. Rouag and R. Penelle, Textures and Microstructures, J_L 203 (1989). I. S.I. Wright, I. of Computer Assisted Microscopy, 5, N03, 207 (1993). 8. P. Paillard, T. Baudin and R. Penelle, Proc. of the 10th ICOTOM, Clausthal, Sept 1993, ed. H.J. Bunge, Materials Science Forum, 157_I62. 1027, Trans Tech Publications (1994). 9. T. Baudii, J. Jura, R. Penelle and J. Pospiech, J. Appl. Cryst., a 582 (1995). 10. P. Paillard, These, Universite de Paris Sud Orsay (1994). 11. J. Harase, R. Shimizu and T. Watanabe, Proc. of the 7th Riso International Symposium on Metallurgy and Materials Science, Roskilde, Denmark, Sept 1986, ed. N. Hansen et al., 343 (1986). 12. Y. Breehet and L. Kubin, Proc. of the NATO Advanced Study Institute on Computer Simulation in Materials Science, Ile d’Oleron, France., June 1995, ed. H.O.K. Kirchner et al., 247-258.3, Kluwer Academic Publishers (1996). 13. R. Penelle, P. Paillard, Y. Liu and T. Baudin, Proc. of the 1lth ICOTOM, Xi’an, China (in press). 14. J.C. Louin, DEA de Physique et Genie des Materiaux, ENSMPAJNSA (19%).