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On the formation of twin grains as a result of grain encounters during the process of recrystallization and grain growth
Scripta METALLURGICA et MATERIALIA
Vol.
25, pp. 1099-1102, 1991 Printed in the U.S.A.
Pergamon Press plc All rights reserved
" O N T H E F O R M A T I O N O F T W Z N G R A I N 8 AS A R E S U L T O F G R A I N E N C O U N T E R S D U R / N G T H B P R O C E 8 8 OF R E C R Y S T A L L I K A T I O N A N D G R A I N G R O W T H H
K.J.Kurzydlowski Institute of Materials science and Engineering Warsaw University of Technology, 0 2 - 5 2 4 Warszawa, Narbutta 85 (Received December 27, 1990) (Revised March 4, 1991)
Twin grains are a common feature of some FCC metals. Twins are characterized by a special orientation of crystal lattices of neighboring grains. Crystal lattices of twins are mirror reflections of one another. In terms of coincidence site lattice theory such an orientation can be described as the orientation Z=3. The process of twin grains formation has been analyzed in a n-mher of models, which can be divided into three general categories: a) grain boundary reactions models (for example [1]) b) grain growth defect models (for example [2]) c) grain encounter models (for example [3]) The last two groups of models are based on the assumption that the twin orientation is a result of an encounter of two grains which come into contact during the recrystallization process, or as a result of g r a i n g r o w t h , and which are in twin orientation. The physical significance of such a model strongly depends on the probability of finding a given grain structure of two grains which do have twin misorientation of their crystal lattices. It is intuitively felt that this probability is relatively low in the case of polycrystals characterized by a random orientation of the grains. This has been substantiated b y t h e calculations of Nielsen [4], who s h o w e d t h a t in such a situation a negligibly small number of grains is expected to have twin orientation. The existence of texture in the orientation of grains forming a polycrystal, no matter how it has been formed, changes the probability of finding two grains in the twin orientation. This effect has been semi-quantitatively studied in [5]. To the best knowledge of the author no systematic studies of the effect of texture on the frequency of occurrence of twin grain have been reported. Garbacz and Grabski [6] recently published results of their studies of the relationship between texture of a polycrystal and misorientation of its grain boundaries. It is the aim of the present paper to show that their results can be used to modify the predictions of Nielsen [4]. Their results allow one to analyze more systematically the effect of texture on the frequency of twin grains in polycrystals. M o ~ e l bY G a r b a c s
and Grabski
r61
The model used b y G a r b a c z and Grabski is based on the assumption that grains have constant volume and each grain in a polycrystal has 14 neighbors. The orientation of the crystal lattice in a given grain is randomly taken from a set of orientations typical of an assumed texture of grain lattice orientations. In the calculations, they analyzed the wire type textures that can be defined by specifying the Miller indices of the crystal axis parallel to the axis of the specimen.
1099 0036-9748/91 $3.00 + .00 Copyright (c) 1991 Pergamon Press pl¢
1100
FORMATION OF TWINS
Vol.
25, No. 5
Textures of different sharpness have been analyzed in the calculations. It has been assumed that an appropriate axis of the crystal lattices must make an angle ~ (smaller than a certain value ~o) with the specimen axis. Three values of ~o were assumed in the calculations: 3, 5 and 15 °. The sharpness of the texture fcr a given value of Qo was examined on the basis of calculated pole figures. These values of ~o yielded a high, medium and low sharpness of the textures, respectively. The m i e o r i e n t a t l o n s b e t w e e n crystal lattices of nearest neighbors were calculated and the miscrientatlons were used tc categorize the character of the grain boundaries in terms of CSL theory. In this formalism, grains characterized by a twin orientation y i e l d the miscrientation described as ~-3. It has been assumed in the calculations that the misorientation of two grains may deviate from the exact twin disorientation by an a n g l e ~ g " that is small enough f o r t he deviation to be accommodated by an array of dislocations. This value is consistent with the criterion proposed b y B r a n d o n [7] and close to the value that was adopted by Nielsen [4]. Two models of a textured polycrystal have b e e n employed: model I and model II. The grains in the model I were assumed to have randomly rotated the other two (nearly perpendicular) axes of the grain lattices. In the model II the grains had a fixed orientation of the other two axes. The results of the calculations are given in Tables I and II. TABLE I. Twin Frequency Calculated for Different Types of Texture Employing The Model I. Ul
TYPE OF TEXTURE
TEXTURE SHARPNESS
TWIN FREQUENCY
%
It]
random
no texture
1.4
100%
low - 15 °
0.5
-
100%
50% , 50~
100%
high
- 3°
9.7
medium - 5 °
5.8
low
1.4
-- 15"
3°
2.6
5"
2.4
15"
1.7
3"
13.6
5°
7.2
15 o
1.6
[]
Table I and Table II show that in textured p o l y c r y s t a l s the frequency of grains characterized by twin orientation is much h i g h e r than predlcted for a random orientation of grains. In certain cases it exceeds one tenth of the total grain misorientations. This means that as much as 10% of the grain boundary population might be characterized by a twin misorientation.
Vol.
25, No. 5
FORMATION
Twin Frequency Calculated
OF TWINS
TABLE II. for Different Textures
TYPE OF TEXTURE
TEXTURE SHARPNESS
ii01
Employing The Model II. TWIN F R E Q U E N C Y
[%]
- uQ
100%
high
-
3°
43.5
medium -
5°
23.6
- 15"
24.7
low 50% , 50%
100%
3°
10.9
5"
5.8
15 °
1.2
3°
0.0
5"
0.0
15"
2.5
The frequency of twin oriented grains is correlated with the texture type and generally increases with increasing sharpness of the texture. The highest frequencies are observed in model I for a <110> texture and in the model II for <111> texture. The frequency also depends on the criterion used to decide whether a given misorientation is close enough to the exact twin misorientation. Additional misorientations angle B.
computations [8 ] proved that the to a first approximation is proportional Discusmlon
frequency of twin to the value of the
and Conoluslons
The majority of metals are characterized by the existence of a crystal orientation texture which is produced during material processing, deformation or annealing. On the other hand, the results of calculations by Garbacz and Grabski show that merely for geometrical reasons textured materials are expected to exhibit a significant number of the grains, which are in twin (or near twin) misorientation one with respect to the other. The fraction of twin misoriented grains in textured materials has been shown to be much higher than was estimated by Nielsen [4] for the case of a randomly oriented grains. In this situation one has to accept that, apart from other possible contributions, grain encountering by itself can account for a significant fraction of twin grains observed in metals and technical alloys. The contribution of the other mechanisms of twin grains formation to the total number of twins can only be properly ascertained if the contribution of the grain encounter case is determined. In practice this would require that the computations of Garbecz and Grabski [6] be repeated for the texture observed in a given material. Without such calculations it is difficult to decide the relative fractions of twins originated for geometrical reasons (grain encounters) and as a result of other processes such as grain boundary dissociation. The need for a proper approach to the twins created as a result of grain encounters is especially evident in the studies of twin frequency variation as a result of grain growth, which in general brings about profound changes in the texture of the material.
1102
FORMATION OF TWINS
Vol.
25,
No.
A~ovledautentJ The
author wishes to thank Prof. Orabski and Dr. unpublished results and for their discussions.
1. 2. 3. 4. 5. 6. 7. 8.
A.Garbaoz
for access to
R.L.Fullman and J.C.Fisher, g _ ~ , 22, 1350 (1951) H.Gellter, ~ , 17, 1421, (1969) W.G.Burgers, Phvslca, IS, 22, (1949) I.P.Nielsen, ~ L ~ J L ~ , 15, 1082, (1967), S.G.Khalyutln, Fizika Miet.Nietalloviedlenle, 1316, 29, (1970) A.Garbacs and M.W.Grabskl, ~ J ~ L _ W ~ J ~ , 23, 1369, (1989) D.G.Brandon, ~JL_~J~LI~., 14, 1479, (1966) A.Garbacz unpubllshed results
5
Vol.
25, pp. 1099-1102, 1991 Printed in the U.S.A.
Pergamon Press plc All rights reserved
" O N T H E F O R M A T I O N O F T W Z N G R A I N 8 AS A R E S U L T O F G R A I N E N C O U N T E R S D U R / N G T H B P R O C E 8 8 OF R E C R Y S T A L L I K A T I O N A N D G R A I N G R O W T H H
K.J.Kurzydlowski Institute of Materials science and Engineering Warsaw University of Technology, 0 2 - 5 2 4 Warszawa, Narbutta 85 (Received December 27, 1990) (Revised March 4, 1991)
Twin grains are a common feature of some FCC metals. Twins are characterized by a special orientation of crystal lattices of neighboring grains. Crystal lattices of twins are mirror reflections of one another. In terms of coincidence site lattice theory such an orientation can be described as the orientation Z=3. The process of twin grains formation has been analyzed in a n-mher of models, which can be divided into three general categories: a) grain boundary reactions models (for example [1]) b) grain growth defect models (for example [2]) c) grain encounter models (for example [3]) The last two groups of models are based on the assumption that the twin orientation is a result of an encounter of two grains which come into contact during the recrystallization process, or as a result of g r a i n g r o w t h , and which are in twin orientation. The physical significance of such a model strongly depends on the probability of finding a given grain structure of two grains which do have twin misorientation of their crystal lattices. It is intuitively felt that this probability is relatively low in the case of polycrystals characterized by a random orientation of the grains. This has been substantiated b y t h e calculations of Nielsen [4], who s h o w e d t h a t in such a situation a negligibly small number of grains is expected to have twin orientation. The existence of texture in the orientation of grains forming a polycrystal, no matter how it has been formed, changes the probability of finding two grains in the twin orientation. This effect has been semi-quantitatively studied in [5]. To the best knowledge of the author no systematic studies of the effect of texture on the frequency of occurrence of twin grain have been reported. Garbacz and Grabski [6] recently published results of their studies of the relationship between texture of a polycrystal and misorientation of its grain boundaries. It is the aim of the present paper to show that their results can be used to modify the predictions of Nielsen [4]. Their results allow one to analyze more systematically the effect of texture on the frequency of twin grains in polycrystals. M o ~ e l bY G a r b a c s
and Grabski
r61
The model used b y G a r b a c z and Grabski is based on the assumption that grains have constant volume and each grain in a polycrystal has 14 neighbors. The orientation of the crystal lattice in a given grain is randomly taken from a set of orientations typical of an assumed texture of grain lattice orientations. In the calculations, they analyzed the wire type textures that can be defined by specifying the Miller indices of the crystal axis parallel to the axis of the specimen.
1099 0036-9748/91 $3.00 + .00 Copyright (c) 1991 Pergamon Press pl¢
1100
FORMATION OF TWINS
Vol.
25, No. 5
Textures of different sharpness have been analyzed in the calculations. It has been assumed that an appropriate axis of the crystal lattices must make an angle ~ (smaller than a certain value ~o) with the specimen axis. Three values of ~o were assumed in the calculations: 3, 5 and 15 °. The sharpness of the texture fcr a given value of Qo was examined on the basis of calculated pole figures. These values of ~o yielded a high, medium and low sharpness of the textures, respectively. The m i e o r i e n t a t l o n s b e t w e e n crystal lattices of nearest neighbors were calculated and the miscrientatlons were used tc categorize the character of the grain boundaries in terms of CSL theory. In this formalism, grains characterized by a twin orientation y i e l d the miscrientation described as ~-3. It has been assumed in the calculations that the misorientation of two grains may deviate from the exact twin disorientation by an a n g l e ~ g " that is small enough f o r t he deviation to be accommodated by an array of dislocations. This value is consistent with the criterion proposed b y B r a n d o n [7] and close to the value that was adopted by Nielsen [4]. Two models of a textured polycrystal have b e e n employed: model I and model II. The grains in the model I were assumed to have randomly rotated the other two (nearly perpendicular) axes of the grain lattices. In the model II the grains had a fixed orientation of the other two axes. The results of the calculations are given in Tables I and II. TABLE I. Twin Frequency Calculated for Different Types of Texture Employing The Model I. Ul
TYPE OF TEXTURE
TEXTURE SHARPNESS
TWIN FREQUENCY
%
It]
random
no texture
1.4
100%
low - 15 °
0.5
-
100%
50% , 50~
100%
high
- 3°
9.7
medium - 5 °
5.8
low
1.4
-- 15"
3°
2.6
5"
2.4
15"
1.7
3"
13.6
5°
7.2
15 o
1.6
[]
Table I and Table II show that in textured p o l y c r y s t a l s the frequency of grains characterized by twin orientation is much h i g h e r than predlcted for a random orientation of grains. In certain cases it exceeds one tenth of the total grain misorientations. This means that as much as 10% of the grain boundary population might be characterized by a twin misorientation.
Vol.
25, No. 5
FORMATION
Twin Frequency Calculated
OF TWINS
TABLE II. for Different Textures
TYPE OF TEXTURE
TEXTURE SHARPNESS
ii01
Employing The Model II. TWIN F R E Q U E N C Y
[%]
- uQ
100%
high
-
3°
43.5
medium -
5°
23.6
- 15"
24.7
low 50% , 50%
100%
3°
10.9
5"
5.8
15 °
1.2
3°
0.0
5"
0.0
15"
2.5
The frequency of twin oriented grains is correlated with the texture type and generally increases with increasing sharpness of the texture. The highest frequencies are observed in model I for a <110> texture and in the model II for <111> texture. The frequency also depends on the criterion used to decide whether a given misorientation is close enough to the exact twin misorientation. Additional misorientations angle B.
computations [8 ] proved that the to a first approximation is proportional Discusmlon
frequency of twin to the value of the
and Conoluslons
The majority of metals are characterized by the existence of a crystal orientation texture which is produced during material processing, deformation or annealing. On the other hand, the results of calculations by Garbacz and Grabski show that merely for geometrical reasons textured materials are expected to exhibit a significant number of the grains, which are in twin (or near twin) misorientation one with respect to the other. The fraction of twin misoriented grains in textured materials has been shown to be much higher than was estimated by Nielsen [4] for the case of a randomly oriented grains. In this situation one has to accept that, apart from other possible contributions, grain encountering by itself can account for a significant fraction of twin grains observed in metals and technical alloys. The contribution of the other mechanisms of twin grains formation to the total number of twins can only be properly ascertained if the contribution of the grain encounter case is determined. In practice this would require that the computations of Garbecz and Grabski [6] be repeated for the texture observed in a given material. Without such calculations it is difficult to decide the relative fractions of twins originated for geometrical reasons (grain encounters) and as a result of other processes such as grain boundary dissociation. The need for a proper approach to the twins created as a result of grain encounters is especially evident in the studies of twin frequency variation as a result of grain growth, which in general brings about profound changes in the texture of the material.
1102
FORMATION OF TWINS
Vol.
25,
No.
A~ovledautentJ The
author wishes to thank Prof. Orabski and Dr. unpublished results and for their discussions.
1. 2. 3. 4. 5. 6. 7. 8.
A.Garbaoz
for access to
R.L.Fullman and J.C.Fisher, g _ ~ , 22, 1350 (1951) H.Gellter, ~ , 17, 1421, (1969) W.G.Burgers, Phvslca, IS, 22, (1949) I.P.Nielsen, ~ L ~ J L ~ , 15, 1082, (1967), S.G.Khalyutln, Fizika Miet.Nietalloviedlenle, 1316, 29, (1970) A.Garbacs and M.W.Grabskl, ~ J ~ L _ W ~ J ~ , 23, 1369, (1989) D.G.Brandon, ~JL_~J~LI~., 14, 1479, (1966) A.Garbacz unpubllshed results
5