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On the symmetrical deformation of tilt bicrystals
Scripta METALLURGICA
Vol. 16, pp. 353-356, 1982 Printed in the U.S.A.
ON THE SYMMETRICAL DEFORMATION OF T I L T
Pergamon Press Ltd. All rights reserved
BICRYSTALS.
J.J. BACMANN, M . O . GAY, R. de TOURNEMINE. Centre d'Etudes Nucl~aires de G r e n o b l e . DMG/SER - 8 5 X - 3 8 0 4 1 GRENOBLE CEDEX.
(Received October 26, 1981) (Revised January 19, 1982) Introduction. I n t e r a c t i o n s between d i s l o c a t i o n s and grain boundaries have been m o d e l e d in a number of cases (I-6). In a bicrystal, a grain boundary d i s l o c a t i o n (GBD} a p p e a r s when a d i s l o c a t i o n of crystal 1 crosses the grain b o u n d a r y and then moves as a crystaI 2 disiocation. The Burgers vector b x of the GBD is related to the Burgers vectors b I and b 2 of the two lattice d i s i o c a t z o n s and is given by (7) : b i = b2 - b I
/I/
Such extrinsic G B D ' s are the basis of some e l e m e n t a r y m e c h a n i s m s which can account for the b e h a v i o u r of stressed or deformed bicrystals : long range stress field (1), grain b o u n d a r y crack initiation (2), bicrystaI d i s o r i e n t a t i o n (3), grain b o u n d a r y siiding (4) and migration ( 4 , 6 ) .
Excepting the work o f G u i l l o p ~ and P o i r i e r ( 6 ) , in which the example o f Na Cl i s c o n s i dered, in p r e v i o u s analyses ( 1 - 5 ) , the c r y s t a l s t r u c t u r e i s assumed t o be simple cubic and the stresses are complex or p o o r l y d e f i n e d . Furthermore, t h e r e i s a lack o f e x p e r i m e n t a l v e r i f i c a t i o n of the model. As a p r e l i m i n a r y study o f b i c r y s t a l creep ( 8 ) , i t is the purpose o f t h i s note t o show t h a t equation / 1 / can be e x p e r i m e n t a l l y i l l u s t r a t e d w i t h f a i r l y good accuracy when the a p p l i e d s t r e s s allows the d e t e r m i n a t i o n o f a couple bI and b2 which i s supposed t o r e a c t . Experiments are c a r r i e d out on Ge b i c n y s t a l s . Symmetrical deformation o f a t i l t
bicr~stal.
We s h a l l r e s t r i c t ourselves t o the case of a symmetrical t i l t b i c r y s t a l stressed along a d i r e c t i o n o f the boundary plane which produces s i n g l e s l i p c o n d i t i o n s in each c r y s t a l . The s l i p systems s a t i s f y the symmetry c o n d i t i o n s and the two c r y s t a l s are deformed s y m m e t r i c a l l y w i t h respect t o the boundary. The s i t u a t i o n i s s c h e m a t i c a i l y represented in f i g u r e 1 where the b i c r y s t a l i s under t e n s i l e s t r e s s . Formally, i t was assumed t h a t the specimen e l o n g a t i o n o r i g i n a t e s from e i t h e r one d i s l o c a t i o n , in each g r a i n , moving towards the g r a i n boundary ( F i g . l a ) or one d i s l o c a t i o n moving across the boundary ( F i g . l b ) . In f i g u r e l a , the Burgers v e c t o r b2 and the sense o f displacement m o f d i s l o c a t i o n 2 are r e s p e c t i v e l y deduced from b~ 2 I and mI o f d i s l o c a t i o n 1 through a symmetry about the boundary plane. The l i n e sense o f a dislocation has an a x i a l c h a r a c t e r . Consequently, by a m i r r o r symmetry o p e r a t i o n , the component o f the l i n e sense which i s p a r a l l e l to the m i r r o r i s reversed ; the senses o f d i s l o c a t i o n s 1 and 2 are o p p o s i t e . In f i g u r e l b , the m2 v e c t o r is reversed w i t h respect to f i g u r e l a . Hence d i s l o c a t i o n 2 should change signs. The Burgers v e c t o r b , corresponding t o i. the same l i n e o r i e n t a t i o n o f the r e s u l t i n g GBD and o f d i s l o c a t i o n 1, xs given by the b balance o f f i g u r e I . I n these c o n d i t i o n s , the homogeneous deformation o f the b i c r y s t a l leads t o a w a l l o f edge GBD added to the i n i t i a l boundary. This introduces a t i l t d e v i a t i o n from the i n i t i a l mutual o r i e n t a t i o n o f the two c r y s t a l s . The a x i s o f t h i s a d d i t i o n a l r o t a t i o n i s p a r a l l e l to the common i n t e r s e c t i o n o f the boundary plane w i t h the g l i d e planes o f d i s l o c a t i o n s I and 2. In f . c . c , l a t t i c e s , an example can be set by the symmetrical ~=g b i c r y s t a l ( 3 8 , g 4 ° , [ 0 1 1 ] boundary plane (122)) submitted t o a t e n s i l e s t r e s s along [26 7 20] which corresponds to the p r o j e c t i o n o f the nearest < 321 > a x i s o f each g~ain on the boundary plane. S i n g l e s l i p c o n d i t i o n s are expected. The primary g l i d e planes (111) i n t e r s e c t the (122) boundary along the
353
0036-9748/82/040353-04503.00/0 Copyright (c) 1982 Pergamon Press Ltd.
354
DEFORMATION OF BICRYSTALS
Vol. 16, No. 4
g.b.
g.b.
f
f
II bi
(a)
a) symmetrical
movement
{b) FIG.1 Symmetrical glide in a tilt bicrystal. of dislocations, b) dislocation moving across
the boundary.
[ 0 1 1 ] common a x i s . I n t h i s s i m p l e example, o n l y a v a r i a t i o n o f t h e 38,940 a n g le i s e x p e c t e d . The s i g n o f t h e r o t a t i o n about ~11 ] a r i s i n g from e l o n g a t i o n can be p r e d e t e r m i n e d . C o n s i d e r t h e f i g u r e 2a which r e p r e s e n t s the c r y s t a l l o g r a p h i c relationship o f t h e ~=9 b o u n d a ry. A s e t o f GBD w i t h l i n e o r i e n t a t i o n [011] and Burgers v e c t o r o r i e n t a t i o n [152] c o r r e s p o n d s t o e x t r a h a l f p l a n e s added upwards and leads t o i n c r e a s e t h e a n g le between the [100] d i r e c t i o n o f each crystal. On the c o n t r a r y , i f the Burgers v e c t o r o f t h e GBD p o i n t s in t h e [ ~ 2 2 ] d i r e c t i o n , the a n g l e between the two c r y s t a l s d e c r e a s e s . In o t h e r r e s p e c t s , i t can be shown ( F i g . 2 b ) t h a t
o,b.
g.b.
I
[414] primary slip plane /
[414] [100111
/ /
[100] I
/
//iX
[011] [1~2]
[11011 J~ z /
I" I
"
, [~
FIG.2 (a)
a) crystallographical
relationship.
[ih]
[211]
7 20]
(b)
=9 bicrystal. b) primary slip using a [26 7 20] common tensile direction.
/
Vol.
16, No.
4
DEFORMATION
OF BICRYSTALS
355
the elongation of c r y s t a l 1, in the [26 7 20 ] d i r e c t i o n , can be obtained by moving 600 d i s l o c a t i o n s ( l i n e ori_entation [011 ] , Burgers vector ~ [~10] , corresponding to the primary s l i p system, in the [ 2 1 1 ] d i r e c t i o n . The b rule of figure 1 gives the Burgers vector bi of the resulting GBD, provided that t h e i r l i n e o r i e n t a t i o n is [ 011 ]. b : Y~ [~10] - 1/18 [-3 3 12] = 6/18 [~22] l
/2/
where 1/18 [3 3 12] is symmetry related to ~ [~1~ about (122). Consequently, the angle between the two c r y s t a l s should decrease and is related to the elongation by the following equation : @ e =
([7101 .~) (2 [7101 . £) d @
131
38.94 in which ~ and ~ are the t e n s i l e d i r e c t i o n and the boundary normal respectively. Experimental r e s u l t s . =9 germanium b i c r y s t a l s have been crept at 4g0C, under a t e n s i l e stress of 20 MPa, along a [26 7 20] common d i r e c t i o n , f o r times varying from ~ to 24 hours, under p u r i f i e d argon atmosphere. X-ray Laue back r e f l e c t i o n patterns have been used to determine the mutual o r i e n t a t i o n of the two grains a f t e r the creep experiment. Examination of strained b i c r y s t a l surfaces has shown traces of two glide planes : (111) and (111), primary and secondary ( c r i t i c a l ) glide plane respectively. A d d i t i o n a l l y , boundaries have been observed, a f t e r deformation, using X-ray topography f o r the very early stages of the process and transmission electron microscopy f o r a higher deformation r a t i o (0.1%.15%). These observations have confirmed that two s l i p systems are operating even during the f i r s t stage of deformation. However, figure 3 shows that the 60° d i s l o c a t i o n s (A) a r i s i n g from the primary s l i p system (~T1) [I-10] are predominant, while the density of dislocations (B) t h a t can be a t t r i b u t e d to the secondary s l i p system (111) ~10] is lower. These two s l i p systems are also activated in the case of single c r y s t a l s in the same stress conditions. The a c t i v a t i o n of the secondary s l i p system does not seem to be related to a grain boundary effect.
Transmission electron direction [26 7 2 0 J .
microscopy
observation
FIG.3 of
a
slightly
deformed
~=9
boundary
(tensile
356
DEFORMATION
OF
BICRYSTALS
Vol.
16,
No.
4
Figures 4 and 5 i l l u s t r a t e the r o t a t i o n accompanying elongation. Figure 4 which represents a cross section of b i c r y s t a l specimen strained about 16% , shows the shape change associated with the r e l a t i v e r o t a t i o n of the c r y s t a l s .
4 0 ~ u~
@
FIG.4 Cross section a f t e r elongation of an i n i t i a l l y f l a t b i c r y s t a l specimen
__
@
calculated from eq.3
IO
1@
f %
FIG.5 Angle between the two c r y s t a l s in function of deformation.
Figure 5 allows the comparison between calculated and measured r o t a t i o n s about[011] in function of strain. Rotations have been calculated from equation (3) assuming (i) single slip conditions (iil constant angle between thai011] common axis and the tensile direction. According to these approximations and experimental errors (the accuracy on @ and C measures is respectively estimated within + I ° and + 1%), in particular because crystal damage occurs during deformation, it can be- considere-d that fairly good agreement is achieved between calculated and measured values of figure 5. Conclusion. The interaction of lattice dislocations with a grain boundary can account for the symmetrical deformation of tilt bicrystals when single slip conditions are satisfied. The model predicts t h a t a mutual r o t a t i o n of the two c r y s t a l s should be associated with the deformation. The expected r o t a t i o n is a c t u a l l y in agreement with the experiment. References. (1) (2) (3) (4)
M.J. Marcinkows~i and W.P. Tseng, Metal. Trans., I , 3397 (Ig70). E.S.P. Das and M.J. Marcinkowski, J. Mater. Sci. Eng., 8, 18g (1971). E.S.P. Das and M.J. Marcinkowski, Acta Met., 20, 199 ( 1 9 7 2 ) . K. Sadananda and M.J. Marcinkowski, J. Appl. Phys., 45, 1533 (1974).
(5 (6 (7
N.K. G i l r a , S c r i p t a Met., 8, 897 (1974). M. Guilopp6 and J.P. P o i r i e r , Acta Met. 28, 163 (1980). J. F r i e d e l , D i s l o c a t i o n s , Addison-Wesley P u b l i s h i n g Compagny, s e t t s 187 (1964). M.O. Gay, Th~se Grenoble (1981).
(8
Inc,
Reading,
Massachu-
Acknowled@ements. The authors Millier,
wish
to
thank
Dr.
A.
George f o r
Petit, Vallais for technical assistance.
stimulating
d i s c u s s i o n s and MM. Debrenne,
Vol. 16, pp. 353-356, 1982 Printed in the U.S.A.
ON THE SYMMETRICAL DEFORMATION OF T I L T
Pergamon Press Ltd. All rights reserved
BICRYSTALS.
J.J. BACMANN, M . O . GAY, R. de TOURNEMINE. Centre d'Etudes Nucl~aires de G r e n o b l e . DMG/SER - 8 5 X - 3 8 0 4 1 GRENOBLE CEDEX.
(Received October 26, 1981) (Revised January 19, 1982) Introduction. I n t e r a c t i o n s between d i s l o c a t i o n s and grain boundaries have been m o d e l e d in a number of cases (I-6). In a bicrystal, a grain boundary d i s l o c a t i o n (GBD} a p p e a r s when a d i s l o c a t i o n of crystal 1 crosses the grain b o u n d a r y and then moves as a crystaI 2 disiocation. The Burgers vector b x of the GBD is related to the Burgers vectors b I and b 2 of the two lattice d i s i o c a t z o n s and is given by (7) : b i = b2 - b I
/I/
Such extrinsic G B D ' s are the basis of some e l e m e n t a r y m e c h a n i s m s which can account for the b e h a v i o u r of stressed or deformed bicrystals : long range stress field (1), grain b o u n d a r y crack initiation (2), bicrystaI d i s o r i e n t a t i o n (3), grain b o u n d a r y siiding (4) and migration ( 4 , 6 ) .
Excepting the work o f G u i l l o p ~ and P o i r i e r ( 6 ) , in which the example o f Na Cl i s c o n s i dered, in p r e v i o u s analyses ( 1 - 5 ) , the c r y s t a l s t r u c t u r e i s assumed t o be simple cubic and the stresses are complex or p o o r l y d e f i n e d . Furthermore, t h e r e i s a lack o f e x p e r i m e n t a l v e r i f i c a t i o n of the model. As a p r e l i m i n a r y study o f b i c r y s t a l creep ( 8 ) , i t is the purpose o f t h i s note t o show t h a t equation / 1 / can be e x p e r i m e n t a l l y i l l u s t r a t e d w i t h f a i r l y good accuracy when the a p p l i e d s t r e s s allows the d e t e r m i n a t i o n o f a couple bI and b2 which i s supposed t o r e a c t . Experiments are c a r r i e d out on Ge b i c n y s t a l s . Symmetrical deformation o f a t i l t
bicr~stal.
We s h a l l r e s t r i c t ourselves t o the case of a symmetrical t i l t b i c r y s t a l stressed along a d i r e c t i o n o f the boundary plane which produces s i n g l e s l i p c o n d i t i o n s in each c r y s t a l . The s l i p systems s a t i s f y the symmetry c o n d i t i o n s and the two c r y s t a l s are deformed s y m m e t r i c a l l y w i t h respect t o the boundary. The s i t u a t i o n i s s c h e m a t i c a i l y represented in f i g u r e 1 where the b i c r y s t a l i s under t e n s i l e s t r e s s . Formally, i t was assumed t h a t the specimen e l o n g a t i o n o r i g i n a t e s from e i t h e r one d i s l o c a t i o n , in each g r a i n , moving towards the g r a i n boundary ( F i g . l a ) or one d i s l o c a t i o n moving across the boundary ( F i g . l b ) . In f i g u r e l a , the Burgers v e c t o r b2 and the sense o f displacement m o f d i s l o c a t i o n 2 are r e s p e c t i v e l y deduced from b~ 2 I and mI o f d i s l o c a t i o n 1 through a symmetry about the boundary plane. The l i n e sense o f a dislocation has an a x i a l c h a r a c t e r . Consequently, by a m i r r o r symmetry o p e r a t i o n , the component o f the l i n e sense which i s p a r a l l e l to the m i r r o r i s reversed ; the senses o f d i s l o c a t i o n s 1 and 2 are o p p o s i t e . In f i g u r e l b , the m2 v e c t o r is reversed w i t h respect to f i g u r e l a . Hence d i s l o c a t i o n 2 should change signs. The Burgers v e c t o r b , corresponding t o i. the same l i n e o r i e n t a t i o n o f the r e s u l t i n g GBD and o f d i s l o c a t i o n 1, xs given by the b balance o f f i g u r e I . I n these c o n d i t i o n s , the homogeneous deformation o f the b i c r y s t a l leads t o a w a l l o f edge GBD added to the i n i t i a l boundary. This introduces a t i l t d e v i a t i o n from the i n i t i a l mutual o r i e n t a t i o n o f the two c r y s t a l s . The a x i s o f t h i s a d d i t i o n a l r o t a t i o n i s p a r a l l e l to the common i n t e r s e c t i o n o f the boundary plane w i t h the g l i d e planes o f d i s l o c a t i o n s I and 2. In f . c . c , l a t t i c e s , an example can be set by the symmetrical ~=g b i c r y s t a l ( 3 8 , g 4 ° , [ 0 1 1 ] boundary plane (122)) submitted t o a t e n s i l e s t r e s s along [26 7 20] which corresponds to the p r o j e c t i o n o f the nearest < 321 > a x i s o f each g~ain on the boundary plane. S i n g l e s l i p c o n d i t i o n s are expected. The primary g l i d e planes (111) i n t e r s e c t the (122) boundary along the
353
0036-9748/82/040353-04503.00/0 Copyright (c) 1982 Pergamon Press Ltd.
354
DEFORMATION OF BICRYSTALS
Vol. 16, No. 4
g.b.
g.b.
f
f
II bi
(a)
a) symmetrical
movement
{b) FIG.1 Symmetrical glide in a tilt bicrystal. of dislocations, b) dislocation moving across
the boundary.
[ 0 1 1 ] common a x i s . I n t h i s s i m p l e example, o n l y a v a r i a t i o n o f t h e 38,940 a n g le i s e x p e c t e d . The s i g n o f t h e r o t a t i o n about ~11 ] a r i s i n g from e l o n g a t i o n can be p r e d e t e r m i n e d . C o n s i d e r t h e f i g u r e 2a which r e p r e s e n t s the c r y s t a l l o g r a p h i c relationship o f t h e ~=9 b o u n d a ry. A s e t o f GBD w i t h l i n e o r i e n t a t i o n [011] and Burgers v e c t o r o r i e n t a t i o n [152] c o r r e s p o n d s t o e x t r a h a l f p l a n e s added upwards and leads t o i n c r e a s e t h e a n g le between the [100] d i r e c t i o n o f each crystal. On the c o n t r a r y , i f the Burgers v e c t o r o f t h e GBD p o i n t s in t h e [ ~ 2 2 ] d i r e c t i o n , the a n g l e between the two c r y s t a l s d e c r e a s e s . In o t h e r r e s p e c t s , i t can be shown ( F i g . 2 b ) t h a t
o,b.
g.b.
I
[414] primary slip plane /
[414] [100111
/ /
[100] I
/
//iX
[011] [1~2]
[11011 J~ z /
I" I
"
, [~
FIG.2 (a)
a) crystallographical
relationship.
[ih]
[211]
7 20]
(b)
=9 bicrystal. b) primary slip using a [26 7 20] common tensile direction.
/
Vol.
16, No.
4
DEFORMATION
OF BICRYSTALS
355
the elongation of c r y s t a l 1, in the [26 7 20 ] d i r e c t i o n , can be obtained by moving 600 d i s l o c a t i o n s ( l i n e ori_entation [011 ] , Burgers vector ~ [~10] , corresponding to the primary s l i p system, in the [ 2 1 1 ] d i r e c t i o n . The b rule of figure 1 gives the Burgers vector bi of the resulting GBD, provided that t h e i r l i n e o r i e n t a t i o n is [ 011 ]. b : Y~ [~10] - 1/18 [-3 3 12] = 6/18 [~22] l
/2/
where 1/18 [3 3 12] is symmetry related to ~ [~1~ about (122). Consequently, the angle between the two c r y s t a l s should decrease and is related to the elongation by the following equation : @ e =
([7101 .~) (2 [7101 . £) d @
131
38.94 in which ~ and ~ are the t e n s i l e d i r e c t i o n and the boundary normal respectively. Experimental r e s u l t s . =9 germanium b i c r y s t a l s have been crept at 4g0C, under a t e n s i l e stress of 20 MPa, along a [26 7 20] common d i r e c t i o n , f o r times varying from ~ to 24 hours, under p u r i f i e d argon atmosphere. X-ray Laue back r e f l e c t i o n patterns have been used to determine the mutual o r i e n t a t i o n of the two grains a f t e r the creep experiment. Examination of strained b i c r y s t a l surfaces has shown traces of two glide planes : (111) and (111), primary and secondary ( c r i t i c a l ) glide plane respectively. A d d i t i o n a l l y , boundaries have been observed, a f t e r deformation, using X-ray topography f o r the very early stages of the process and transmission electron microscopy f o r a higher deformation r a t i o (0.1%.15%). These observations have confirmed that two s l i p systems are operating even during the f i r s t stage of deformation. However, figure 3 shows that the 60° d i s l o c a t i o n s (A) a r i s i n g from the primary s l i p system (~T1) [I-10] are predominant, while the density of dislocations (B) t h a t can be a t t r i b u t e d to the secondary s l i p system (111) ~10] is lower. These two s l i p systems are also activated in the case of single c r y s t a l s in the same stress conditions. The a c t i v a t i o n of the secondary s l i p system does not seem to be related to a grain boundary effect.
Transmission electron direction [26 7 2 0 J .
microscopy
observation
FIG.3 of
a
slightly
deformed
~=9
boundary
(tensile
356
DEFORMATION
OF
BICRYSTALS
Vol.
16,
No.
4
Figures 4 and 5 i l l u s t r a t e the r o t a t i o n accompanying elongation. Figure 4 which represents a cross section of b i c r y s t a l specimen strained about 16% , shows the shape change associated with the r e l a t i v e r o t a t i o n of the c r y s t a l s .
4 0 ~ u~
@
FIG.4 Cross section a f t e r elongation of an i n i t i a l l y f l a t b i c r y s t a l specimen
__
@
calculated from eq.3
IO
1@
f %
FIG.5 Angle between the two c r y s t a l s in function of deformation.
Figure 5 allows the comparison between calculated and measured r o t a t i o n s about[011] in function of strain. Rotations have been calculated from equation (3) assuming (i) single slip conditions (iil constant angle between thai011] common axis and the tensile direction. According to these approximations and experimental errors (the accuracy on @ and C measures is respectively estimated within + I ° and + 1%), in particular because crystal damage occurs during deformation, it can be- considere-d that fairly good agreement is achieved between calculated and measured values of figure 5. Conclusion. The interaction of lattice dislocations with a grain boundary can account for the symmetrical deformation of tilt bicrystals when single slip conditions are satisfied. The model predicts t h a t a mutual r o t a t i o n of the two c r y s t a l s should be associated with the deformation. The expected r o t a t i o n is a c t u a l l y in agreement with the experiment. References. (1) (2) (3) (4)
M.J. Marcinkows~i and W.P. Tseng, Metal. Trans., I , 3397 (Ig70). E.S.P. Das and M.J. Marcinkowski, J. Mater. Sci. Eng., 8, 18g (1971). E.S.P. Das and M.J. Marcinkowski, Acta Met., 20, 199 ( 1 9 7 2 ) . K. Sadananda and M.J. Marcinkowski, J. Appl. Phys., 45, 1533 (1974).
(5 (6 (7
N.K. G i l r a , S c r i p t a Met., 8, 897 (1974). M. Guilopp6 and J.P. P o i r i e r , Acta Met. 28, 163 (1980). J. F r i e d e l , D i s l o c a t i o n s , Addison-Wesley P u b l i s h i n g Compagny, s e t t s 187 (1964). M.O. Gay, Th~se Grenoble (1981).
(8
Inc,
Reading,
Massachu-
Acknowled@ements. The authors Millier,
wish
to
thank
Dr.
A.
George f o r
Petit, Vallais for technical assistance.
stimulating
d i s c u s s i o n s and MM. Debrenne,