HAL Id: jpa-00230028
https://hal.archives-ouvertes.fr/jpa-00230028
Submitted on 1 Jan 1990
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
TRIPLE LINE DISCLINATIONS
W. Bollmann
To cite this version:
W. Bollmann. TRIPLE LINE DISCLINATIONS. Journal de Physique Colloques, 1990, 51 (C1),
pp.C1-95-C1-96. �10.1051/jphyscol:1990112�. �jpa-00230028�
COLLOQUE DE PHYSIQUE
Colloque Cl, supplement au n O l , Tome 51, janvier 1990
TRIPLE LINE DISCLINATIONS
W. BOLLMANN
22, Chemin Vert, Pinchat, CH-1234 Vessy, Geneva, Switzerland
R∑
-
Nous d o n n o n s i c i une b r e v e d e s c r i p t i o n d u c o n c e p t d e s " d i s c l i n a t i o n s d e s l i g n e s t r i p l e s " q u i p e u v e n t se f o r m e r & 1 8 j o n c t i o n de t r o i s j o i n t s de g r a i n s d a n s d u m a t e r i e l p o l y c r i s t a l l i n .A b s t r a c t
-
A b r i e f d e s c r i p t i o n i s g i v e n o f t h e c o n c e p t o f " t r i p l e l i n e d i s c l i n a t i o n s "w h i c h c a n f o r m a t t h e j u n c t i o n o f t h r e e g r a i n b o u n d a r i e s i n a p o l y c r y s t a l l i n e m a t e r i a l .
I n o r d e r t o c a l c u l a t e t h e d i s l o c a t i o n n e t w o r k o f a g r a i n o r p h a s e b o u n d a r y we need a t r a n s - f o r m a t i o n w h i c h r e l a t e s a u n i t c e l l o f t h e f i r s t l a t t i c e w i t h a u n i t c e l l o f t h e s e c o n d one /l/. Many r e l a t i o n s a r e m a t h e m a t i c a l l y p o s s i b l e b u t we assume t h a t t h e c r y s t a l w i l l c h o o s e t h e one w h i c h i n t r o d u c e s t h e l e a s t d i s t o r t i o n i n t h e b o u n d a r y , i . e . t h a t w h i c h r e l a t e s t h e n e a r e s t n e i g h b o u r p o s i t i o n s ( n e a r e s t n e i g h b o u r r e l a t i o n NNR) i n t h e t w o l a t t i c e s a n d w h i c h g i v e s t h e r e f o r e t h e w i d e s t s p a c e d d i s l o c a t i o n n e t w o r k .
U s u a l l y t h r e e g r a i n b o u n d a r i e s m e e t i n a l i n e , t h e t r i p l e l i n e ( b o u n d a r y " a " b e t w e e n g r a i n 1 a n d 2 a n d c o r r e s p o n d i n g l y "b" b e t w e e n 2 a n d 3 a n d " c " b e t w e e n 3 a n d 1). S t a r t i n g w i t h a u n i t c e l l i n g r a i n 1 we c a n f o l l o w t h e s e q u e n c e t o g r a i n 2 ( t r a n s f o r m a t i o n A ) t o g r a i n 3
(E)
t o a r r i v e a g a i n a t g r a i n 1(cf.
T h e r e a r e now t w o p o s s i b i l i t i e s :1 . We a r r i v e a t t h e same u n i t c e l l f r o m w h i c h we s t a r t e d .
CBA
= - I ( t h e i d e n t i t y ) . We c a l l t h i s t y p e o f t r i p l e l i n e a n I - l i n e .2. Due t o t h e NNR we a r r i v e i n g r a i n 1 a t a u n i t c e l l w h i c h i s d i f f e r e n t f r o m t h e s t a r t i n g one.
-
CBA =9.
We c a l l t h i s a U - l i n e . U i s a u n i m o d u l a r t r a n s f o r m a t i o n ( d e t ( U ) = 1 ) s i n c e-
t h e v o l u m e o f t h e t w o u n i t c e l l s ( w h i c h compose t h e same l a t t i c e ) i s t h e same.
I t h a s b e e n shown f o r t h e I - l i n e t h a t t h e d i s l o c a t i o n b a l a n c e ( i . e . F r a n k ' s node c o n d i t i o n ) i s a l w a y s s a t i s f i e d a t t h e t r i p l e l i n e . I n t h e U - l i n e , h o w e v e r , t h i s c o n d i t i o n d o e s n o t h a v e t o b e s a t i s f i e d anymore. D i s l o c a t i o n s c a n e n d o r s t a r t t h e r e (/l/ c h a p t e r 1 8 ) . I t h a s f u r t h e r been shown t h a t t h e U - l i n e s p r e s e n t t h e c h a r a c t e r o f d i s c l i n a t i o n s / 2 / , i . e . o f o r i e n t e d l i n e s w i t h a t e n s o r of r a n k 2 a t t r i b u t e d t o them, a n d t h i s t e n s o r i s
9
( w h i l e a d i s l o c a t i o n i s a n o r i e n t e d l i n e w i t h a t e n s o r o f r a n k 1, t h e B u r g e r s v e c t o r , a t t r i b u t e d t o i t ) . These " t r i p l e l i n e d i s c l i n a t i o n s " ( T - d i s c l i n a t i o n s ) a r e c o n t i n u o u s t h r o u g h o u t t h e p o l y c r y s t a l l i n e m a t e r i a l a n d u n d e r g o r e a c t i o n s among e a c h o t h e r i n a way a n a l o g o u s t o d i s - l o c a t i o n s i n a s i n g l e c r y s t a l / 3 , 4 / .W h i l e t h e m a t h e m a t i c s o f d i s l o c a t i o n g e o m e t r y i s d e t e r m i n e d b y v e c t o r c a l c u l a t i o n , t h a t o f T - d i s c l i n a t i o n s i s , i n e s s e n c e , m a t r i x c a l c u l a t i o n . I n c o n t r a s t t o t h e v e c t o r a d d i t i o n m a t r i x m u l t i p l i c a t i o n i s n o t c o m m u t a t i v e , i . e . t h e r e s u l t d e p e n d s o n t h e s e q u e n c e o f t h e f a c t o r s . W h i l e a d i s l o c a t i o n i s g i v e n b y t w o r e p r e s e n t a t i o n s a c c o r d i n g t o t h e l i n e s e n s e , a T - d i s c l i n a t i o n c a n have u p t o s i x r e p r e s e n t a t i o n s a c c o r d i n g t o t h e l i n e s e n s e ( s e q u e n c e a s r i g h t hand s c r e w ) a n d t h e g r a i n i n w h i c h t h e s e q u e n c e s t a r t s a n d ends.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990112
COLLOQUE DE PHYSIQUE
The T - d i s c l i n a t i o n s a r e h i g h e n e r g y l i n e s . T h e r e e x i s t e n c e h a s b e e n p r o v e d e x p e r i m e n t a l l y b y Palumbo a n d A u s t / 5 , 6 / b y r e l a t i n g , i n h i g h p u r i t y n i c k e l , t h e e t c h i n g b e h a v i o u r o f t h e t r i p l e node w i t h t h e r e l a t i v e o r i e n t a t i o n s o f t h e a d j o i n i n g g r a i n s a s d e t e r m i n e d b y SEM e l e c t r o n c h a n n e l i n g p a t t e r n s . T h e s e a u t h o r s showed t h a t o f 1 6 n o d e s w h i c h w e r e i d e n t i - f i e d a s U-nodes 1 3 w e r e e t c h e d p r e f e r e n t i a l l y a t t h e node w h i l e
df
52 I - n o d e s none showed p r e f e r e n t i a l e t c h i n g .T h e s e t r i p l e l i n e d i s c l i n a t i o n s b r i n g a n a d d i t i o n a l a s p e c t i n t o t h e m a t e r i a l s s c i e n c e o f p o l y c r y s t a l l i n e m a t e r i a l . They may i n f l u e n c e e.g. d i f f u s i o n p r o p e r t i e s , t h e b e h a v i o u r o n r e c r y s t a l l i z a t i o n o r c r a c k f o r m a t i o n e t c . . A l s o on t h e t h e o r e t i c a l s i d e t h e y open new a s p e c t s .
REFERENCES
/l/ B o l l m a n n , W., " C r y s t a l L a t t i c e s , I n t e r f a c e s , M a t r i c e s " , p u b l i s h e d b y t h e a u t h o r , Geneva 1 9 8 2 .
/ 2 / B o l l m a n n , W . , P h i l o s . Mag. A,49 ( 1 9 8 4 ) 7 3 . / 3 / B o l l m a n n , W., P h i l o s . Mag. A,57 ( 1 9 8 8 ) 637.
/4/ B o l l m a n n , W., M a t . Sc. Eng. A,1134 ( 1 9 8 9 ) 1 2 9 .
/ 5 / Palumbo, G. a n d A u s t , K.T., S c r . M e t a l l . , 22 ( 1 9 8 8 ) 847.
/ 6 / Palumbo, G. a n d A u s t , K.T., M a t . Sc. Eng., A,113?: ( 1 9 8 9 ) 1 3 9 . ( A , 1 1 3 4 : b o o k "Low E n e r g y D i s l o c a t ~ o n S t r u c t u r e s