INTERPRETATION OF GRAIN BOUNDARY NONEQUILIBRIUM IN TERMS OF GEOMETRICALLY NECESSARY AND STATISTICALLY STORED DISLOCATIONS

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INTERPRETATION OF GRAIN BOUNDARY NONEQUILIBRIUM IN TERMS OF

GEOMETRICALLY NECESSARY AND STATISTICALLY STORED DISLOCATIONS

W. Lojkowski, J. Wyrzykowski, J. Kwieci�ski

To cite this version:

W. Lojkowski, J. Wyrzykowski, J. Kwieci�ski. INTERPRETATION OF GRAIN BOUNDARY NONEQUILIBRIUM IN TERMS OF GEOMETRICALLY NECESSARY AND STATISTICALLY STORED DISLOCATIONS. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-239-C1-244.

�10.1051/jphyscol:1990137�. �jpa-00230295�

COLLOQUE DE PHYSIQUE

Colloque Cl, suppl6ment au nol, Tome 51, janvier 1990

INTERPRETATION OF GRAIN BOUNDARY NONEQUILIBRIUM IN TERMS OF GEOMETRICALLY NECESSARY AND STATISTICALLY STORED DISLOCATIONS

W. LOJKOWSKI , J

.

UNIPRESS, High Pressure Research Centre, Polish Academy of Sciences,

~ o k o d o w s k a 29, PL-01-142 Warsaw, Poland

Institute of Materials Science and Engineering, Warsaw University of Technology, Narbutta 85, Warsaw, Poland

BBSTR9C;L

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T h e purpose of t h e present paper is to present s o m e recent progress in understanding t h e g r a i n boundary ( G B ) nonequilibrium induced by s p r e a d i n g of Trapped L a t t i c e D i s l o c a t i o n s CTLDs) ( 1 - 7 ) in c o n n e c t i o n with dynamical recovery mechanism in polycrystals

( 8 ) . The T L D spreading is understood either a s c o n t i n u o u s

spreading of its c o r e parallel t o the GB plane ( 9 - 1 2 ) or splitting of t h e T L D s into Extrinsic G r a i n Boundary D i s l o c a t i o n s CEGBDs) moving apart under mutual repulsion < 1 3 , 1 4 1 . G B s will be

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990137

Cl-240 COLLOQUE DE PHYSIQUE

considered a s "active" i f at the g i v e n temperature the T L D s a r e a b l e t o spread or dissociate. Hence, t h e a c t i v e bcrundaries a r e characterised by a high diffusion coefficient (18). T h e other boundaries, in which T L D s do not spread or dissociate in a measurable time, will be called 'epassive". T h e problem of calculating the energy of a G B in the nonequilibrium a t a t e c a u s e d by trapping of t h e T L D s or their dissociation. will be solved by combining the Grabski ( 1 ) model of recovery of d i s l o c a t i o n s a t G B s during superplastic deformation and the theory of geometrically necessary d i s l o c a t i o n s of Ashby ( 1 5 ) . In that respect E G B O s will be divided into t w o groups: S S G B D s

-

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T h e a d v a n t a g e of the proposed model is that t h e energy of nonequilibrium G B s c a n be calculated. Its application to real experimental s i t u a t i o n s will be illustrated for the c a s e o f s t r a i n induced G B migration ( 7 ) .

2

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A comprehensive description of the experimental procedure is

given in ref . ( 7 > . Aluminium s p e c i m e n s of 9 9 . 9 9 % purity and g r a i n s i z e 5 4 pm have been annealed at 6 B 0 K for 1 hour and divided into two groups. S p e c i m e n s A have been deformed at room temperature by

2% and subsequent1 y annealed a t 58BK for 1 hour. Optical

microscopic investigations have s h o w n that only a few g r a i n boundaries in s p e c i m e n s A could migrate CFig.1a). T h e s p e c i m e n s B have been deformed by 2% at 5 8 0 K . In these s p e c i m e n s almost all G B s h a v e migrated. T h e average migration distance h a s been 14 p m <Fig.lb).

Fig.1. S t r a i n induced GB migration in Al. a: Only few G B migrated i f deformation preceded annealing. b: Host G B migrated i f deformation w a s a t 588K. Note shrinking of a g r a i n like in fig.2.

For both types of s p e c i m e n s the fraction of "active" G B s w a s investigated by Transmission Electron Microscopy (TEM). It h a s been found that 4% of t h e G B s have been a c t i v e at room temperature f o r spreading time 1 hour while for spreading times of 18 min. at 588K 9 6 % of G B s h a v e been a c t i v e ,

It a p p e a r s t h a t in s p e c i m e n s w h e r e o n l y f e w b o u n d a r i e s m i g r a t e d o n l y f e w o n e s a r e a c t i v e in s p r e a d i n g of T L D s . H e n c e s p r e a d i n g a n d e x t e n t of g r a i n g r o w t h may b e i n t e r r e l a t e d . F o r .specimens A, o n e may c o m e t o t h e c o n c l u s i o n t h a t o n l y t h o s e b o u n d a r i e s m i g r a t e d d u r i n g a n n e a l i n g a t 5 8 8 K w h i c h w e r e " a c t i v e " a t 293 K. F o l l o w i n g V a l i e v e t a l . C4,5), t h e i n t e r p r e t a t i o n w o u l d b e t h a t t h e s p r e a d i n g t r a n s f o r m s G B s t r u c t u r e i n t o a m o r e m o b i l e o n e , s o t h a t a f t e r s p r e a d i n g G B s m i g r a t e f a s t e r . However. t h i s i n t e r p r e t a t i o n is not s a t i s f a c t o r y in t h e p r e s e n t c a s e . T h e r e a s o n is t h a t in s p e c i m e n s A a f t e r a n n e a l i n g a t 5 8 8 K s p r e a d i n g m u s t h a v e o c c u r r e d o n a l l G B s . T h i s i s b e c a u s e a f t e r 2% d e f o r m a t i o n all G B s h a v e t r a p p e d a d e n s i t y of a t least

1e7

It will b e f u r t h e r s h o w n t h a t it is not t h e c h a n g e of G B s t r u c t u r e c a u s e d by s p r e a d i n g o f T L D s t h a t is c r u c i a l for t h e i n t e r p r e t a t i o n of s t r a i n i n d u c e d G B m i g r a t i o n . T h e e x t e n d of G B m i g r a t i o n d e p e n d s o n t h e level of s t r a i n in t h e p o l y c r y e t a l a n d o n t h e d e g r e e of r e l a x a t i o n of t h a t s t r a i n t h a t is p o s s i b l e o w i n g t o G B s l i d i n g a n d m i g r a t i o n . It will b e s h o w n t h a t t h e f r a c t i o n of s t r a i n e n e r g y t h a t c a n u n d e r g o r e l a x a t i o n by G B s l i d i n g a n d m i g r a t i o n d e p e n d s o n t h e d e n s i t y of T L D s a b s o r b e d . T h e a b o v e i n t e r p r e t a t i o n is b a s e d o n t h e G r a b s k i model of d y n a m i c a l r e c o v e r y of d i s l o c a t i o n s a t G B s d u r i n g s u p e r p l a s t i c d e f o r m a t i o n ( l , 7 1 . I n t h e c o u r s e of d e f o r m a t i o n , a h i g h e r d e n s i t y of T L D s c a n b e a b s o r b e d by t h e a c t i v e G B s t h a n by t h e p a s s i v e o n e s . T h i s is b e c a u s e T L D s a c c u m u l a t e d o n p a s s i v e G B s g e n e r a t e a s t r e s s f i e l d b a r r i e r for d i s l o c a t i o n s a p p r o a c h i n g t h e GBs. O n a c t i v e GBs, h o w e v e r , t h i s b a r r i e r may b e r e l a x e d o w i n g t o t h e d i s s o c i a t i o n of T L D s . A d y n a m i c a l e q u i l i b r i u m may b e a c h i e v e d b e t w e e n t h e f l u x of d i s l o c a t i o n s a p p r o a c h i n g t h e G B s t o i n c r e a s e of t h i s b a r r i e r a n d t h e r e c o v e r y p r o c e s s e s r e s p o n s i b l e t o r e l a x t h i s s t r a i n f i e l d . H e n c e , t h e d e n s i t y of T L D s a b s o r b e d by t h e a c t i v e G B 5 is h i g h e r t h a n t h e d e n s i t y a b s o r b e d by p a s s i v e o n e s . F o l l o w i n g G r a b s k i (l), t h e r e c o v e r y m e c h a n i s m i s p r o p o s e d t o i n c l u d e r e l a x a t i o n of pile- u p s f o r m e d by E O B D s a t g r a i n b o u n d a r y t r i p l e j u n c t i o n s . S u c h pile-ups may lead t o i n c o m p a t i b i l i t y s t r a i n s w h i c h a s well r e p e a l l a t t i c e d i s l o c a t i o n s from g r a i n b o u n d a r i e s ( f i g . 3 ) . A s i l l u s t r a t e d in f i g s . 2 a n d 3, G B m i g r a t i o n a n d s l i d i n g may c o n t r i b u t e t o r e l a x a t i o n of s u c h s t r a i n s . I n t h e p r e s e n t p a p e r , a q u a n t i t a t i v e a p p r o a c h t o t h e model of G r a b s k i C l ) is d e v e l o p e d .

3

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G B n o n e q u i l i b r i u m will be d e f i n e d h e r e a s a s t a t e of t h e G B w h e n i t s e n e r g y is h i g h e r t h a n t h e m i n i m u m e n e r g y posssible for t h e g i v e n c r y s t a l l o g r a p h i c p a r a m e t e r s . F i g , 4 i l l u s t r a t e s t h e e n e r g e t i c s t a t e of a n o n e q u i l i b r i u m G B , t h e t r a n s f o r m a t i o n l e a d i n g t o e q u i l i b r i u m a n d a t r a n s f o r m a t i o n of t h e G B s t r u c t u r e t a k i n g p l a c e in e q u i l i b r i u m . I n s e t s i l l u s t r a t e w h a t k i n d of r e a c t i o n s b e t w e e n G B d i s l o c a t i o n s w o u l d lead t o t h e a b o v e c h a n g e s of G B e n e r g y i n t h e e n e r g y v s . m i s o r i e n t a t i o n d i a g r a m , a s s u m i n g t h a t long r a n g e s t r a i n f i e l d s o f G B d i s l o c a t i o n s a r e n e g l e c t e d . T h e p a t h a-b c o r r e s p o n d s t o t h e r e c o v e r y of S S G B D s w h i l e t h e p a t h b-c r e p r e s e n t s t h e r e c o v e r y of G N G B D s . R e c o v e r y of G N G B D s is e q u i v a l e n t t o r e l a x a t i o n of i n c o m p a t i b i l i t y s t r a i n s of a b o v e

COLLOQUE DE PHYSIQUE

F i g . 2 . a : A g r a i n w h e r e a h i g h d e n s i t y o f g e o m e t r i c a l l y d ) n e c e s s a r y d i s l o c a t i o n s ( G N G B D s )

C) is p r e s e n t a t g r a i n b o u n d a r i e s ,

b : T h a n k s t o G B m i g r a t i o n t h e

. /

F i g . 3 . a : D i s l o c a t i o n s r u n i n t o t h e G B a n d T L D s a r e c r e a t e d , b:

T L D s d i s s o c i a t e a n d m a k e p l a c e for n e w d i s l o c a t i o n s . c : T h e c a p a b i l i t y of t h e G B for a b s o r b i n g new d i s l o c a t i o n s is e x h a u s t e d . d : T h a n k s t o g r a i n b o u n d a r y m i g r a t i o n t h e s u r p l u s d e n s i t y of g r a i n b o u n d a r y d i s l o c a t i o n s c a n m o v e a w a y a t r e l a t i v e l y low t e m p e r a t u r e s b Y p u r e g l i d e and r e a c t w i t h d i s l o c a t i o n s o n o t h e r g r a i n b o u n d a r i e s . T h e g r a i n b o u n d a r y is a g a i n c a p a b l e of a b s o r b i n g

l a t t i c e d i s l o c a t i o n s .

F i g . 4 . T h e G B e n e r g y v.s m i s o r i e n t a t i o n r e l a t i o n s h i p . E n e r g y of long r a n g e s t r a i n f i e l d s o f G B d i s l o c a t i o n s w a s n e g l e c t e d , T h e i n s e t s d i s p l a y s c h e m a t i c a l l y t h e transf or- m a t i o n s of G B s t r u c t u r e . T h e p o i n t a c o r r e s p o n d s t o G B non- e q u i l i b r i u m . T h e t r a j e c t o r y a + b c o r r e s p o n d s t o m u t u a l a n n i h i - lation of d i s l o c a t i o n s w i t h a n t i p a r a l l e l B u r g e r s v e c t o r s I S S G B D s )

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m e n t i o n e d p i l e - u p s of E G B D s a t G B t r i p l e j u n c t i o n s . T r a n s f o r m a t i o n s of a r r a y s of G B d i s l o c a t i o n s s h o w n in i n s e t s in F i g s . 4 h a v e b e e n f r e q u e n t l y o b s e r v e d by T E M < 1 3 , 1 4 , 1 9 > .

T L D s may g i v e t h e f o l l o w i n g c o n t r i b u t i o n t o G B e n e r g y : e n e r g y of T L D c o r e s , e n e r g y of t h e i r s h o r t r a n g e s t r a i n f i e l d s a n d e n e r g y of t h e long r a n g e s t r a i n f i e l d s . T h e s h o r t r a n g e s t r a i n f i e l d s a r e d e f i n e d a s s t r a i n f i e l d s a c t i n g o v e r d i s t a n c e s s m a l l e r t h a n t h e s p a c i n g of d i s l o c a t i o n s . T h e c o r e e n e r g y and s h o r t r a n g e s t r a i n f i e l d s e n e r g y i s m a x i m u m b e f o r e T L D s d i s s o c i a t i o n t a k e s p l a c e . T h e s u m of t h e s e t w o f a c t o r s c o r r e s p o n d s t o t h e l e n g t h a-b in f i g . 4 a n d c a n be e x p r e s s e d as:

w h e r e AyQe

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T L D c o r e w i d t h b e f o r e s p r e a d i n g a n d R is g r a i n s i r e . F o r t y p i c a l v a l u e s of p,,, a f t e r 2 % d e f o r m a t i o n , i.e, 1 0 7 m-' 1 6 , 1 7 3 t h e m a x i m u m p o s s i b l e e x c e s s e n e r g y of G B s c a u s e d by T L D s , n e g l e c t i n g their long r a n g e s t r a i n f i e l d s , is of t h e o r d e r of 10% of t h e a v e r a g e G B e n e r g y . T h e m a g n i t u d e of t h i s e n e r g y r e f l e c t s t h e e x t e n t of p o s s i b l e c h a n g e s of G B p r o p e r t i e s c a u s e d by t h e p r e s e n c e of T L D s or t h e i r d i s s o c i a t i o n p r o d u c t s . U s i n g t h e B o r i s o v r e l a t i o n s h i p f o r t h e c o r r e l a t i o n b e t w e e n G B e n e r g y a n d d i f f u s i o n ( s e e for i n s t a n c e t h e work of G u p t a : 18), it c a n b e s h o w n t h a t a 18% i n c r e a s e of G B e n e r g y c a n c a u s e a n i n c r e a s e of G B d i f f u s i o n c o e f f i c i e n t by a f a c t o r 3. O n l y s h o r t r a n g e s t r a i n f i e l d s h a v e b e e n t a k e n i n t o a c c o u n t i n t h a t c a l c u l a t i o n b e c a u s e t h e long r a n g e s t r a i n f i e l d s a r e of m u c h s m a l l e r a m p l i t u d e t h a n t h e s h o r t r a n g e o n e s a n d c a n n o t a f f e c t s i g n i f i c a n t l y t h e G B s t r u c t u r e o r c a u s e a n i n c r e a s e o f v a c a n c y c o n c e n t r a t i o n in G B s .

F u r t h e r , it may b e s h o w n t h a t t h e l o n g r a n g e s t r a i n f i e l d s a r e c o n n e c t e d w i t h m u c h h i g h e r e n e r g i e s t h a n t h e s h o r t r a n g e o n e s . T h e e n e r g y of t h e s e s t r a i n f i e l d s d e p e n d s o n t h e d e n s i t y of G N G B D s . t o j k o w s k i e t a l . ( 7 ) , b a s e d o n t h e t h e o r y of g e o m e t r i c a l l y n e c e s s a r y d i s l o c a t i o n s of A s h b y ( 1 5 3 , h a v e d e r i v e d t h e f o l l o w i n g e q u a t i o n s :

1 . F o r t h e d e n s i t y cif G N G B D s :

w h e r e : E

-

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=

2 . F o r t h e e n e r g y of long r a n g e s t r a i n f i e l d s of G N G B D s :

In a l u m i n i u m , i f E = 1 or 2%, R = 20 pm, A Y L E = 1 4 8 - 5 7 0 m J / m 2 . T h i s i s a b o u t t h e a v e r a g e G B e n e r g y .

3. F o r t h e r e p e l l i n g f o r c e b e t w e e n G N G B D s a n d l a t t i c e d i s l o c a t i o n s :

If E = 2%, Q R E P ~ 2 5 M P a s w h i c h is c l o s e t o t h e y i e l d s t r e s s .

Cl-244 COLLOQUE DE PHYSIQUE

4 . B e s i d e s , t h e e q u a t i o n for t h e r a t e of r e c o v e r y of T L D s is:

w h e r e -c is t h e e f f e c t i v e T L D r e l a x a t i o n t i n e , i n c l u d i n g all t h e p o s s i b l e r e l a x a t i o n m e c h a n i s m s , t is t h e d e f o r m a t i o n r a t e a n d A i s t h e t i m e in w h i c h t h e m a x i m u m p o s s i b l e d e n s i t y of d i s l o c a t i o n s c l o s e t o t h e G B w o u l d b e a c h i e v e d i f n o r e l a x a t i o n t a k e s p l a c e . It f o l l o w s f r o m e q . C 5 ) t h a t t h e s m a l l e r t h e d e n s i t y of T L D s a t G B s for c o n s t a n t d e f o r m a t i o n r a t e , t h e m o r e t h e n u m b e r of T L D s a b s o r b e d by G B s . O n t h e o t h e r h a n d , t h e m o r e t h e n u m b e r of a b s o r b e d T L D s , t h e larger is t h e d e n s i t y of G N G B D s a n d h i g h e r a r e t h e i n c o m p a t i b i l i t y s t r a i n s . A s a c o n s e q u e n c e , G B s a c t i v e in s p r e a d i n g h a v e a h i g h e r d r i v i n g f o r c e for s l i d i n g a n d m i g r a t i o n . T h u s , a s i m p l e e x p l a n a t i o n for t h e o f t e n e n c o u n t e r e d p h e n o m e n o n of h i g h m o b i l i t y of t h e G B s d u r i n g h i g h t e m p e r a t u r e d e f o r m a t i o n ( 2 8 ) w a s p r o p o s e d . H e n c e , i n o r d e r t o e x p l a i n s t r a i n i n d u c e d G 8 m i g r a t i o n a n d s l i d i n g , t h e r e is n o n e e d t o i n t r o d u c e n e w c o n c e p t s l i k e h i g h m o b i l i t y , n o n e q u i l i b r i u m G B s t r u c t u r e s c a u s e d by t h e s p r e a d i n g of T L D s l i k e t h e s e p r o p o s e d by V a l i e v et a 1 . < 4 , 5 1 .

1 . B o u n d a r i e s w h e r e m e c h a n i s m s of T L D r e c o v e r y a r e a c t i v e a r e a b l e t o a b s o r b h i g h e r d e n s i t i e s of TLDs t h a n t h e p a s s i v e o n e s . A b s o r p t i o n of T L D s a n d t h e i r d i s s o c i a t i o n c a u s e s i n c o m p a t i b i l i t y s t r a i n s e n h a n c i n g G B m i g r a t i o n a n d s l i d i n g . In o r d e r t o e x p l a i n s t r a i n i n d u c e d G B m i g r a t i o n t h e r e is n o n e e d t o u s e t h e c o n c e p t of s p r e a d i n g i n d u c e d G B n o n e q u i l i b r i u m s t r u c t u r e s .

2 . T h e t h e o r y of g e o m e t r i c a l l y n e c e s s a r y d i s l o c a t i o n s p r o v i d e s a useful m a t h e m a t i c a l a p p r o a c h t o t h e q u e s t i o n of G B n o n e q u i l i b r i u m c a u s e d by i n t e r a c t i o n of G B s a n d d i s l o c a t i o n s .

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