CONTINUOUS-DISTRIBUTION DISLOCATION MODEL OF INTERNAL FRICTION ASSOCIATED WITH THE INHOMOGENEOUS SLIDING ALONG HIGH-ANGLE GRAIN BOUNDARIES

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CONTINUOUS-DISTRIBUTION DISLOCATION

MODEL OF INTERNAL FRICTION ASSOCIATED

WITH THE INHOMOGENEOUS SLIDING ALONG

HIGH-ANGLE GRAIN BOUNDARIES

Z. Sun, T. Kê

To cite this version:

Z. Sun, T. Kê.

CONTINUOUS-DISTRIBUTION DISLOCATION MODEL OF INTERNAL

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JOURNAL DE PHYSIQUE

CoZloque C5, supplgment au nOIO, Tome 42, octobre 1981 page C5-451

CONTINUOUS-DISTRIBUTION DISLOCATION M O D E L O F INTERNAL FRICTION ASSOCIATED WITH T H E INHOMOGENEOUS SLIDING ALONG HIGH-ANGLE GRAIN BOUNDARIES

2.9. Sun and T . S . KC*

I n s t i t u t e of Solid State Physics (Hefeiland I n s t i t u t e o f Metal Research (Shenyang), Academia Sinica, China

Abstract.- Basing on a c o n t i n u o u s - d i s t r i b u t i o n d i s l o c a t i o n model o f high-angle g r a i n b o u n d a r i e s , a n i n t e g r a l - d i f f e r e n t i a 1 e q u a t i o n governing t h e inhomogeneous s l i d i n g a l o n g t h e b o u n d a r i e s was s e t up. T h i s e q u a t i o n c o n s i s t s o f t h r e e terms. The f i r s t term i s t h e s h e a r s t r e s s produced by t h e d i s l o c a t i o n s w i t h a g i v e n c o n t i n u o u s d i s t r i b u t i o n l i n e a r d e n s i t y , t h e second term i s t h e a p p l i e d s h e a r s t r e s s and t h e t h i r d term i s t h e Newtonian v i s c o u s r e s i s t a n t s t r e s s . T h i s e q u a t i o n w a s solved approximately and t h e approximate formulae f o r t h e i n t e r n a l f r i c t i o n and modulus d e f e c t were o b t a i n - ed. For h i g h - p u r i t y i s o t r o p i c m e t a l s , t h e v i s c o s i t y f o r g r a i n - boundary s l i d i n g i s c o r r e l a t e d w i t h t h e d i f f u s i o n c o e f f i c i e n t

a l o n g grain-boundary, D

,

by an e x p r e s s i o n similar t o E i n s t e i n - S t o k e s formula. The opt?mum t e m p e r a t u r e of grain-boundary i n t e r n a l f r i c t i o n peak f o r a number o f pure m e t a l s c a l c u l a t e d a c c o r d i n g t o t h i s model and s l i d i n g mechanism a r e f a i r l y c l o s e t o t h e c o r r e s - ponding e x p e r i m e n t a l l y observed v a l u e s . I t i s shown t h a t f o r i m - p u r e m e t a l s , t h e v i s c o s i t y o f some g r a i n b o u n d a r i e s i s c o n s i d e r - a b l y changed by t h e s e l e c t i v e s e g r e g a t i o n o f i m p u r i t i e s a l o n g them, s o t h a t a n o t h e r i n t e r n a l - f r i c t i o n peak ( t h e s o l u t e peak) a p p e a r s a t a d i f f e r e n t t e m p e r a t u r e . Also, i n a n i s o t r o p i c p u r e m e t a l s and i n t h e p r e s e n c e o f i n t e r n a l s t r e s s , t h e m i g r a t i o n o f g r a i n b o u n d a r i e s may cause a n o t h e r h i g h t e m p e r a t u r e i n t e r n a l - f r i c - t i o n peak o r a h i g h e r i n t e r n a l - f r i c t i o n background i n a d d i t i o n t o t h e grain-boundary peak a s s o c i a t e d w i t h grain-boundary s l i d i n g .

I. I n t r o d u c t i o n . - Using a t o r s i o n pendulum, ~k observed i n p o l y c r y s t a l - l i n e 99.991 A 1 an i n t e r n a l - f r i c t i o n ( I F ) peak around 2 8 5 ' ~ ( f = 0.8 H Z ) which w a s a b s e n t i n f u l l y annealed bamboo s t r u c t u r e " s i n g l e c r y s t a l s " / l / . He t h e r e f o r e a t t r i b u t e d t h i s peak ("grain-boundary peakt1) t o a r e l a x a t i o n p r o c e s s i n t h e g r a i n boundaries. R e c e n t l y , Woirgard e t a 1 /2/ r e p o r t e d t h a t a n o t a b l e r e l a x a t i o n e f f e c t i s p r e s e n t i n s l i g h t l y s t r a i n e d Cu s i n g l e c r y s t a l s i n t h e same t e m p e r a t u r e r a n g e a s t h e g r a i n - boundary (GB) peak i n p o l y c r y s t a l s and a l s o a v e r y weak r e l a x a t i o n e f f e c t i n u n s t r a i n e d A 1 s i n g l e c r y s t a l s . They concluded t h a t t h e GB peaks must be e x p l a i n e d by mechanisms which a r e n o t s p e c i f i c o f t h e GB i t s e l f , b u t i n v o l v e more g e n e r a l l y t h e climb and g l i d e o f l a t t i c e d i s - l o c a t i o n s . S i m i l a r p r o p o s i t i o n b a s been made by Gondi e t a1 / 3 / . They r e p o r t e d t h a t s i n g l e c r y s t a l s o r m a c r o c r y s t a l l i n e s h e e t s o f 99.6 % A 1

--p

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C5-452 JOURNAL DE PHYSIQUE

show I F i n s t a b i l i t i e s i n t h e temperature range of t h e K& peak, and s l i g h t deformations of t h e s e specimens cause t h e K& peak t o appear. The p o s s i b i l i t y i s considered t h a t t h e

K8

peak depends on f r e e o r polygon- i z e d d i s l o c a t i o n s i n s i d e t h e g r a i n s . The p r e s e n t r e p o r t w i l l suggest a c o n c r e t e d i s l o c a t i o n model i l l u s t r a t i n g t h a t t h e GB peak ( o r t h e so- c a l l e d "K& can be a s s o c i a t e d with a r e l a x a t i o n p r o c e s s i n t h e GB i t s e l f .

It i s evident t h a t any r e a l i s t i c mechanism of t h e GB peak must con- form t o t h e s t r u c t u r e of t h e high-angle g r a i n boundaries. The coinci- dence s u p p e r l a t t i c e model of high-angle GB r e c e n t l y proposed w a s very s u c c e s s f u l i n e x p l a i n i n g many p r o p e r t i e s of high-angle g r a i n boundaries /4/. A c l o s e approach t o t h e coherency c o n d i t i o n can be achieved f o r high-angle boundaries w i t h a p e r i o d i c segmented s t r u c t u r e and t h e r e i s

a r e g i o n where t h e " f i t 1 ' i s r e l a t i v e l y good and one where it i s poor. T h i s i s b a s i c a l l y s i m i l a r t o t h e e a r l y models put forward by K&

/5/

and by Mott /6/ f o r e x p l a i n i n g t h e assumed viscous behavior of high-angle g r a i n bomdary i n a t o m i s t i c terms. It h a s been pointed out t h a t t h e co-

herency c o n d i t i o n i s achieved by t h e c o n s t r a i n t a p p l i e d t o each l a t t i c e by t h e r e g i o n s of *good f i t n between t h e two l a t t i c e s . Disturbance of t h e r e g u l a r i t y of t h i s h i g h l y c o n s t r a i n e d s t r u c t u r e , by t h e i n t r o d u c t i o n of " m i s f i t n segments, can be e f f e c t i v e l y described i n terms of d i s l o c a - t i o n s with Burgers v e c t o r n o t of a l a t t i c e t r a n s l a t i o n r e c t o r but of t h e a p p r o p r i a t e h i g h index i n t e r p l a n e r apacings /7/. T h i s l e a d s natu- r a l l y t o a c o n t i n u o u s - d i s t r i b u t i o n d i s l o c a t i o n model of high-angle g r a i n boundaries.

11.

-

Continuous-Distribution D i s l o c a t i o n Model of High-Angle Grain Boundaries and t h e Mechanism of Inhomogeneous S l i d i n g along t h e Grain Boundaries.- Let u s consider an a r b i t r a r y high-angle GB as shown i n Fig. 1 i n which AB i s t h e GB plane s e p a r a t i n g g r a i n 1 and g r a i n 2. Assume t h e t h i c k n e s s of GB l a y e r be d and ro t h e atomic r a d i u s . We can imagine t h a t when g r a i n 1 and g r a i n 2 approach each o t h e r , t h e f r o n t i e r atoms of one g r a i n w i l l be a c t e d by a m i s f i t f o r c e from t h e f r o n t i e r atoms of t h e o t h e r g r a i n , so t h a t t h e s e atoms w i l l be d i s p l a c e d through

X l i

and ( i r e p r e s e n t s t h e i - t h atom) and give r i s e t o a c e r t a i n in- t e r n a l streser. E v e n t u a l l y a s t a b l e high-angle GB i s formed a s an e q u i l i - brium s t a t e i s reached. The GB l a y e r i s t h e r e f o r e e q u i v a l e n t t o a con- t i n u o u s d i s t r i b u t i o u of a l t e r n a t i v e l y +ve and -ve ( i n t h e sense of s t a t i s t i c a l average) edge-type and screw-type d i s l o c a t i o n s .

According t o Kr8ner

/a/,

t h e continuous d i s t r i b u t i o n d i s l o c a t i o n d e n s i t y t e n s o r i n t h e GB l a y e r may be expressed as

d =

-

n X Grad

(3

-

,U,),

(4)

where

g

i s t h e normal u n i t v e c t o r . Although t h e s p e c i f i c form o f

-

o(.

i s unknown, b u t t h e mean v a l u e o f 6 a l o n g GB should be z e r o a c c o r d i n g t o

_-

s i m p l e p h y s i c a l arguments, s o t h a t

& =

0. Moreover,o( depends on t h e

_-

d i r e c t i o n and magnitude o f t h e B u r g e r s v e c t o r and t h e d i r e c t i o n o f d i s - l o c a t i o n l i n e which a l l have a random d i s t r i b u t i o n . We may c l a s s i f y t h e GB d i s l o c a t i o n s i n t o f o u r t y p i c a l t y p e s a s shown i n Fig. 2. ( a ) +ve and -ve edge d i s l o c a t i o n s w i t h B u r g e r s v e c t o r (B) a l o n g i and d i s l o c a t i o n l i n e (D) a l o n g j ; ( b ) +ve and -ve screw d i s l o c a t i o n s w i t h B and D b o t h a l o n g j; ( c ) +ve and -ve edge d i s l o c a t i o n s w i t h B and D a l o n g k a n d j; ( d ) GB v a c a n c i e s which can be c o n s i d e r e d as d i p o l e s composed o f a p a i r o f d i s l o c a t i o n s o f +ve and -ve c-type.

It can be seen from Fig. 2 t h a t t h e elementary p r o c e s s f o r GB s l i d - i n g i s t h e s l i p o f t h e d i s l o c a t i o n s o f a- o r b-type and t h a t f o r GB m i - g r a t i o n i s t h e climb o f t h e d i s l o c a t i o n s o f a-type o r t h e s l i p o f c- t y p e . The e l e m e n t a r y p r o c e s s o f GB d i f f u s i o n i s t h e climb o f d i s l o c a t i o n d i p o l e s o f d-type.

111. I n t e r n a l F r i c t i o n g i v e n r i s e by t h e I n h o m o ~ e n e o u s S l i d i n g a l o n g Grain Boundaries.- When a s h e a r s t r e s s 7 i s a p p l i e d a l o n g t h e GB i n t h e d i r e c t i o n X, some o f t h e e l e m e n t a r y d i s l o c a t i o n s o f a-type o r b-type can overcome t h e p o t e n t i a l b a r r i e r t o move i n t o a n e i g h b o u r i n g v a l l e y by t h e r m a l a c t i v a t i o n . T h i s w i l l f a c i l i t a t e t h e n e i g h b o u r i n g e l e m e n t a r y d i s l o c a t i o n s t o overcome t h e p o t e n t i a l b a r r i e r , s o t h a t t h e s l i p t a k e s p l a c e i n t h e form of an "avalancheH i n v o l v i n g t h e group motion o f a c h a i n o f e l e r n e t a r y d i s l o c a t i o n s . Such a c o o p e r a t i v e p r o c e s s w a s r e c e n t - l y proposed by I s h i t a e t a1 / 9 / b a s i n g on t h e i r s t u d i e s o f GB d i f f u s i o n i n c o l l o i d p o l y s t y r e n e l a t e x and gold s o l c r y s t a l s .

Consider t h e inhomogeneous s l i d i n g o f a

flat

and smooth GB l a y e r e x t e n d i n g i n d e f i n i t e l y a l o n g t h e y - d i r e c t i o n and h a v i n g a w i d t h

( ~ i g . 2). Consider o n l y t h e c a s e o f i s o t r o p i c m e t a l s s o t h a t t h e s l i d - i n g ~ on b o t h s i d e s o f t h e boundary a r e e q u a l and o p p o s i t e . The inhomogeneous s l i d e u ( x , t ) a t t h e p o i n t X a l o n g GB s a t i s f i e s t h e e q u a t i o n :

w h e r e h i s t h e s h e a r m o d u l u s , D i s t h e P o i s s o n r a t i o ,

CO

i s t h e a p p l i e d s t r e s s amplitude, i s t h e a n g u l a r f r e q u e n c y and i s t h e assumed v i s - c o s i t y c o e f f i c i e n t a s s o c i a t e d w i t h t h e GB s l i d i n g . The f i r s t term on t h e l e f t - h a n d s i d e of Eqn. ( 2 ) i s t h e s h e a r s t r e s s produced a t X by t h e d i s - l o c a t i o n s w i t h a c o n t i n u o u s d i s t r i b u t i o n l i n e a r d e n s i t y o f

3

( 2 u ) b x , t h e second term i a t h e a p p l i e d s t r e s s , and t h e - t h i r d t e r m i s t h e Newtonian viscotas r e s i s t a n t s t r e s s . The boundary c o n d i t i o n f o r u(x) i s u(*R/2)=0.

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CS-454 JOURNAL DE PHYSIQUE and t h e n V =

[ ~ N T (

l-~)J?%]ue - i m t ;

c

= - n ( l - ' ) l ) ~ ~ r / / ~ d , ( 4 , 5 ) Eqn. ( 2 ) can be w r i t t e n a s I d v dg'

-

_l

₊

_{C l i}₌_{0 and} _V ₍₂₁₎₌_0. d.g'

3 -

5'

Now l e t u s t r y t o f i n d an approximate s o l u t i o n of t h e form

V = q2/=/(1+~,,/ l - c i ). (8)

Obviously t h i s s o l u t i o n s a t i s f i e s t h e boundary c o n d i t i o n s given by Eqn.

(7). S u b s t i t u t i n g Eqn. (8) i n t o Eqn. ( 6 ) and l e t

5

= 0 , we g e t on put- t i n g t h e r e a l p a r t and t h e imaginary p a r t e q u a l t o zero r e s p e c t i v e l y :

2 ~ + ( l / , , / 7 ) l o g c ( - t + p 2 + p , J T ) / ( l + p 2 - p

43))

=

c,

( 9 ) ( ~ + p ~ ) ' / ~ / ( i ~ + p c , , / T ) = q. i i o ) The a n e l a s t i c s t r a i n &a a s s o c i a t e d with t h e GB s l i d i n g i s

&a The t o t a l s t r a i n i s

E

2udx = TT (1-2')q

E;

+ E ; ,

where 1

/v

=

4

Vd?jeiWt. ( 1 1 ) =

z0/',

( t h e e l a s t i c s t r a i n ) .

E ,

The i n t e r n a l f r i c t i o n and modulus d e f e c t can be obtained from t h e ima-

(I l

g i n a r y and t h e r e a l p a r t r e s p e c t i v e l y from (f2/E1) = fa/(%/&) and We have

Q-'

= -1m(EaM/C,) = [IT ( I - y ) / 2

1 [

( 4 ~ - c ) 4 / P 2 ]

,

( 1 2 )

(

IS)

I n Fig. 3, Q-' and AM/M ( o r ~p/ht) axe p l o t t e d a s f u n c t i o n of C, t h e u n i t of t h e o r d i n a t e has been chosen a s

y(

1-21) /2. The d o t t e d curves r e p r e s e n t t h e corresponding curves f o r s i n g l e r e l a x a t i o n time. It can be seen t h a t a peak appeaxs a t

C = ~ ( l - ~ ) , f u ) ~ / ~ d = 4.269, ( 3 4 )

and t h e h e i g h t of t h e peak i s

Q&

c

0.2971(1-V) /2. (15)

The r e l a x a t i o n s t r e n g t h A l y can be determined from t h e extreme values t a k e n from t h e AM/M-C curve i n Fig. 3 a s

A M

= A M / M

( f o r GO) -AM/M ( f o r C=oo) = 0.5(1-Y) /2. ( 1 6 )

R o r t h i s r e g e t

Q,L

=

$ h M

= 0.25 (I-Y) 12, which is c l o s e t o t h e value determined d i r e c t l y from t h e

Q-'-c

curve.

1V.- Comparison with Experiments. Assuming t h a t only t h e GB d i s l o c a t i o n s n e a r t h e GB vacancies a r e f e a s i b l e t o s l i d i n g , then t h e f o r c e a c t i n g on a d i s l o c a t i o n segment of average l e n g t h a i s Z a b where b i s t h e t o t a l Burgers v e c t o r of t h e s e mobile d i s l o c a t i o n s . I f we apply E i n s t e i n ' s r e l a t i o n s h i p approxilnately t o t h e p r e s e n t case, t h e n we have t h e average

(6)

= 'iTd/v,, = k T d / a b ~ ~ = kT/2roDb, (17) on assuming t h a t a = b = d = 2 r 0 . Thus t h e v i s c o s i t y f o r GB s l i d i n g i s c o r r e l a t e d w i t h t h e d i f f u s i o n c o e f f i c i e n t a l o n g g r a i n boundaries. I f Dbo i s t h e d i f f u s i o n c o n s t a n t and H t h e a c t i v a t i o n e n t h a l p y a s s o c i a t e d w i t h GB d i f f u s i o n , t h e n we have Tpexp(H/Tp) = 4 . ~ 6 9 ~ ( 2 r o ) 2 ~ b o / ~ ( ~ - ~ ) k ( 2 n f ) . (18) where T i s t h e peak t e m p e r a t u r e o f t h e GB I F peak and f i s f r e q u e n c y o f

P

v i b r a t i o n . The T v a l u e s c a l c u l a t e d a c c o r d i n g t o Eqn. (18) f o r a number D

o f pure m e t a l s &e summerieed i n Table 1 w i t h e x p e r i m e n t a l v a l u e s . The v a l u e s o f Dbo and H were i n g e n e r a l t a k e n from Ref./lO/. D a t a f o r Al,Cu, Au were e s t i m a t e d a c c o r d i n g t o t h e e m p i r i c a l f o r m u l a g i v e n i n ~ e f . / l l / . I n c a s e s when

1

and f a r e n o t known, we assumed t h a t f = 1Hz and

1

=

0.05 mm t a k i n g account o f t h e f a c t t h a t

Q

may be s m a l l e r t h a n t h e a c t u a l g r a i n s i z e i f t h e boundary i s n o t f l a t and smooth enough so t h a t l e d g e s and p r o t r u s i o n s may a c t as o b s t r u c t i n g s i t e s f o r GB s l i d i n g . A s t h e o r i - g i n s of t h e e x p e r i m e n t a l d a t a a r e q u i t e d i v e r s i f i e d and t h e e x p e r i m e n t a l v a l u e s a r e v a r i e d a c c o r d i n g t o e x p e r i m e n t a l c o n d i t i o n s and specimen pur- i t y , t h e agreement between e s t i m a t e d v a l u e s and e x p e r i m e n t a l v a l u e s shown i n Tab. 1 seems t o be q u i t e s a t i s f a c t o r y . For A l , Sn and some o t h e r m e t a l s , t h e e s t i m a t e d v a l u e s a r e much lower t h a n t h e e x p e r i m e n t a l v a l u e s and t h e s e may be connected w i t h an i m p u r i t y e f f e c t .

V.- D i s c u s s i o n . I n m e t a l s c o n t a i n i n g s o l u b l e i m p u r i t i e s , it i s w e l l known t h a t t h e r e i s a s e l e c t i v e s e g r e g a t i o n a l o n g c e r t a i n b o u n d a s i e s

/12/. If t h e v i s c o s i t y a s s o c i a t e d w i t h t h e inhomogeneous s l i d i n g i n t h e Tab. l . Comparison between e s t i m a t e d and e x p e r i m e n t a l v a l u e s o f T

P

p u r e s o l v e n t r e g i o n of t h e boundary i s

q l ,

and t h a t i n t h e s o l u t e r e g i o n i s

T 2 ,

t h e n Eqn. ( 6 ) s t i l l h o l d s b u t C w i l l depend upon

4 .

T h i s w i l l

-

g i v e r i s e t o a s o l u t e I F peak a t a t e m p e r a t u r e above t h e o r i g i n a l s o l - v e n t peak i f i s l a r g e r t h a n

ql.

The s o l u t e peak w i l l appear at a tem- p e r a t u r e below t h a t of t h e s o l v e n t peak o n l y when p r e c i p i t a t i o n h a s o c c u r r e d a t t h e g r a i n boundaries i n which c a s e

q2

may be s m a l l e r t h a n

7,.

T h i s may be t h e c a s e observed i n Cu c o n t a i n i n g B i /13/.

Although d i s l o c a t i o n s o f a-type and c-type shown i n Fig. 2 can

(7)

C5-456 JOURNAL DE PHYSIQUE

s t r e s s a r e d i f f e r e n t a t t h e two s i d e s o f t h e b o u n d a r i e s and it can be shown t h a t t h e m i g r a t i o n r a t e o f t h e boundary toward t h e s i d e h a v i n g a

s m a l l e r s h e a s modulus (e.g./$) under t h e a c t i o n o f a s h e a r s t r e s s i s

v, = ( D b h T ) (1/M1- 'C2ab

*

v/, (l/,U1- i / & ) 7 < < v

.

( 1 9 ) T h i s s t r e s s - a s s i s t e d GB m i g r a t i o n r a t e v, may cause a h i g h t e m p e r a t u r e I F background o r a n o t h e r GB peak which i s a m p l i t u d e dependent. But i t

can be s e e n from Eqn.(lg) t h a t t h e c o n t r i b u t i o n o f GB m i g r a t i o n t o GB i n t e r n a l f r i c t i o n should be much s m a l l e r t h a n t h a t o f GB s l i d i n g . R e f e r e n c e s

/ l / T.S. K k , Phys. Rev., 71,533(1947).

/2/ J. Woirgard, J. ~ m i r a r i t and J. de Pouquet, Proc. 5 - I C I F U A C S (Aachen, 19751, Vol.1, p.392.

/3/ E. B o n e t t i , E. Evangelists, P. Gondi and R. T o m a t o , I1 Nuovo Cimento, 3JB,408(1976).

/4/ H. G l e i t e r and B. Chalmers, High-Angle Grain Boundaries, i n "Pro- g r e s s i n M a t e r i a l s S c i e n c e (Pergamon P r e s s , 1 9 7 2 ) , Vol. 16.

/5/ T.S.

xQ,

J. appl. Phys., 20, 274(19 9 ) . /6/ R.F. Mott, Proc. Phys.

SO^,

60,991q1948). /7/ ~ e f . / 4 / , p.12.

/8/ E. KriSner, Kontinumstheorie d e r Verzetzung und I n n e r Spannung ( s p r i n g e r Verlag, 1955).

/ g / P. I s h i d a , S. Okamoto and S. Hachiza, Acta m e t a l l . , 2 , 6 5 1 (1978). /10/ ~ e f . / 4 / , p.93, p:222.