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MODEL FOR THE ULTRASONIC ATTENUATION AND VELOCITY DEFECT DUE TO GEOMETRIC AND THERMAL KINKS UNDER THE ACTION OF
A SLOWLY VARYING BIAS STRESS
M. Bujard, G. Gremaud
To cite this version:
M. Bujard, G. Gremaud. MODEL FOR THE ULTRASONIC ATTENUATION AND VELOC-
ITY DEFECT DUE TO GEOMETRIC AND THERMAL KINKS UNDER THE ACTION OF A
SLOWLY VARYING BIAS STRESS. Journal de Physique Colloques, 1983, 44 (C9), pp.C9-673-C9-
678. �10.1051/jphyscol:19839101�. �jpa-00223334�
MODEL FOR THE ULTRASONIC ATTENUATION AND VELOCITY DEFECT DUE TO GEOMETRIC AND THERMAL KINKS UNDER THE ACTION OF A SLOWLY VARYING BIAS STRESS
M. Bujard and G. Gremaud
I n s t i t u t de Ge'nie Atomique, Swiss FederaZ I n s t i t u t e o f Technology, PHB-EcubZens, CH-1015 Lausanne, Switzerland
Rdsumd
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L ' a t t d n u a t i o n e t l e d d f a u t de v i t e s s e u l t r a s o n o r e s s o n t c a l c u l d s en f o n c t i o n du nombre de ddcrochements e t d'une c o n t r a i n t e s t a t i q u e dans l e mo- d e l e de l a chaene de dbcrochements. L ' i n f l u e n c e de l a f o r m a t i o n de p a i r e de dbcrochements ( r e l a x a t i o n de Bordoni) s u r l e s parametres u l t r a s o n o r e s e s t considdrde.A b s t r a c t
-
The u l t r a s o n i c a t t e n u a t i o n and v e l o c i t y d e f e c t a s a f u n c t i o n of k i n k number and b i a s s t r e s s a r e c a l c u l a t e d from t h e kink c h a i n model. Thermal k i n k s a r e considered, which allows t o p r e d i c t t h e changes of t h e u l t r a s o n i c parameters when k i n k p a i r f o r m a t i o n mechanism occurs (Bordoni's r e l a x a t i o n ) .1.
INTRODUCTION
The low frequency Bordoni's r e l a x a t i o n has been s t u d i e d f o r a long time. I t i s c u r r e n t l y e x p l a i n e d by a mechanism of t h e r m a l l y a c t i v a t e d f o r m a t i o n of k i n k p a i r s ( l a t e r r e f e r r e d a s thermal k i n k s ) on t h e d i s l o c a t i o n s [1,21.
An experimental v e r i f i c a t i o n of t h i s i n t e r p r e t a t i o n could b e p o s s i b l e by u s i n g a measurement technique r e c e n t l y developped by Gremaud
r31.
T h i s technique c o n s i s t s i n a p p l y i n g t o t h e sample a low frequency s i n u s o i d a l s t r e s s i n o r d e r t o s t i m u l a t e t h e mechanism of k i n k p a i r formation, which i s simultaneously measured by t h e indu- ced changes of t h e a t t e n u a t i o n (a) and t h e v e l o c i t y d e f e c t (Avlv) of u l t r a s o n i c wa- v e s ( b i a s s t r e s s experiment).U n t i l now, t h e measurements of u l t r a s o n i c a t t e n u a t i o n and v e l o c i t y d e f e c t due t o t h e d i s l o c a t i o n kinks were i n t e r p r e t e d by t h e model of Suzuki and a l . [41 o r t h e model of A l e f e l d !51, which a r e both based on t h e geometric k i n k c h a i n model of See- g e r and a l . r61. The S u z u k i ' s model can be a p p l i e d o n l y when t h e kink c h a i n i s s t a - t i c s t r e s s f r e e ; t h e A l e f e l d ' s one c a n t a k e a s t a t i c a p p l i e d s t r e s s i n t o account, b u t t h e c a l c u l a t i o n s u s e a mean d i s t a n c e between t h e kinks of t h e c h a i n , what i s on- l y v a l i d f o r v e r y small s t a t i c s t r e s s e s .
The purpose of t h i s paper i s t o p r e s e n t a new c a l c u l a t i o n of t h e u l t r a s o n i c a t t e n u a t i o n and v e l o c i t y d e f e c t t a k i n g more r e a l i s t i c d i s t a n c e s between k i n k s i n t o account, when a b i a s s t r e s s i s a p p l i e d on t h e k i n k c h a i n . One c o n s i d e r s a l s o t h e c a s e where thermal kinks a r e coming on. T h i s allows t o p r e d i c t t h e u l t r a s o n i c a t t e - n u a t i o n and v e l o c i t y d e f e c t a s a f u n c t i o n of a low frequency s i n u s o i d a l s t r e s s ap- p l i e d on a sample near t h e temperature of t h e low frequency Bordoni's r e l a x a t i o n .
F i r s t , one develops t h e geometric k i n k c h a i n model (52) by e s t a b l i s h i n g t h e d i s t a n c e s between kinks when a s t a t i c s t r e s s i s a p p l i e d and by showing how a d i s l o - c a t i o n w i t h thermal k i n k s can be d e s c r i b e d as composed by two independent k i n k c h a i n s . Secondly (53) t h e u l t r a s o n i c a t t e n u a t i o n and v e l o c i t y d e f e c t a r e c a l c u l a t e d a s a f u n c t i o n of t h e k i n k number N and of t h e r e s o l v e d s t a t i c a p p l i e d s t r e s s
u .
I n f a c t , t h e k i n k number N and t h e a p p l i e d s t r e s s cs a r enot
independent v a r i a b l e s i n t h e case of thermal k i n k s ; b u t i t i s e a s i e r t o c o n s i d e r them a s independent f o r t h e c a l c u l a t i o n of t h e u l t r a s o n i c a t t e n u a t i o n and v e l o c i t y d e f e c t and t o t a k e t h e r e l a - t i o n between them i n t o account only when t h e a p p l i c a t i o n of t h e low frequency Bordo-Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19839101
C9-674 JOURNAL DE PHYSIQUE
n i ' s r e l a x a t i o n i s considered ( 5 4 ) .
2 . KINK CHAIN MODEL FOR GEOMETRIC AND T H E W N , KINKS
2 . 1 Kink c h a i n model - I n t h i s model, t h e k i n k s a r e p a r t i c l e s w i t h a mass which can only move p a r a l l e l t o t h e c l o s e packed d i r e c t i o n s and between which e l a s t i c f o r c e s a c t .
S e v e r a l a u t h o r s [6,7,81 . . have determined t h e e l a s t i c i n t e r a c t i o n f o r c e between two k i n k s ; w i t h good approximation, one can use :
G ~ ' R a 2
f ( d ) =
--
47TZF
= *R?;Iz a 2w i t h G : s h e a r modulus'; b :Burger's v e c t o r ; a i s p a c i n g between neighbouring P e i e r l s v a l l e y s ; #3 = ( ( l + v ) cos2@ + (1-V) s i n 2 @ ) / ( l - v f ;
v
: P o i s s o n ' s r a t i o ;$ : a n g l e of b a g a i n s t c l o s e packed d i r e c t i o n s ; d : k i n k d i s t a n c e ; E = ~b'#3/47~.
I n t h e s o - c a l l e d nearest-neighbour approximation, t h e e q u i l i b r i u m e q u a t i o n of R t h e Kth kink of a c h a i n submitted t o a s t a t i c s t r e s s cfstat i s t h e n g i v e n by :
~ ~ a ~ ~ a ~
- - - = o. b a
s t a t
2% %+l
: d i s t a n c e between t h e Kth and (K-1)th k i n k s .
rom ( 2 ) , t h e d i s t a n c e s between k i n k s may be c a l c u l a t e d ; t h i s i s f i r s t done f o r t h e geometric k i n k s . Then t h e r e s u l t i s a p p l i e d t o thermal k i n k s .
2.2 Geometric kink c h a i n
-
I n f i g . 1, a geometric kink c h a i n submitted t o a s t a t i c s t r e s s i s r e p r e s e n t e d . S t a r t i n g from t h e e q u i l i b r i u m e q u a t i o n (eq. 2 ) , Lems [ 91
has shown t h a t t h e e q u i l i b r i u m kink d i s t a n c e d may be expressed by : K
w i t h ' t h e geometric r e l a t i o n : Cd = L
K (4)
dN+l : d i s t a n c e from t h e Nth k i n k t o t h e kink on t h e p i n n i n g p o i n t
L : d i s l o c a t i o n l e n g t h
dN+l w i l l become r a p i d l y l a r g e when in- c r e a s i n g t h e s t a t i c s t r e s s ; t h u s L t h e p o s i t i o n of t h e kinks w i l l r a p i d l y
-
d i f f e r from t h o s e occupied when t h e k i n kc h a i n i s s t r e s s f r e e . F i g . 1
- Geometric kink chain submitted t o a s t a t i c s t r e s s .
2.3 Thermal k i n k c h a i n
-
I n f i g . 2, two thermal k i n k c h a i n s a r e used t o d e s c r i b e t h e e q u i l i b r i u m c o n f i g u r a t i o n of a d i s - l o c a t i o n under t h e a c t i o n of a s t a t i c a p p l i e d s t r e s s . The number of k i n k p a i r s i s f i t t e d i n o r d e r t o o b t a i n t h e same s t a t i c deformation t h a n t h a t o b t a i n e d w i t h t h e s t r i n g model of t h e d i s l o c a t i o n . Neglecting t h e a t t r a c t i v e f o r c e between t h e kinks of t h e l a s t c r e a t e d p a i r , which means t h a t t h e two c h a i n s a r e considered a s independent from one a n o t h e r , t h e e- q u i l i b r i u m k i n k d i s t a n c e c a n be expressed by :Fig.
2- Thermal kink ehains; t h e continuous l i n e represents t h e d i s l o c a t i o n Line
configuration i n t h e s t r i n g model.
kink c h a i n s w i t h d i f f e r e n t numbers of k i n k s . 3. ATI'ENUATION AND VEMCITY DEFECT OF ULTRASONIC WAVES
The k i n k c h a i n used i n t h e c a l c u l a t i o n s of t h e u l t r a s o n i c a t t e n u a t i o n and ve- l o c i t y d e f e c t i s a c h a i n of N geometric o r thermal k i n k s . The e q u i l i b r i u m p o s i t i o n r e s u l t s from t h e s t a t i c s t r e s s 0 a p p l i e d t o t h e c h a i n (eq. 3 o r eq. 5 ) . The dynamic s t r e s s
u
a s s o c i a t e d w i t h t h e u l t r a s o n i c wave l e a d s t o a small v i b r a t i o n % of t h e kink dyn around i t s e q u i l i b r i u m p o s i t i o n .Assuming
-
EK
<< dK+l, t h e e q u a t i o n of motion of t h e Kth kink i n t h enearest-neighbour approximation can be expressed a s :
~ a+ ~i ~a= b i a ~ a + E a Z k E K - 1
dyn
R
w i t h M : k i n k mass p e r u n i t l e n g t h ; B : k i n k phonon drag c o e f f i c i e n t p e r u n i t l e n g t h This s e t of N coupled e q u a t i o n s can be reduced t o one by u s i n g t h e a r e a A = a C
%
swept by a l l t h e kinks :M X + B A + K A = N b 0 dyna (7)
i n which K i s a r e s t o r i n g c o e f f i c i e n t due t o t h e e l a s t i c i n t e r n a l energy s t o r e d du- r i n g t h e dynamic k i n k c h a i n motion; i t i s found by a f i r s t o r d e r development of t h e e q u i l i b r i u m e q u a t i o n (eq. 3 f o r geometric kinks, eq. 5 f o r thermal k i n k s ) :
w i t h : i i
X. = C (N-K+1)
q
and Z = C$;
ZN+l + m f o r thermal k i n k s .K = 1 i
Eq. 7 l e a d s immediately t o t h e u l t r a s o n i c a t t e n u a t i o n
a
(eq. 9) and v e l o c i t y d e f e c t Av/v (eq. 10),
i n whicha
i s i n d ~ / y s .AG/G = A tl-W2/W;) = 2 Av/v (10)
( ~ - W ~ / W ; ) ~ + ( W T ) ~
w i t h A : r e l a x a t i o n s t r e n g t h ; T : r e l a x a t i o n time;
wo
a n g u l a r frequency of resonnan- c e ;w
: u l t r a s o n i c a n g u l a r frequency; L : d i s l o c a t i o n l e n g t h ; A : d i s l o c a t i o n den- s i t y .Ab2a
A = G - T = - B =
K
KL K M
For geometric k i n k s , K i s g i v e n by eq. 8; f o r thermal k i n k s , K i s given by eq. 8 w i t h ZN+l 9
The u l t r a s o n i c a t t e n u a t i o n due t o t h e geometric kinks d e c r e a s e s immediately when t h e s t a t i c a p p l i e d s t r e s s i n c r e a s e s . This i s v e r y d i f f e r e n t from t h e r e s u l t s of A l e f e l d i n which t h e r e i s no change f o r small s t a t i c a p p l i e d s t r e s s . This d i f f e - rence of behaviour i s e s s e n t i a l l y due t o t h e f a c t t h a t A l e f e l d c o n s i d e r s only a mean d i s t a n c e d (d=L/(N+l)) between t h e k i n k s i n s t e a d of t h e r e a l d i s t a n c e
4
(eq.3 ) , as done i n t h i s paper.
The most o r i g i n a l p a r t of t h e p r e s e n t model i s t h e t r e a t m e n t , u s i n g t h e k i n k c h a i n model, of t h e u l t r a s o n i c a t t e n u a t i o n and v e l o c i t y d e f e c t due t o thermal k i n k s .
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DE
PHYSIQUEFor. t h e overdamped case (w T << I ) , f i g 3 shows t h e u l t r a s o n i c a t t e n u a t i o n a as a f u n c t i o n of t h e s t a t i c a p p l i e d s t r e s s 5 and t h e number of thermal k i n k s N.
s t a t
For a f i x e d v a l u e of t h e s t a - t i c s t r e s s , t h e i n c r e a s e of t h e a t t e n u a t i o n a s a f u n c t i o n of N i s explained by an i n c r e a - s e of both t h e swept a r e a and t h e r e l a x a t i o n time.
For a f i x e d v a l u e of t h e kink number, t h e d e c r e a s e of t h e a t - t e n u a t i o n a s a f u n c t i o n of 5 i s e x p l a i n e d by an hard-
s t a t
enlng mechanism due t o t h e de- c r e a s e of t h e d i s t a n c e s b e t - ween k i n k s .
With eq. 9,one c a n e a s i l y show t h a t t h i s d e c r e a s e i s propor- t i o n n a l t o
0-3
s t a t '
A s i m i l a r behaviour i s o b t a i n - ed f o r t h e v e l o c i t y d e f e c t Av/v, w i t h a d e c r e a s e propor- t i o n n a l t o f o r a f i x e d
s t a t '
v a l u e of t h e k i n k number.
Comparing q u a l i t a t i v e l y t h e u l - t r a s o n i c a t t e n u a t i o n a r i s i n g from t h e s t r i n g model, one c a n s e e , a t t h e c o n t r a r y . o f t h e ge- n e r a l l y admitted i d e a , t h a t i n both c a s e s t h e a t t e n u a t i o n in- c r e a s e s w i t h i n c r e a s i n g v a l u e s of t h e s t a t i c s t r e s s i f t h e temperature i s h i g h enough t o have a kink p a i r f o r m a t i o n me- chanism.
Fig. 3
-
Ultrasonic attenuation a versus thermal kink numberN
and applied static stressa
feq. 91.stat
4. APPLICATION OF THE MODEL TO THE STUDY OF THE BORDONI'S RELAXATION
I n o r d e r t o o b t a i n t h e curves a ( a ) and Av/v(a) which have t o be experimental- l y observed near t h e low frequency Bordoni's r e l a x a t i o n , a r e l a t i o n N ( a , T, t ) must b e i n t r o d u c e d i n t h e g e n e r a l r e l a t i o n s a(N, a ) and Av/v (N,
a)
e s t a b l i s h e d f o rthermal k i n k s i n t h e p r e c e d i n g paragraph.
N(o, T, t ) i s t h e number of thermal k i n k s expressed a s a f u n c t i o n of t h e ap- p l i e d s t r e s s
a,
t h e temperature T and t h e time t ; i t could b e deduced from t h e works of Stadelman [ I ] and Esnouf [ 2 ] . But t h e complexity of t h i s r e l a t i o n l e a d s ust o p r e f e r a q u a l i t a t i v e approach by chosing a l i n e a r f u n c t i o n between N and t h e a p p l i e d s t r e s s
a .
Taking s t i l l i n t o account t h e f a c t t h a t , f o r u l t r a s o n i c f r e q u e n c i e s s m a l l e r t h a n 30 MHz, experiments show t h a t t h e d i s l o c a t i o n s a r e overdamped (w T << 1 ) 1101, one o b t a i n s t h e p l o t s of f i g . 4.
Three s t a g e s a s a f u n c t i o n of t h e s t r e s s a can be d i s t i n g u i s h e d : I) A t t e n u a t i o n i s d e c r e a s i n g , v e l o c i t y i s i n c r e a s i n g .
This i s t h e r e s u l t of t h e hardening mechanism b e t - ween p r e - e x i s t i n g geometric k i n k s .
11) A t t e n u a t i o n and v e l o c i t y a r e b o t h d e c r e a s i n g . This s i n g u l a r s t a g e i s due t o t h e f a c t t h a t t h e harden-
111) A t t e n u a t i o n i s i n c r e a s i n g , v e l o c i t y i s d e c r e a s i n g . Here t h e i n c r e a s e of t h e kink p a i r number over-
comes t h e hardening e f f e c t s .
For i n c r e a s i n g temperatures, t h e s l o o p of t h e curve N=N(a) has t o i n c r e a s e due t o t h e thermal a c t i v a t i o n of t h e k i n k p a i r c r e a t i o n mechanism, s o t h a t t h e c r i t i c a l s t r e s s e s , f o r which t h e a t t e n u a t i o n b e g i n s t o in- c r e a s e o r t h e v e l o c i t y b e g i n s t o d e c r e a s e , w i l l b e lowered.
D e t e r r e [I11 h a s made experiments on 6N aluminium w i t h b i a s s t r e s s i n c r e a s i n g l i n e a r l y w i t h time. His r e s u l t s a r e q u i t e s i m i l a r t o those p l o t t e d i n f i g . 4 : t h e extrema of t h e curves Aa(a) and A ( A v l v > ( ~ ) have been observed and t h e c r i t i c a l s t r e s s e s of t h e s e extrema d e c r e a s e when t h e temperature i n c r e a s e s .
Fig.
4Behaviour o f
Aa5
a ( a l-
a ( o o )and
A(&v/v)+
Au/v(o)-
Av/v(oo)versus s t a t i c applied s t r e s s a i n pre-
s t a t sence o f
t h e kink pair creation mechanism.
No :
pre-existing geometric kink number on t h e d i s l o c a t i o n a.
:pre-existing applied s t a t i c
s t r e s s
5.
CONCLUSION
The good agreement between t h e experimental r e s u l t s of D e t e r r e and t h e theo- r e t i c a l p r e d i c t i o n s of t h e above-described model shows t h a t t h i s model i s q u i t e a- dapted t o i n t e r p r e t t h e u l t r a s o n i c parameter changes due t o b o t h t h e geometric k i n k s o r t h e thermal kinks d u r i n g b i a s s t r e s s experiments.
But t h e u s e of a low frequency s i n u s o i d a l b i a s s t r e s s i n s t e a d of a time l i n e - a r l y v a r y i n g one's ( a s done by D e t e r r e ) , w i l l be more i n t e r e s t i n g f o r two reasons :
1 ) t h e measurements a ( a ) and Av/v(a) a r e made i n a s t a t i o n n a r y s t a t e ; 2) d i r e c t comparison w i t h t h e i n t e r n a l f r i c t i o n r e s u l t s a r e t h e n p o s s i b l e . By i n t r o d u c i n g t h e p r e c i s e r e l a t i o n between t h e thermal k i n k number and a harmonic a p p l i e d s t r e s s , which h a s been o b t a i n e d by Esnouf and Stadelmann, t h i s model i s a b l e t o p r e d i c t a c c u r a t e l y t h e " s i g n a t u r e r r curves Aa(a) and A(Av/v) ( a ) which have t o b e observed on t h e Bordoni's r e l a x a t i o n d u r i n g low frequency harmonic b i a s s t r e s s experiments (coupling t e c h n i q u e ) .
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