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STUDIES ON EARLY AND LATE STAGES OF FORMATION OF δ’ PHASES IN Al-Li ALLOYS BY
NEUTRON SMALL ANGLE SCATTERING
S. Fujikawa, M. Furusaka, M. Sakauchi, K. Hirano
To cite this version:
S. Fujikawa, M. Furusaka, M. Sakauchi, K. Hirano. STUDIES ON EARLY AND LATE STAGES OF FORMATION OF δ’ PHASES IN Al-Li ALLOYS BY NEUTRON SMALL ANGLE SCATTERING.
Journal de Physique Colloques, 1987, 48 (C3), pp.C3-365-C3-372. �10.1051/jphyscol:1987342�. �jpa-
00226573�
JOURNAL DE PHYSIQUE
Colloque C3, supplkment au n09, Tome 48, septembre 1987
STUDIES ON EARLY AND LATE STAGES OF FORMATION OF 6 ' PHASES IN A1-Li ALLOYS BY NEUTRON SMALL ANGLE SCATTERING
S. FUJIKAWA
,
M. FURUSAKA",
M. SAKAUCHI and K. HIRANOTohoku
University, Departmento f
Materials Science, J-Sendai 980, Japan" ~ o h o k u
University, Physics Department, J-Sendai 980, JapanA b s t r a c t
The phase s e p a r a t i o n process i n A1-9.5 and 11.4 a t . % L i a l l o y s a ed a t 423 K has been s t u d i e d o v e r a wide q ( = ( h s i n ~ ) / i ) range ( 0 . 0 6 r q r 6 nm-1) w i t h tRe p u l s e d c o l d n e u t r o n s m a l l a n q l e s c a t t e r i n q i n s t r u m e n t a t N a t i o n a l L a b o r a t o r v f o r Hiqh Enerqv Physics. A t v e r y e a r l y s t a e- c a t t e r j n g f u n c t i o n p t t h e h i g h
$
s i d eii
n o t o i i y p r o p o r t i o n a l t o q-2, q-4 an!4-6.
b u t i t obeys n o n - ~ n t e g e r p we 0 g . The power changes c o n t i n u o u s l y from q-1.5 o r q-2-t.o q - 6 - i n v e r y s h o r t t i m e . A t t h e l a t e s t a g e , t h e s c a t t e r i n g f u n c t i o n c o n s i s t s o f q-4and qadependences. The v e r y i n t e r e s t i n g r e s u l t s can n o t be s a t i s f a c t o r i l y e x p l a i n e d by t h e e x i s t i n g t h e o r i e s . The phase s e p a r a t i o n a t t h e v e r y e a r l y s t a g e t a k e s p l a c e by t h e s p i n o d a l mechanism, b u t some d e v i a t i o n s a r e observed f r o m t h e l i n e a r s p i n o d a l t h e o r y . A t t h e l a t e stage, t h e dynamic s c a l i n g l a w proposed by Furukawa and L e b o w i t z d e s c r i b e s w e l l t h ee x p e r i m e n t a l r e s u l t s .
I n t r o d u c t i o n
I t i s w e l l known t h a t t h e m e t a s t a b l e A13Li ( 6 ' ) phases i n A1-Li a l l o y s a r e r a p i d l y formed and t h e y t e n d t o be s h e r i c a l because o f t h e s m a l l coherency s t r a i n . However, i t i s s t i l l u n c e r t a i n whether 6 ' phases a r e formed b y s p i n o d a l decomposition, by n u c l e a t i o n and growth, o r by t r a n s f o r m a t i o n o f o t h e r m e t a s t a b l e phases i n
c o n t r o v e r s y . G e n e r a l l y , i t i s d i f f i c u l t t o i n v e s t i g a t e q u a n t i t a t i v e l y t h e mechanism o f t h e e a r l y s t a g e o f f o r m a t i o n o f 6 ' phases u s i n g u s u a l methods such as a n e l e c t r o n microscopy, c a l o r i m e t r y and r e s i s t i v i t y measurement. n a d d i t i o n , t h e l a t e s t a g e has been o f t e n s t u d i e d by X-ray s m a l l a n g l e s c a t t e r i n g f l ),(2), b u t i t i s n o t easy t o s t u d y t h e e a r l y s t a g e by t h e method. On t h e o t h e r hand, n e u t r o n s m a l l a n g l e
s c a t t e r i n g i s v e r y u s e f u l f o r t h e s t u d y o f t h e e a r l y s t a g e because o f t h e l a r g e d i f f e r e n c e between n e u t r o n s c a t t e r i n g l e n g t h b ~ 1 and b ~ i . Recently, t h e k i n e t i c s o f phase s e p a r a t i o n o f quenched a l l been o f t e n i s t i a t e d by t h e dyanamical s c a l i n g l a w proposed by Furukawa
?Y?,?n
and ~ebowitzi'F'7.?
t i s v e r y i n t e r e s t i n g t o s t u d y by n e u t r o n s m a l l a n g l e s c a t t e r i n g whether t h e dynamic s c a l i n g l a w h o l d s t o t h e l a t e s t a g e o f f o r m a t i o n o f 6 ' phases o r n o t .I n t h e p r e s e n t work, we g i v e t h e r e s u l t s c o n c e r n i n g t h e t i m e e v o l u t i o n o f t h e s c a t t e r i n g i n t e n s i t y from t h e v e r y e a r l y s t a g e t o t h e l a t e s t a g e o f t h e phase s e p a r a t i o n i n A1-Li a l l o y s u s i n g a s m a l l a n g l e n e u t r o n i n s t r u m e n t . The r e s u l t s were a n a l y z e d f r o m t h e v i e w p o i n t of power l a w o f s c a t t e r i n g f u n c t i o n , t h e s p i n o d a l t h e o r y and t h e dynamic s c a l i n g law.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987342
JOURNAL DE PHYSIQUE
Experimental procedures
The c a s t i n g o t s o f A1-Li a l l o y s c o n t a i n i n g 2.65 mass% (9.5 a t . % ) and 3.24 mass%
(11.4 a t . % ) L i were r e p e a t e d l y h o t r o l l e d and f i n a l l y c o l d r o l l e d t o 2 mm i n t h i c k n e s s . The p l a t e was c u t i n t o s m a l l specimens w i t h t h e s i z e o f 2 x 1 0 x 30 mm f o r t h e s c a t t e r i n g e x p e r i m e n t . These a l l o y s c o n t a i n e d 0.09 mass% Fe, 0.03 mass% S i
,
3 ppm Na, 20 ppm Ca and <1 ppm Na as i m p u r i t i e s . The s c a t t e r i n x p e r i m e n t s were c a r r i e d o u t u s i n g t h e s m a l l a n g l e n e u t r o n s c a t t e r i n g i n s t r u m e n t 1 6 y i n s t a l l e d a t t h e p u l s e d c o l d n e u t r o n s o u r c e a t t h e N a t i o n a l L a b o r a t o r y f o r High Energy Physics (KEK).
The s a m p l e - t o - d e t e c t o r d i s t a n c e was s e l e c t e d t be 1 m f o r t h e e a r l y - t i m e e v o l u t i o n e x p e r i m e n t t o c o v e r t h e q r a n g e o f O..?<q<6 nm-?, 5 m f o r t h e l a t e - t i m e e v o l u t i o n e x p e r i m e n t t o c o v e r 0.06<q<l nm-1. a w i d e range o f wave l e n g t h ( ~ = 0 . 3 % 1 .l nm) o f i n c i d e n t n e u t r o n spectrum was u t i l i z e d t o c o v e r t h e s e wide q ranges. The s c a t t e r i n g d a t a were c o r r e c t e d f o r background, t r a n s m i s s i o n and d e t e c t o r s e n s i t i v i t y as d e s c r i b e d i n e l sewhere(6).
S o l u t i o n t r e a t m e n t was c a r r i e d o u t a t 773 K f o r A1-9.5 a t . % L i a l l o y and a t 823 K f o r A1-11.4 a t . % L i a l l o y i n an a r g o n atmosphere. The specimens were water-quenched and aged a t 423 K i n a n o i l b a t h i n t h e r a n g e o f a g e i n g t i m e between 60 s and 6.06 x 105 s. One measurement was c a r r i e d o u t a t room t e m p e r a t u r e u s i n g 3 o r 4 specimens w h i c h were h e a t t r e a t e d s i m u l t a n e o u s l y . A t s h o r t e r a g e i n g t i m e t h a n 7.2 x 104 s t h e specimens were r e p l a c e d i n an o i l b a t h a t 423 K a f t e r measurements were made, w h i l e a t l o n g e r a g e i n g t i m e t h a n 1.05 x
l o 5
s each specimen was used t o o b t a i n o n l y one datum p o i n t . The a g e i n g t e m p e r a t u r 423 K was s e l e c t e d f r o m t h e f o l l o w i n g reason.A c c o r d i n g t he c a l o r i m e t r i c study?7) and n e u t r o n h i g h - r e s o l u t i o n powder
d i f f r a c t i o n y 8 j on t h e p r e c i p i t a t i o n process i n A l - L i a l l o y s , t h e 6 ' phases a r e m a i n l y formed i n t h e m a t r i x a t 423 K.
R e s u l t s and d i s c u s s i o n
F i g . 1 and F i g . 2 show t h e change o f s c a t t e r i n g i n t e n s i t y a t t h e e a r l y s t a g e o f phase s e p a r a t i o n a t 423 K i n A1-9.5 and 11.4 a t . % i
t
a l l o y s , r e s p e c t i v e l y . Here, t h e i n c r e a s e o f i n t e n s i t y a t s m a l l q r e g i o n (q<0.4 nm- ) i s a t t r i b u t e d t o a s m a l l amount o f l a r g e p r e c i p i t a t e s on g r a i n b o u n d a r i e s . F i g . 3 shows t i m e e v o l u t i o n o f t h e maxi- mum p o s i t i o n qmax and t h e maximum i n t e n s i t y Imax Obtained f r o m t h e r e s u l t s shown i n F i g . 1 and F i g . 2. The q u a l i t a t i v e f e a t u r e s a r e as f o l l o w s : ( 1 ) The maximum p o s i t i o n i n A1-9.5 a t . % L i a l l o y i s s h i f t e d s l o w l y towards s m a l l e r q w i t h a g e i n g t i m e : ( 2 ) The maximum p o s i t i o n i n A1-11.4 a t . % L i a l l o y i s s h i f t e d s l o w l y towards l a r g e r q w i t h a g e i n g t i m e and t a k e s t h e b r o a d maximum; ( 3 ) The maximum i n t e n s i t y i n A1-11.4 a t . % L i a l l o y i s l a r g e r t h a n t h a t i n A1-9.5 a t . % L i a l l o y . I n A1-11.4 a t . % L i a l l o y , a c l e a r peak i s observed even f o r as-quenched s t a t e , i n d i c a t i n g t h a t t h e phase s e p a r a t i o n o c c u r s v r y r a p i d l y ;e
( 4 ) Crossover e f f e c t s p r e d i c t e d by t h e Cahn's s p i n o d a l t h e o r y ( 9 a r e n o t c l e a r l y observed and r a t h e r t h e h i g h q t a i l appears t o merge t o g e t h e r . F i g . 4 shows t h e t y p i c a l r e s u l t s o f t h e l a t e - t i m e e v o l u t i o n o f s c a t t e r i n g i n t e n s i t y , showing t h a t t h e maximum p o s i t i o n and t h e maximum i n t e n s i t y much d i f f e r f r o m t h o s e a t e a r l y stage. F i g . 5 shows t h e Log-Log p l o t o f t h e t i m e e v o l u t i o n o f t h e maximum p o s i t i o n and t h e max~,mum i n t e n s i t y . B o t h parameters obey t h e power law,% x i t ) he a g e i n g t i m e exceeding a b o u t 3.6 x 10 s.
-.
aad I m a x ( t )-
t - a.
I t i s no ed t h a t t h e v a l u e s o f a ' and a " change5
F i g . 6 shows t h e t y p i c a l G u i n i e r p l o t s i n A1-9.5 a t . % L i a l l o y s . F i g . 7 shows t i m e e v o l u t i o n o f G u i n i e r r a d i u s ( R ~ ) p l o t t e d on t h e b a s i s o f t h e t1I3 power l a w . I t i s i n t e r e s t i n g t h a t t h e dependence i s d i v i d e d i n t o two p o r t i o n s .F i g . 8 shows t h e t y p i c a l Log-Log p l o t s o f t h e q dependence o f s c a t t e r i n g i n t e n s i t y a t e a r l y s t a g e i n A1-9.5 a t . % L i a l l o y s . A4 shown i n F i g . 8, t h e s c a t t e r i n g i n t e n s i t y a t h i g h q s i d e obeys t h e f o l l o w i n g e q u a t i o n ,
I ( q ) c q - a ( t ) .
. . . . .. . . . . .. . .
( 11
F i g . 9 shows t h e t i m e e v o l u t i o n o f power a ( t ) i n eq. ( 1 ) a t t h e e a r l y s t a g e o f phase s e p a r a t i o n i n A1-9.5 and 1 1 . 4 a t . % L i a l l o y s . I t i s gound t h a t t h e s c a t t e r i n g f u n c t i o n a r e n o t o n l y p r o p o r t i o n a l t o q-2, q-4 and q- a t h i g h q s i d e o f them, b u t t h e y obey n o n - i n t e g e r fowers o f q. As shown i n F i g . 9, t h e power changes c o n t i n u - o u s l y f r o m q-1.5 o r q- t o q-6 i n v e r y s h o r t t i m e . These v e r y i n t e r e s t i n g r e s u l t s can n o t be now s a t i s f a c t o r i l y e x p l a i n e d by t h e e x i s t i n g t h e o r i e s . The s c a t t e r i n g f u n c t i o n s f o r t h e specimens aged f o r a b o u t 120 s e x h i b i t a q-2 dependence f o r h i g h q values, s u g g e s t i n g t h a t t h e q-2 dependence i s a c h a r a c t e r i s t i c o f t h e e a r l y s t a g e o f phase s e p a r a t i o n . I n t h e A1-11 .4 a t . % L i a l l o y t h e q-2 dependence i s c l e a r l y observed even f o r as-quenched s t a t e , i n d i c a t i n g t h a t t h e e a r l y s t a g e proceeds v e r y r a p i d l y i n con A1-Li a l l o y s . The q-2 dependence o f t h e s c a t t e r i n g f u n c t i o n can be derivedrs0f:TiP9 from t h e k i n e t i c e q u a t i o n o f L a n g e r - B a r - o n - H i l l e r r s
t h e o r y ( 1 2 ) . I t i s found t h a t t h e s c a t t e r i n g f u n c t i o n s i n b o t h a l l o y s aged f o r 600 s obey q-4 dependence, c o r r e s p o n d i n g t o t h e s i t u a t i o n t h a t i n t h i s s t a g e t h e p r e c i p i - t a t e s have d e f i n i t e i n t e r f a c e w i t h t h e m a t r i x . However, t h e q-4 dependence i s r e - p l a c e d by t c a t t e r i n g f u n c t ' w i t h h i g h e r power f o r a g e i n g t i m e t h a n 600 s. I n A1 -Zn a l l o y P P 1
f
and Fe-Cr a l l o y l p 8 ) , q-4 dependence i s observed f o r s u f f i c i e n t l y l o n g p e r i o d , T h e r e f o r e , t h e p r e s e n t r e s u l t seems t o b e unus a l . The s c a t t e r i n g u n c t i o n s o f t h e specimens aged f o r a b o u t 3.6 x 103 s obey t h e q-8 dependence. The q'&dependence o f t h e s c a t t e r i n g f u n c t i o n i s t h o r e t i c a l l y p r e d i c t e d by ~ u r u k a w a ( ~ ) * ( ~ ) , i n d i c a t i n g t h a t c l u s t e r s have i n t r i c a t e s u r f a c e . The e x p ~ ~ ~ ~ e n t a l r e s u l t o f
9-
6 dependence has been r e p o r t e d o n l y i n Fe-34 a t . % Cr a l l o y aged f o r l o n g time a t 773 K. F i g . 1 0 shows t h e t y p i c a l r e s u l t o f Log-Log p l o t o f I ( q ) f o r l a e s t a g e The cI-4 dependence appears a g a i n and t h e s c a t t e r i n g f u n c t ' o n c o n s i s t s o f q-' and q-' dependences. U i t h ageing t i m e t h e c o n t r i b u t i o n o f q-4 dependence i n c r e a s e s , c o r r e s p o n d i n g p r o b a b l y t o t h e s i t u a t i o n t h a t t h e f o r m o f 6 ' phases approaches more s p h e r i c a l . Thus, t h e change o f t h e power l a w o f s c a t t e r i n g f u n c t i o n i n A1-Li a l l o y s i s much more c o m p l i c a t e d t h a n t h o r e t i c a l l y exoected. T h e r e f o r e , t h e more d e t a i l e d i n v e s t i g a t i o n i s necessary.0.8 I I
0
0.7-
Al-9.5 atit.% Li alloy -
0.8 I
40 min I
I
F i g . 1 E a r l y - t i m e e v o l u t i o n of s c a t t e r i n r l F i g . 2 Early-time e v o l u t i o n o f
i n t e n s i t y i n A1-9.5 a t . % L i a l l o p a r l e d s c a t t e r i n q i n t e n s i t y i n A1-11.4 a t . % L i
a t 423 K. a l l o y s aged a t 423 K.
C3-368 JOURNAL DE PHYSIQUE
Fig.. 3 Time e v o l u t i o n o f maximum p o s i t i o n and maximum i n t e n s i t y a t e a r l y s t a g e a t 423 K i n A1-9.5 and 11.4 a t . % L i a l l o y s .
F i g . 4 L a t e - t i m e e v o l u t i o n o f s c a t t e r i n g i n t e n s i t y i n A1-11.4 a t . % L i a l l o y s aged a t 423 K.
As shown i n F i 9, t h e s c a t t e r i n g i n t e n s i t y o f t h e specimens aged f o r 600 s a t 423 K shows t h e q-l'dependence. A c c o r d i n g t o t h e s o - c a l l e d P o r o d ' s law, i f t h e s h a r p i n t e r f a c e i s e s t a b l i s h e d between t h e m a t r i x and p r e c i p i t a t e s , f o r l a r g e v a l u e o f q, t h e p r o d u c t q 4 . 1 ( ~ ) approaches c o n s t a n t , such
l i m q 4 . 1 ( q , t ) = constant:...
Qt" ( 2 )
F i g . 11 shows t h e p l o t s o f q 4 . ~ ( q , t ) versus q4 a t t h e v e r y e a r l y s t a g e o f phase s e p a r a t i o n i n AT-9.5 a t . % L i a l l o y s . W i t h a g e i n g t i m k t h e q 4 . 1 ( q , t ) i n c r e a s e s s t e a d i l y . P o r o d ' s l a w i s s a t i s f i e d a f t e r 600 s . The r e s u l t i n d i c a t e s t h a t t h e i n t e r f a c e between 6 ' phases and m a t r i x i s i n i t i a l l y d i f f u s e
"""",
t 1 " " " I """"I " "[
AI-9.5=tPI0 Li alloy1
AI-9.5 at.'/. Li alloy
F i g . 6 T y p i c a l G u i n i e r p l o t s f o r A1-9.5 a t . % L i a l l o y s aged a t 423 K.
Timels
F i g . 5 Time e v o l u t i o n o f qmax. and Imax i n A1-9.5 a t . % L i a l l o y s aged a t 423 K i n Log-Log p l o t .
F i g . 7 Time e v o l u t i o n o f G i n i e r r a d i u s f o r 1 A1-9.5 a t . % L i a l l o y s i n t Y / 3 p l o t .
n w
-
and sharpens up g r a d u a l l y i n t h e e a r l y cource o f .I phase s e p a r a t i o n t i 1 1 600 s. S i m i l a r r e s u l t s
were obtained f o r t h e A1-11.4 a t . % L i a l l o y s . The r e s u l t i s one o f evidences f o r t h e spinodal decomposition i n A1 - L i a1 l o y s
.
Therefore, we apply the 1 in e a r spinodal t h e o r y ( 9 ) 1 q/nm-I 10
t o our r e s u l t s on t h e very e a r l y stage and
e s t i m a t e t h e i n t e r d i f f u s i o n c o e f f i c i e n t
(8)
a t Fig- O f 423 K i n A1-Li a l l o y . According t o t h e theory, s c a t t e r i n g i n t e n s i t y i n A1-9.5 a t . % t h e time-dependent s c a t t e r i n g i n t e n s i t y has a L i a l l o y s i n Log-Log p l o t .form
I ( q , t ) = I(q,O) exp ( 2 R ( q ) . t ) , . . . - ( 3 )
where I(q,O) i s t h e i n i t i a l s c a t t e r i n g i n t e n s i t y and R(q) i s t h e a m p l i f i c a t i o n fac- t o r g i v e n by
where f " i s t h e second d e r i v a t i v e o f t h e Helmholz f r e e energy o f t h e system p e r u n i t volume. 0 i s t h e f r a c t i o n a l change, ( l / a ) / ( a a / a c), o f t h e l a t t i c e parameter o f t h e s o l i d s o l u t i o n , evaluated a t t h e average composition c
.
K i s t h e energy g r a d i e n t c o e f f i c i e n t . If a l i n e a r spinodal process occurs, t h e aaximum p o s i t i o n qm and t h e crossover p o i n t qc should be i n v a r i a n t w i t h ageing time, having a r e l a t i o n s h i p o f qc/qm = 42. I n t h i s case, a p l o t o f ~ ( q ) / ~ ~ , should be a s t r a i g h t l i n e , w i t h an i n - t e r c e p t , -?f(q) and slope, - ( 2 ~ 8 / f ~ ) . J t h e l i n e a r spinodal the0 y i s a p p l i c a b l e ,f F
i t i s p o s s i b l e t o o b t a i n t h e values of and K, p r o y i d e d f " and
n
Y are known. F i g . 12 shows the l o g a r i t h m o f I ( q ) versus time p l o t t e d f o r d i f f e r e n t q values. It i s noteworthy t h a t a l i n e a r r e l a t i o n s h i p i s observed a t v e r y e a r l y stage. The d e v i a t i o n from t h e l i n e a r i t y occurs a t l o n g e r ageing t i m e than 300 o r 600 s. F i g . 13 shows t h e R(q) versus q p l o t as obtained on t h e b a s i s o f eq. (3) by a least-square method.T h i s curve shows a maximum and extends t o t h e negative value o f R(q), having a v a l u e o f R(q) = 0 a t q = q
.
I t shows t h e c h a r a c t e r i s t i c f e a t u r e s o f spinodal decomposi- tion.The r a t i o q /qmcin t h e present work i s about 2, which excleds t h e value o f 42 p r e d i c t e d by l i n g a r theory. F i g . 14 shows t h e R(q)/q 2 versus q p l o t as obtainedC3-370 JOURNAL DE PHYSIQUE
from F i g . 13.The p l o t c o n s i s t s o f a l i n e a r p 2 r t i o n and,a curved p o r t i o n . I f eq. ( 4 ) i s used f o r t h e l i n e a r o r i o n o f the R(q)/q versus2qk p l o t , t h e value o f -i)(q) 1s equal t o 3.17 x 10-21 m4 i f t h e term o f ( 1
+
( 2 n Y/fl')) i s considered,b
i sn e a r l y equal t o -3 x 0-'I i 2 s - l ; which i s we1 1 c o n s i s t e n t w i t h t h e r e c e n t data
(i)
= 5.0 x 10:21 m2s-I a t 425 K) obtained by e l a s t i c r e c o i l d e t e c t i o n a n a l y s i s (14).Thus, from t h e p r e l i m i n a r y a n a l y s i s we have obtained the experimental evidences t h a t t h e phase separation a t v e r y e a r l y stage i n A l - t i a l l o y s takes p l a c e by the spinodal mechanism. This evidence included: ( a ) a l i n e a r time dependence o f Log I ( q ) , ( b ) t h e occurence o f a negative d i f f u s i o n c o e f f i c i e n t and t h e p l a u s i b l e value o f
B,
( c ) t h e e x i s t e n c e o f t h e d i f f u s e i n t e r f a c e , (d) observation o f t h e c h a r a c t e r i s t i c d5pendeye o f t h e a m p l i f i c a t i o n f a c t o r R(q) on q and ( d ) a l i n e a r dependence o f R(q)/q on q.
However, some d e v i a t i o n s f r o m ' t h e l i n e a r spinodal t h e o r y were observed. Such d e ~ i a - t i o n s from l i n e a r i t y are r e f l e c 3 e d i n the2form o f t h e peak s h i f t o f I ( q ) and t h e c u r v a t u r e i n t h e p l o t o f R(q)/q versus q
.
I n a d d i t i o n , qc/qm r a t i o i s higher than J2. Such a n o n - l i n e a r behavior o f t h e s p i n rocess i L i a110 s must be reex- amined on the s t r i c t t h e o r i e s o f F u r u k a ~ a ( ~ f : 1 ~ 7 , LangerPlfl;nd Cookf15).I n order t o compare t h e s c a t t e r i n g s p e c t r a i n t h e present work w i t h theory, we have c a l c u l a t e d t h e normalized s c a l i n g f u n c t i o n F as f o l l o w ,
5
=
initial stage-
1
-
Al-11.4 at.%Li alloy1
AI-9.5al.%Li alloy. I
Fig. 9 Time e v o l u t i o n of power a ( t ) Fig. 10 Late-time e v o l u t i o n o f s c a t t e r i n g o f eq. (1) f o r A1-9.5 and 11.4 i n t e n s i t y i n A1-9.5 a t . % L i a l l o y s aged a t 423 K at.% L i a l l o y s aged a t 423 K. i n Log-Log p l o t .
Fig. 11 V a r i a t i o n of q o . I ( q , t ) a s a
f u n c t i o n q4 a t t h e v e r y e a r l y s t a g e a t
q lnm-'
423 K i n Al-9.5 a t . % L i a l l o y sF i g . 1 3 A m p l i f i c a t i o n f a c t o r R(q) a s a f u n c t i o n o f q i n A1-9.5 a t . % L i a l l o y s aged a t 423 K.
L
Al -9.5 at.% 4U
alloyj
Fig. 12 Time dependence o f Log I ( q , t )
Fig. 1 4 P l o t o f ~ ( q ) / q Z v e r s u s q 2 f o r i n A1-9.5 a t . % L i a l l o y s aged a t
424 K. A]-9.5 a t . % L i a l l o y s aged a t 423
K.
JOURNAL DE PHYSIQUE
where q l b e i n g t h e f i r s t moment o f t h e s c a t t e r i n g f u n c t i o n , d e f i n e d as
Thus, a p l o t o f F ( q / q l ( t ) ) versus q / q l ( t ) f o r t h e samples aged a t same temperature s h o u l d be independent o f a g e i n g t i m e . F i g . 15 shows t h e t y p i c a l r e s u l t s o f s c a l i n g a n a l y s i s f o r l a t e s t a g e i n A1-9.5 and 1 1 . 4 * a t . % L i a l l o y s . I t i s n o t e d t h a t t h e s c a l i n g l a w h o l d s w e l l and independent o f a e i n g t i m e - a n d L i c o n c e n t r a t i o n . I n t h e case t h e s c a t t e r i n g f u n c t i o n c o n s i s a f ~ ) ~ f q-a and q-6 dependences as shown i n F i g . 10.
A t t h e v e r y e a r l y s t a g e ( ~ ( q ) Q, q-
,
a ( t ! < 4 ) , t h e s c a l i n g l a w does n o t h o l d because o f t h e d i f f u s e i n t e r f a c e . The d e t a i l e d r e s u l t s o f a n a l y s i s w i l l be p u b l i s h e d i n e l sewhere.F i g . 15 S c a l i n g f u n c t i o n F(q/ql ( t ) ) a p p l i e d t o t h e l a t e s t a g e i n A1-9.5 a t . % L i a l l o y s aged a t 423 K.
0 0.5 1 1.5 2 2.5
(14
1Acknowledgements
We a r e g r a t e f u l t o Sumitomo L i g h t M e t a l s I n d . L t d . f o r t h e p r e p a r a t i o n o f specimens.
The p r e s e n t work was supported by G r a n t - i n A i d f o r General Research C (No.62550468) from t h e Japanese M i n i s t r y o f Education, Science and C u l t u r e .
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