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MARTENSITIC TRANSFORMATION AND ELASTIC
CONSTANTS
T. Suzuki
To cite this version:
JOURNAL DE PHYSIQUE
Colloque ClO, supplement au n012, Tome 46, ddcembre 1985 page C10-581
MARTENSITIC TRANSFORMATION AND ELASTIC CONSTANTS
T. SUZUKI
Institute
of
Applied Physics, Universityof
Tsukuba, Sakura, Ibaraki305,
JapanA b s t r a c t
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The d e f o r m a t i o n energy b a r r i e r f o r t h e m a r t e n s i t i c t r a n s f o r m a t i o n i s e s t i m a t e d f r o m t h e d a t a o f t h e e l a s t i c c o n s t a n t s . The d e f o r m a t i o n energy b a r r i e r f o r B.C.C + F.C.C m a r t e n s i t i c t r a n s f o r m a t i o n i s , a t most, o f t h e o r d e rof t h e t h e r m a l energy a t room temperature. The d e f o r m a t i o n energy b a r r i e r between B.C.C. and F.C.C can be surmounted by thermal energy a t t e m p e r a t u r e s h i g h e r t h a n MS. The m a r t e n s i t i c t r a n s f o r m a t i o n i s proposed t o be u n d e r s t o o d as t h e f r e e z i n g process o f t h e n o n - l i n e a r l a t t i c e v i b r a t i o n , i.e. s o l i t o n s , i n s t e a d o f t h e s o f t phonons.
I
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INTRODUCTIONM a r t e n s i t i c t r a n s f o r m a t i o n can be d e s c r i b e d as t h e s e l f generated f i n i t e d e f o r m a t i o n of t h e l a t t i c e . The second o r d e r e l a s t i c c o n s t a n t s a r e d e f i n e d as t h e second
d e r i v a t i v e o f t h e d e f o r m a t i o n energy w i t h r e s p e c t t o t h e d e f o r m a t i o n parameter o f t h e l a t t i c e . Hence, t h e anomaly o f t h e second o r d e r e l a s t i c c o n s t a n t s a s s o c i a t e d w i t h t h e m a r t e n s i t i c t r a n s f o r m a t i o n may be expected. The c o m b i n a t i o n o f e l a s t i c con- s t a n t s , C'=(C11-C12)/2, o f some a l l o y s such as In-Ta i s known t o decrease as t h e temperature approaches Ms, t h e m a r t e n s i t i c t r a n s f o r m a t i o n s t a r t temperature, i t s t i l l s t a y s p o s i t i v e a t Ms. Then, t h e q u e s t i o n a r i s e s why t h e l a t t i c e deforms i t s e l f , w h i l e t h e second o r d e r e l a s t i c c o n s t a n t i s d e f i n i t e l y p o s i t i v e .
I t must be remembered t h a t t h e second o r d e r e l a s t i c c o n s t a n t s o n l y g i v e t h e t e s t c r i t e r i o n f o r t h e s t a b i l i t y o f t h e l a t t i c e w i t h r e s p e c t t o t h e i n f i n i t e s i m a l and u n i f o r m d e f o r m a t i o n . The c r i t e r i o n f o r t h e s t a b i l i t y o f t h e l a t t i c e w i t h r e s p e c t t o t h e i n f i n i t e s i m a l and n o n - u n i f o r m d e f o r m a t i o n i s g i v e n by t h e phonon d i s p e r s i o n r e l a t i o n s h i p . The d i f f u s i o n - l e s s second o r d e r s t r u c t u r a l t r a n s f o r m a t i o n observed i n such c r y s t a l s as S r T i 0 3 o r BaTiOg has been e s t a b l i s h e d t o be due t o t h e anomaly i n t h e phonon d i s p e r s i o n r e l a t i o n s h i p , t h e phonon f r e q u e n c y o f a s p e c i f i c wave v e c t o r and p o l a r i z a t i o n become z e r o as t h e t e m p e r a t u r e approaches t h e c r i t i c a l temperature: f r e e z i n g of a s o f t phonon. However, i n t h e case o f m a r t e n s i t i c t r a n s f o r m a t i o n , t h e search f o r a s o f t phonon has always y i e l d e d n e g a t i v e answers /1,2,3/.
A c c o r d i n g l y , t h e t e s t c r i t e r i o n f o r t h e s t a b i l i t y o f t h e m a r t e n s i t i c a l l o y must n o t be t h e one f o r t h e i n f i n i t e s i m a l d e f o r m a t i o n b u t t h e one f o r t h e f i n i t e deformation, u n i f o r m o r non-uniform. T h i s i s c o n s i d e r e d t o be c o n s i s t e n t w i t h t h e f a c t t h a t t h e l a t t i c e d e f o r m a t i o n a s s o c i a t e d w i t h t h e m a r t e n s i t i c t r a n s f o r m a t i o n always has a f i n i t e magnitude.
(30-582 JOURNAL
DE
PHYSIQUEI 1
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FINITE UNIFORM MARTENSITIC TRANSFORMATIONThe c r i t e r i o n f o r t h e s t a b i l i t y o f t h e l a t t i c e w i t h r e s p e c t t o a f i n i t e deformation i s o b t a i n e d by s t u d y i n g how t h e l a t t i c e energy increases w i t h the deformation and by comparing t h e maximum value o f t h e deformation energy w i t h a thermal f l u c t u a t i o n energy o r w i t h a s t r a i n energy associated w i t h l a t t i c e d e f e c t . Clapp has c a r r i e d o u t t h e l a t t e r procedure and introduced t h e i d e a o f t h e s t r a i n spinodal /4/. I n t h i s section, t h e s t a b i l i t y o f t h e l a t t i c e w i t h respect t o a h y p o t h e t i c a l u n i f o r m m a r t e n s i t i c t r a n s f o r m a t i o n i s studied.
Suppose t h a t every atom which i s o r i g i n a l l y l o c a t e d a t ( ~ 1 . ~ 2 ~ ~ 3 ) i s d i s p l a c e d by (ul,u2,u3). The displacements (ul,u2,u3) a r e d e f i n e d i n terms o f t h e t r a n s f o r m a t i o n m a t r i x aij and t h e t r a n s f o r m a t i o n parameter E~ as f o l l o w s :
The usual summation convention w i t h respect t o t h e c o o r d i n a t e s u f f i x e s i s assumed i n Eq. (2.1) as w e l l as i n t h e r e s t o f t h e paper. When E ~ = O , t h e t r a n s f o r m a t i o n parameter represents t h e parent phase. When
~ ~ r 1 ,
i t represents t h e m a r t e n s i t i c phase. When O<E,<~, i t represents t h e i n t e r m e d i a t e s t r u c t u r e between t h e parent phase and t h e m a r t e n s i t i c phase. Equation (2.1) w i t h a given m a t r i x am.. d e f i n e s a h y p o t h e t i c a l u n i f o r m m a r t e n s i t i c . t r a n s f o r m a t i o n . For example, t h e BaihJdeformation from t h e B.C.C. s t r u c t u r e t o t h e F.C.C. s t r u c t u r e i s d e f i n e d i n terms o f t h e f o l l o w - i n g t r a n s f o r m a t i o n m a t r i x a..
1.i
-
\0
where ab and af i n d i c a t e the l a t t i c e constants o f t h e B.C.C. and t h e F.C.C. s t r u c - tures, r e s p e c t i v e l y .
The deformation energy p e r u n i t volume i s g i v e n by t h e continuum e l a s t i c i t y theory i n terms o f the s e r i e s expansion o f t h e Lagrangian deformdtion parameter qij and t h e second CijkR and h i g h e r order e l a s t i c constants Cijkemn--- as f o l l o w s :
where t h e Lagrangian deformation parameter i s d e f i n e d as
The Lagrangian deformation parameter f o r t h e h y p o t h e t i c a l u n i f o r m martensi t i c t r a n s f o r m a t i o n i s c a l c u l a t e d by i n s e r t i n g Eq.(2.1) i n t o Eq.(2.4). I t i s given by
s t r u c t u r e i s e s t i m a t e d t o be 9 . 2 k ~ /5/. However, t h i s i s p r a c t i c a l o n l y when a s i m p l e and r e 1 i a b l e l o c a l p s e u d o p o t e n t i a l i s a v a i l a b l e . A r e a s o n a b l e c o n j e c t u r e a b o u t t h e b e h a v i o r o f t h e d e f o r m a t i o n energy A F ( E ~ ) f o r t h e h y p o t h e t i c a l u n i f o r m m a r t e n s i t i c t r a n s f o r m a t i o n w i l l be made by use o f t h e f o l l o w i n g expression: The f a c t o r s c o n t a i n i n g u i j ' s a r e chosen so t h a t t h e e x p r e s s i o n i s e x a c t f o r t h e h y p o t h e t i c a l u n i f o r m m a r t e n s i t i c d e f o r m a t i o n as f a r as O<cO<<l. The f a c t o r c o n t a i n - i n g E and B i s chosen i n such a way t h a t t h e e x p r e s s i o n has minima a t cO=O and c O = l .
dhen B=2, b o t h minima have equal values. When 6>2, A F ( O ) > A F ( ~ ) . F o r t h e B a i n d e f o r m a t i o n d e f i n e d by Eq.(2.2), Eq.(2.6) i s reduced t o
Thus, t h e h e i g h t o f t h e b a r r i e r d e f o r m a t i o n energy ( p e r u n i t volume) between t h e B.C.C. s t r u c t u r e and t h e F.C.C. s t r u c t u r e i s g i v e n as t h e maximum between E=O and E = l :
For t h e B a i n d e f o r m a t i o n w h i c h conserves volume, a f / a b = 3 f l . Then, t h e h e i g h t o f t h e b a r r i e r ( p e r u n i t volume) between t h B.C.C. and F.C.C. s t r u c t u r e when t h e energy o f b o t h s t r u c t u r e s i s equal, ( c o r r e s p o n d i n g @=2), i s e s t i m a t e d t o be
I n t h i s s e c t i o n , t h e v a l u e o f t h e h e i g h t o f t h e d e f o r m a t i o n energy b a r r i e r i s e s t i m a t e d f o r t h e h y p o t h e t i c a l u n i f o r m m a r t e n s i t i c t r a n s f o r m a t i o n . Any c r y s t a l does n o t t r a n s f o r m u n i f o r m l y i n a macroscopic s c a l e . The h e i g h t o f t h e d e f o r m a t i o n energy b a r r i e r f o r u n i t volume i s enormously l a r g e compared w i t h t h e t h e r m a l f l u c t u a t i o n energy a t room temperature, 3 0 0 k ~ . However, t h e h e i g h t o f t h e d e f o r m a t i o n energy b a r r i e r p e r u n i t c e l l i s n o t l a r g e and sometimes even s m a l l e r t h a n 3 0 0 k ~ . I f one does n o t know how many u n i t c e l l s t r a n s f o r m s s i m u l t a n e o u s l y , one cannot know which d e f o r m a t i o n energy b a r r i e r s h o u l d be compared w i t h t h e thermal f l u c t u a t i o n energy o r w i t h t h e s t r a i n energy a s s o c i a t e d w i t h l a t t i c e d e f e c t . I n o r d e r t o s o l v e t h i s problem, i t becomes necessary t o c a l c u l a t e t h e d e f o r m a t i o n energy i n c r e a s e asso- c i a t e d w i t h t h e f i n i t e n o n - u n i f o r m d e f o r m a t i o n . T h i s i s c a r r i e d o u t i n t h e n e x t s e c t i o n .
I 1 1
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FINITE NON-UNIFORM MARTENSITIC TRANSFORMATIONC10-584 JOURNAL DE PHYSIQUE
i s used t o c a l c u l a t e t h e deformation energy. T h i s i s e s s e n t i a l l y equal t o t h e t r i a l f u n c t i o n used by Green and Cahn i n t h e i r unpublished work. The m a t e r i a l o f t h i s s e c t i o n i s based on t h e unpublished work by M. Green and J.W. Cahn.
The deformation energy increase p e r ' u n i t volume due t o t h e non-uniform martensi t i c t r a n s f o r m a t i o n can be estimated i n t h e continuum approximation by c a l c u l a t i n g t h e Lagrangian deformation parameters
(2.4)
from t h e displacement d e f i n e d byEq. (3.1)
and i n s e r t i n g them i n t o
Eq.(2.3).
The Lagrangian deformation parameter i swhere ci=Xi/A. I n s e r t i n g
(3.2)
i n t o(2.3)
and i n t e g r a t i n g over t h e t o t a l volumeo f
the specimen which c o n t a i n s t h e whole area a f f e c t e d by t h e non-uniform m a r t e n s i t f c t r a n s f o r m a t i o n d e f i n e d by
Eq.(3.1),
we o b t a i nwhere t h e e x p l i c i t expressions f o r f21,f22,f23,f31,---kan be w r i t t e n down b u t they are n e i t h e r necessary n o r u s e f u l , because t h e numerical values o f h i g h e r order e l a s t i c constants a r e n o t known experimentally. What i s e s s e n t i a l i s t h e f u n c t i o n a l dependence o f t h e i n t e g r a l on A and c0. When A+-, t h e displacement d e f i n e d by
Eq. (3.1)
i s reduced t o t h e h y p o t h e t i c a l u n i f o r m martensi t i c t r a n s f o r m a t i o n d e f i n e dby
Eq. (2.1).
Accordingly, t h e i n t e g r a t e d expression o fEq. (3.3)
should be essen-t i a l l y i d e n t i c a l t o
Eq.(2.3).
I t should have a maximum f o r a c e r t a i n v a l u e o f Eand then decrease f o r f u r t h e r increase o f E, i f t h e displacement d e f i n e d by
Eq.(3.1)
represent t h e m a r t e n s i t i c transformation. Hence, the i n t e g r a t e d expression o f
Eq.(3.3)
should behave as a f u n c t i o n o f A and e as shown s c h e m a t i c a l l y i n Fig.1.
The p a r e n t phase i s represented by t h e o r i g i n o f t h e A and E coordinate and t h e
m a r t e n s i t i c phase i s represented by
~ = 1
and A+- i n Fig.1.
The h e i g h t o f t h e b a r r i e r between t h e p a r e n t phase and the m a r t e n s i t i c phase depends on p a t h between them. We can choose a p a r t i c u l a r p a t h as i n d i c a t e d by t h e b o l d l i n e i n Fig.1.
The deformation energy b a r r i e r does n o t e x i s t f o r t h i s path. T h i s may sound l i k e a paradox, because t h e m a r t e n s i t i c t r a n s f o r m a t i o n i s considered t o be a t y p i c a l f i r s t order phase transformation.The argument discussed above i s based on t h e continuum t h e o r y o f e l a s t i c i t y , where the deformation energy increase due t o a f i n i t e deformation i s d e f i n e d by
Eq.(2.3).
S i m i l a r l y , t h e b e h a v i o r o f t h e f r e e energy as shown i n Fig. 1 s h o u l d be a t l e a s t q u a l i t a t i v e l y r i g h t e x c e p t t h e r e g i o n where A becomes t h e o r d e r o f t h e l a t t i c e spacing. Hence, i t i s concluded t h a t t h e s t a b i l i t y o f t h e l a t t i c e w i t h r e s p e c t t o t h e f i n i t e d e f o r m a t i o n a s s o c i a t e d w i t h t h e m a r t e n s i t i c t r a n s f o r m a t i o n s h o u l d be e s t i m a t e d f r o m t h e b a r r i e r h e i g h t f o r t h e h y p o t h e t i c a l u n i f o r m d e f o r m a t i o n p e r u n i t c e l l . A c c o r d i n g l y , a t l e a s t some cases o f t h e B.C.C+F.C.C. m a r t e n s i t i c t r a n s f o r m a - t i o n , t h e d e f o r m a t i o n energy b a r r i e r i s d e f i n i t e l y s m a l l e r t h a n t h e thermal f l u c t u a - t i o n energy. The q u e s t i o n why t h e m a r t e n s i t i c t r a n s f o r m a t i o n can be f i r s t o r d e r even i n such a case i s d i s c u s s e d i n t h e f o l l o w i n g s
.
Zener has p o i n t e d o u t t h a t t h e B.C.C. c o n f i g u r a t i o n i s a s o r t o f saddle p o i n t con- f i g u r a t i o n between t h e F.C.C o r a n o t h e r c l o s e packed c o n f i g u r a t i o n s /6/. The deformation energy b a r r i e r f r o m t h e 5.C.C t o t h e F.C.C. s t r u c t u r e i s expected t o be small f r o m t h e p i c t u r e i n Z e n e r ' s book. The d i s c u s s i o n i n t h e p r e v i o u s paragraph i n d i c a t e s t h a t i t i s l e s s t h a n t h e thermal f l u c t u a t i o n energy a t l e a s t f o r some o f t h e B.C.C. metal and a l l o y s . I n o t h e r words, t h e B.C.C. c o n f i g u r a t i o n i s m a i n t a i n e d s o l e l y because of t h e l a r g e v i b r a t i o n a l e n t r o p y . When a l a t t i c e t a k e s t h e B.C.C c o n f i g u r a t i o n , atoms i n t h e l a t t i c e have a f i n i t e p r o b a b i l i t y o f t a k i n g b o t h B.C.C and F.C.C. m i c r o s c o p i c c o n f i g u r a t i o n s . These c o n f i g u r a t i o n s o f atoms corresponds t o t h e e x c i t a t i o n o f t h e l a r g e amp1 i tude-non-1 i n e a r - t r a n s v e r s e l a t t i c e waves. The s t a t i s t i c a l mechanics o f t h e n o n - l i n e a r t r a n s v e r s e l a t t i c e wave has been f o r m u l a t e d by Krumhansl and S h r i e f f e r i n t h e i r s o l i t o n model / 7 / . The s o l i t o n i s p i c t u r e d as a t r a n s i t i o n r e g i o n between two e q u i v a l e n t s t r u c t u r e s . I n t h e above cases, t h e s o l i t o n s h o u l d be t h e t r a n s i t i o n r e g i o n between t h e two s t a b l e F.C.C. c o n f i g u r a t i o n s . T h i s i s e x a c t l y t h e t w i n i n t h e m a r t e n s i t i c phase. The f o r m u l a t i o n o f t h e t w i n s t r u c t u r e i n terms o f a model, which i s now i d e n t i f i n e d as t h e Sine-Gordon s o l i t o n e q u a t i o n , has been c a r r i e d o u t by Frenkel and Kontrova / 8 / .
However, one m o d i f i c a t i o n i s necessary b e f o r e t h e s o l i t o n model o f Krumhansl and S h r i e f f e r (K-S s o l i t o n model ) i s a p p l i e d t o t h e B.C.C.+F.C.C. m a r t e n s i t i c t r a n s - f o r m a t i o n . Because t h e two n e i g h b o r i n g s t a b l e c o n f i g u r a t i o n s a r e assumed t o have equal p o t e n t i a l energy and t h e s e c o n f i g u r a t i o n s a r e arranged i n a l i n e a r chain, t h e s t r u c t u r a l t r a n s f o r m a t i o n cannot t a k e p l a c e a t a f i n i t e temperature as d i s c u s s e d i n t h e t e x t o f Landau and L i f s c h i t z
/9/.
I n o t h e r words, e i t h e r . o f t h e t w i n cannot be s t a b l e except a t t h e a b s o l u t e z e r o temperature. I n o r d e r t o remedy t h i s s i t u a t i o n , t h e symmetric double w e l l p o t e n t i a l i n t h e K-S s o l i t o n model i s r e p l a c e d by t h e asymmetric double w e l l p o t e n t i a l . The H a m i l t o n i a n o f t h e model used f o r t h e s t u d y of t h e m a r t e n s i t i c t r a n s f o r m a t i o n i s g i v e n byHere, m i s t h e mass o f t h e atom i n t h e model and E i n d i c a t e s t h e s t r e n g t h o f t h e
b i a s f i e l d . I t i s t o be n o t i c e d t h a t when E=O, t h e assymmetric p o t e n t i a l o f Eq.(4.1) i s reduced t o t h e symmetric d o u b l e w e l l p o t e n t i a l w i t h a minimum a t Ui=O and a n o t h e r a t Ui=l.
The p a r t i t i o n f u n c t i o n f o r t h i s model Z i s g i v e n as a p r o d u c t o f t h e k i n e t i c energy p a r t Zp and t h e c o n f i g u r a t i o n a l energy p a r t Zu. The k i n e t i c energy p a r t Zp i s g i v e n by
JOURNAL
DE
PHYSIQUEwhere 0 i s l / k T, Y n t h e n - t h e i g e n f u n c t i o n , En t h e n - t h e i g e n v a l u e s o f t h e t r a n s f e r i n t e g r a l equatBon and a i s t h e l a t t i c e s p a c i n g o f t h e model. The f r e e energy o f t h e F i s c a l c u l a t e d f r o m Zp g i v e n by Eq. (4.3)
The e i g e n v a l u e s o f t h e t r a n s f e r i n t e g r a l Equation, Eq. (4.3), a r e o b t a i n e d by use o f t h e a l g o r i t h m d e s c r i b e d by Schneider and S t o l l
/ l o / .
The f r e e energy o f t h e model i s n u m e r i c a l l y c a l c u l a t e d as a f u n c t i o n o f t h e v a l u e s o f E between 0 and 2.0 f o r t h ev a l u e o f 8=0.2,0.26,0.3 and 0.4 and shown i n F i g . 2. A t s u f f i c i e n t l y h i g h tempera- t u r e , t h e f r e e energy o f t h e model i s minimum when t h e b i a s i s zero. A t l o w e r temperature, t h e model w i t h a f i n i t e v a l u e o f t h e b i a s becomes s t a b l e w i t h r e s p e c t t o t h e model w i t h o u t t h e b i a s . Hence, t h e r e e x i s t s a range o f t e m p e r a t u r e where t h e f r e e energy o f t h e model w i t h o u t b i a s i s equal t o t h a t o f t h e model w i t h t h e b i a s . The model, whose f r e e energy i s shown i n F i g . 2, i s proposed t o r e p r e s e n t t h e mar- t e n s i t i c metal o r a l l o y s . The model w i t h t h e symmetric d o u b l e w e l l (E=O) r e p r e s e n t s t h e p a r e n t phase i n t h e B.C.C.+F.C.C. m a r t e n s i t i c t r a n s f o r m a t i o n . The maximum between symmetric w e l l s corresponds t o t h e B.C.C. c o n f i g u r a t i o n , w h i l e t h e symmetric w e l l s correspond t o t h e F.C.C. c o n f i g u r a t i o n . S t r i c t l y speaking, a s h a l l o w
m e t a s t a b l e minimum may e x i s t around t h e B.C.C. c o n f i g u r a t i o n . B u t t h i s i s con- s i d e r e d t o be n o t so e s s e n t i a l f o r t h e s t a b i l i t y o f t h e B.C.C. c o n f i g u r a t i o n a t h i g h temperature, because t h e d e f o r m a t i o n energy b a r r i e r i s l o w e r t h a n t h e t h e r m a l f l u c t u a t i o n energy. T h i s has been checked o u t b y t h e c a l c u l a t i o n b y M G l l e r and Wilmanski /11/. Hence, i n t h e p r e s e n t d i s c u s s i o n , t h e e x i s t e n c e o f t h e p o s s i b l e p o t e n t i a l minimum i s n e g l e c t e d f o r t h e sake of s i m p l i c i t y . On t h e o t h e r hand, t h e model w i t h t h e asymmetric d o u b l e w e l l ( ~ 4 0 ) corresponds t o t h e twinn'ed m a r t e n s i t i c phase. E i t h e r o f t h e two e q u i v a l e n t F.C.C. c o n f i g u r a t i o n i s s t a b l i t e d by t h e f i n i t e v a l u e o f t h e b i a s E . The b i a s i s c o n s i d e r e d t o be i n t r o d u c e d by t h e m a r t e n s i t i c
t r a n s f o r m a t i o n i t s e l f , because o f t h e f i n i t e amount o f t h e B a i n deformation. T h i s s i t u a t i o n i s c o n s i d e r e d t o be c l o s e l y r e l a t e d t o t h e shape memory e f f e c t o f t h e m a r t e n s i t i c a l l o y s .
A c c o r d i n g t o t h e i n t e r p r e t a t i o n d i s c u s s e d above, t h e b e h a v i o r o f t h e f r e e energy i n Fig. 2 i n d i c a t e s t h a t a t s u f f i c i e n t l y h i g h temperature, t h e p a r e n t phase w i t h o u t s t a b i l i z e d t w i n n e d s t r u c t u r e i s s t a b l e . I t i s t o be n o t e d t h a t a l t h o u g h t h e s t a b i l - i z e d t w i n b o u n d a r i e s a r e n o t p r e s e n t i n t h e p a r e n t phase, t h e s o l i t o n , w h i c h can be i n t e r p r e t e d as t h e moving t w i n boundary, s h o u l d be p r e s e n t i n t h e p a r e n t phase and r e s p o n s i b l e f o r t h e d i f f r a c t i o n e f f e c t such as c e n t r a l peaks d i s c u s s e d i n t h e K-S s o l i t o n model /7/.
w i t h t h e n u c l e a t i o n d i f f i c u l t y .
On t h e o t h e r hand, t h e a p p r e c i a b l e d i f f e r e n c e between A and M i n t h e F.C.C.+B.C.C. m a r t e n s i t i c t r a n s f o r m a t i o n as observed i n Fe-Ni a l l o y i$ c l e a r f y due t o t h e n u c l e a - t i o n d i f f i c u l t y , which has i t s o r i g i n i n t h e d e f o r m a t i o n energy d i s c u s s e d i n sec- t i o n s 2 and 3. The e x p e r i m e n t a l evidence t h a t t h i s n u c l e a t i o n d i f f i c u l t y i s s u r - mounted by a thermal a c t i v a t i o n process has been p r o v i d e d by t h e c l a s s i c a l e x p e r i - ment by Cech and T u r n b u l l /14/, a l t h o u g h t h e t r a d i t i o n a l e x p l a n a t i o n i s e n t i r e l y d i f f e r e n t .
I n c o n c l u s i o n , t h e m a r t e n s i t i c t r a n s f o r m a t i o n i s proposed t o be understood as t h e s t a b i l i z a t i o n process o f s o l i t o n s , w h i c h e x i s t i n t h e h i g h t e m p e r a t u r e as t h e non- l i n e a r l a t t i c e v i b r a t i o n s and become t h e t w i n boundaries a f t e r s t a b i l i z e d . The c e n t r a l peaks on t h e n e u t r o n i n e l a s t i c s c a t t e r i n g experiment have a l r e a d y been proposed t o be a s s o c i a t e d w i t h t h e presence o f s o l i t o n s . The n o n - l i n e a r i n t e r n a l f r i c t i o n experiment, t h e a u t o m o d u l a t i o n phenomena, has been observed i n m a r t e n s i t i c a l l o y s and has been i n t e r p r e t e d t o t h e t w i n n i n g process a s s o c i a t e d w i t h t h e marten- s i t i c t r a n s f o r m a t i o n /15/.
ACKNOWLEDGEMENT
T h i s paper i s completed w h i l e t h e a u t h o r i s v i s i t i n g t h e Department o f M e t a l l u r g i c a l Engineering, U n i v e r s i t y o f M i s s o u r i - R o l l a w i t h t h e s u p p o r t from t h e N a t i o n a l Science Foundation. The a u t h o r acknowledges t h e s t i m u l a t i n g d i s c u s s i o n w i t h Dr. J.W. Cahn and Dr. M. W u t t i g .
JOURNAL
DE
PHYSIQUEFig. 2
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The dependence o f t h e f r e e energy on t h e b i a s f i e l d E f o r d i f f e r e n ttemperatures.
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