HAL Id: jpa-00222131
https://hal.archives-ouvertes.fr/jpa-00222131
Submitted on 1 Jan 1982
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MOLECULAR DYNAMICS STUDIES OF
MARTENSITIC NUCLEATION AND GROWTH IN TWO DIMENSIONS
J. Rifkin, Ph. Clapp
To cite this version:
J. Rifkin, Ph. Clapp. MOLECULAR DYNAMICS STUDIES OF MARTENSITIC NUCLEATION
AND GROWTH IN TWO DIMENSIONS. Journal de Physique Colloques, 1982, 43 (C4), pp.C4-157-
C4-162. �10.1051/jphyscol:1982417�. �jpa-00222131�
(Accepted 9 August 1982)
A b s t r a c t . - S i m u l a t i o n s of f i r s t o r d e r 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 from a s q u a r e t o a t r i a n g u l a r l a t t i c e u s i n g a Morse i n t e r a t o m i c p o t e n t i a l augmented w i t h a n o s c i l l a t o r y t a i l a r e r e p o r t e d . H e t e r o g e n e o u s n u c l e a t i o n s from two t y p e s o f de- f e c t s and a l s o homogeneous n u c l e a t i o n of a p e r f e c t c r y s t a l was s t u d i e d a t two d i f f e r e n t t e m p e r a t u r e s . S u b s t a n t i a l l y d i f f e r e n t n u c l e a t i o n t i m e s and t r a n s - f o r m a t i o n growth p a t t e r n s were o b s e r v e d amongst t h e v a r i o u s c a s e s , and o n e showed t h e development o f a " s t r a i n embryo" p e r c u r s o r .
I n t r o d u c t i o n . - S e v e r a l r e s e a r c h g r o u p s ( 1 - 3 ) have r e c e n t l y s u c c e e d e d i n s i m u l a t i n g f i r s t - o r d e r m a r t e n s i t i c p h a s e t r a n s f o r m a t i o n s u s i n g m o l e c u l a r dynamics methods and c e n t r a l p a i r - w i s e i n t e r a t o m i c p o t e n t i a l s t h a t r e s e m b l e t h o s e of r e a l m e t a l s . Rahman e t a l . ( 1 , 2 ) a c h i e v e d f c c t o b c c t r a n s f o r m a t i o n s ( a s w e l l a s t h e r e v e r s e c a s e ) by a b r u p t l y c h a n g i n g t h e form o f t h e i n t e r a t o m i c p o t e n t i a l a t some moment i n t i m e . They d i d n o t a t t e m p t t o s t u d y t h e n u c l e a t i o n p r o c e s s which o c c u r r e d e x t r e m e l y q u i c k l y . C l a p p and R i f k i n ( 3 ) s y s t e m a t i c a l l y v a r i e d t h e t e m p e r a t u r e t o i n d u c e t h e t r a n s f o r m a t i o n , w h i l e k e e p i n g t h e i n t e r a t o m i c p o t e n t i a l f i x e d . Both BCC + S h e a r e d BCC (Orthorhombic) and BCC -t FCC 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 were s i m u l a t e d , u s i n g a Morse p o t e n t i a l augmented w i t h a s m a l l o s c i l l a t o r y t a i l ( t o p r o v i d e a c e r t a i n mimickry o f F r i e d e l o s c i l l a t i o n s ) . The p o t e n t i a l s used were somewhat d i f f e r e n t i n e a c h c a s e . The a r r a y s i n v o l v e d were a b o u t 400 a t o m s i n s i z e and i n c o r p o r a t e d a s t a c k i n g f a u l t t o s e r v e a s a " n u c l e a t i o n d e f e c t " .
B e f o r e p r o c e e d i n g t o s u b s t a n t i a l l y l a r g e r t h r e e d i m e n s i o n a l a r r a y s and i n t e r - a t o m i c p o t e n t i a l s f u l l y c h a r a c t e r i s t i c o f a c t u a l t r a n s f o r m i n g s y s t e m s , we f e l t t h e s e n s i t i v i t y o f t h e t r a n s f o r m a t i o n b e h a v i o r t o n u c l e a t i n g d e f e c t , t o a r r a y s i z e , and t o t e m p e r a t u r e ( f o r a f i x e d p o t e n t i a l ) s h o u l d b e e v a l u a t e d i n some f a s h i o n . The s i m p l e s t ( a n d most e c o n o m i c a l ) way t o a c c o m p l i s h t h i s a n d , a t t h e same t i m e , t o e a s i l y comprehend t h e r e s u l t s , was t o c a r r y o u t t h e s i m u l a t i o n s i n two-dimensional a r r a y s
( s u r f a c e m a r t e n s i t e ? ) . T h e s e " s u r f a c e m a r t e n s i t e " s t u d i e s a r e r e p o r t e d below.
R e s u l t s . - The t r a n s f o r m a t i o n s t u d i e d was from t h e s q u a r e l a t t i c e t o t h e t r i a n g u l a r ( o r h e x a g o n a l ) l a t t i c e a t c o n s t a n t a r e a f a t o m shown i n F i g . l ( a ) u s i n g t h e i n t e r a t o m i c p o t e n t i a l o f F i g . l ( b ) . T h i s p o t e n t i a l produced t h e v a r i a t i o n of l a t t i c e e n e r g y w i t h homogeneous s t r a i n ( a t OK) d i a g r a m e d i n F i g . l ( c )
.
The homogeneous s t r a i n used was a s h e a r i n g o f e v e r y o t h e r ( 1 0 ) row t o maximum o f o n e - h a l f t h e i n t e r a t o m i c s p a c i n g , w h i l e s i m u l t a n e o u s l y c o m p r e s s i n g t h e d i s t a n c e between rows and e x p a n d i n g t h e s p a c i n g w i t h i n rows t o u l t i m a t e l y a c h i e v e e q u i l a t e r a l t r i a n g l e c o o r d i n a t i o n a t c o n s t a n t a t o m i c a r e a . The p a r a m e t e r s o f t h e p o t e n t i a l were a d j u s t e d t o p r o d u c e a m e t a s t a b l e e n e r g y minimum a t t h e s q u a r e l a t t i c e c o n f i g u r a t i o n , a s t a b l e e n e r g y minimum a t t h e t r i a n g u l a r l a t t i c e c o n f i g u r a t i o n a b o u t k85K p e r atom lower i n e n e r g y , and a n e n e r g y b a r r i e r a l o n g t h i s c o n f i g u r a t i o n a l p a t h o f a b o u t k55K p e r atom a t OK ( k is t h e B o l t z - mann c o n s t a n t ) . T h i s e n e r g y p l o t (and t h e s i m u l a t i o n s ) a r e done s t r i c t l y c l a s s i c a l l y and s o quantum e f f e c t s ( s u c h a s z e r o p o i n t e n e r g i e s ) h a v e been e x c l u d e d .Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982417
JOURNAL DE PHYSIQUE
"OK
I N T E R ~ M ~ C -2'350K
POTENTIAL
i O O K -2550K
L A T ~ I c E ENERGY
U.0 4.0 W A R E TRIANGLE
i b RADIUS
( c ) F i g . 1 ( a ) Two-dimensional s q u a r e and t r i a n g u l a r l a t t i c e s . (b) I n t e r a t o m i c poten-
t i a l used i n s i m u l a t i o n . ( c ) L a t t i c e e n e r g y a l o n g homogeneous t r a n s f o r m a t i o n a t O°K.
The s t a r t i n g c o n f i g u r a t i o n u s e d had 200 atoms i n t h e s q u a r e l a t t i c e e i t h e r ( a ) w i t h o u t d e f e c t s , ( b ) w i t h a s i n g l e vacancy, o r ( c ) w i t h a u n i f o r m d i s p l a c e m e n t o f one h a l f o f t h e a r r a y b y one q u a r t e r o f t h e a t o m i c r e p e a t d i s t a n c e i n t h e d i r e c t i o n of a s q u a r e d i a g o n a l , a " s h e a r d e f e c t " . I n c a s e ( b ) s t a t i c r e l a x a t i o n was c a r r i e d o u t a t OK b e f o r e t h e s i m u l a t i o n was s t a r t e d a t a f i n i t e t e m p e r a t u r e . S i m u l a t i o n s were performed a t 30K and 50K u s i n g t h e same t e c h n i q u e s a s p r e v i o u s l y r e p o r t e d ( 2 ) .
Although we d i d n o t h o w t h e f r e e e n e r g y r e l a t i o n s h i p s a t f i n i t e t e m p e r a t u r e , we e x p e c t e d t h a t t h e OK v a r i a t i o n of e n e r g y w i t h c o n f i g u r a t i o n c o u l d b e used a s a n a p p r o x i m a t i o n of t h e f r e e e n e r g y a t t e m p e r a t u r e s a few t e n s of d e g r e e s above OK. The o n l y u s e we made o f t h i s a s s u m p t i o n was t o e x p e c t t h a t homogeneous n u c l e a t i o n would b e v e r y d i f f i c u l t a t 30K b u t might b e o b s e r v a b l e a t 50K.
We a l s o r e a s o n e d t h a t s i n c e t h e s q u a r e l a t t i c e i s a " l o o s e r packed l a t t i c e "
t h a n t h e t r i a n g u l a r l a t t i c e , t h e f o r m e r would h a v e more v i b r a t i o n a l e n t r o p y , t h u s c a u s i n g t h e s q u a r e l a t t i c e t o become t h e l o w e s t f r e e e n e r g y c o n f i g u r a t i o n of t h e two a t some h i g h t e m p e r a t u r e . T h i s c r o s s - o v e r t e m p e r a t u r e To c o u l d b e e s t i m a t e d by de- t e r m i n i n g t h e phonon f r e q u e n c i e s o f e a c h l a t t i c e a t OK and u s i n g t h e s t a n d a r d harmon- i c v i b r a t i o n a l f r e e e n e r g y formula. T h i s w e p l a n t o do f o r t h e t h r e e d i m e n s i o n a l s i m u l a t i o n s of r e a l s y s t e m s , b u t d i d n o t do f o r t h e s t u d i e s a t hand. The p o i n t o f t h i s d i g r e s s i o n i s t h a t t h e s i m u l a t i o n s c a n b e viewed a s mimicking a s y s t e m t h a t h a s been quenched e x t r e m e l y r a p i d l y from a t e m p e r a t u r e above To i n t h e s q u a r e l a t t i c e
" a u s t e n i t e " regime t o a t e m p e r a t u r e w e l l below To i n t h e t r i a n g u l a r l a t t i c e "marten- s i t e " regime, and t i m e g i v e n f o r t h e s y s t e m t o a t t e m p t t o n u c l e a t e t h e " m a r t e n s i t e "
phase.
F i g . 2 Vacancy a t 30°K f o r ( a ) s t e p 1 0 0 , ( b ) s t e p 3000.
between any p a i r o f atoms h a v i n g a d i s t a n c e between 0.90 and 1.15, which s p a n s t h e s q u a r e l a t t i c e n e a r e s t n e i g h b o r d i s t a n c e (1.0) and t h e t r i a n g u l a r (-1.07). b u t e x c l u d e s second n e i g h b o r s o f e i t h e r l a t t i c e . "Dangling t a i l s " a t t a c h e d t o atom s i t e s i n d i c a t e where t h a t atom was a t s t e p z e r o . It may b e s e e n from F i g . 2 ( a ) t h a t even a t s t e p 1 0 0 a few atoms n e a r t h e v a c a n c y have moved i n t o a t r i a n g u l a r l y c o o r d i n a t e d c o n f i g u r a t i o n . T h i s c o n f i g u r a t i o n p r o p a g a t e s f a i r l y r a p i d l y o u t w a r d s from t h e vacancy a l o n g a s i n g l e atom l i n e r e a c h i n g t h e b o u n d a r i e s o f t h e a r r a y i n 800 s t e p s o r l e s s (not shown), b u t n e v e r s u c c e e d s i n s p r e a d i n g l a t e r a l l y , as i s a p p a r e n t from F i g . 2 ( b ) and t h e l a t t i c e p o t e n t i a l e n e r g y p l o t f o r t h i s c a s e i n F i g . 7. A l l t h e atom p o s i t i o n maps from s t e p 8 0 0 t o s t e p 3000 show o n l y minor f l u c t u a t i o n s i n l o c a l c o n f i g u r a t i o n s . T h i s r e s u l t i s v e r y r e m i n i s c e n t o f o n e o f t h e m a j o r c o n j e c t u r e s o f " s t r a i n embryo t h e o r y " (4) of m a r t e n s i t i c n u c l e a t i o n , i . e . t h a t a r e a r r a n g e m e n t of atom p o s i t i o n s f a v o r a b l e t o 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 n t h e s t r a i n f i e l d o f a s u i t a b l e d e f e c t p r i o r t o a c t u a l growth of t h e m a r t e n s i t i c phase. T h a t t h e " s t r a i n embryo" o b s e r v e d i n F i g . 2 ( b ) c a n a c t u a l l y s e r v e t o t r a n s f o r m t h e e n t i r e l a t t i c e was t e s t e d by upquenching t h e a r r a y t o 50K ( a t 3000 s t e p s ) and s e e t h a t t h e l a t t i c e was more o r l e s s c o m p l e t e l y t r a n s f o r m e d t o t h e t r i a n g u l a r p h a s e i n a b o u t 1000 s t e p s o r more.
F i g u r e s 3 ( a ) and 3 ( b ) shows e a r l y and l a t e s t a g e s o f t h e s i n g l e v a c a n c y a r r a y s i m u l a t e d a t 50K. Although n o t much d i f f e r e n c e i s a p p a r e n t i n comparing t h e e a r l y s t a g e s , t h e t r a n s f o r m a t i o n h a s r u n most of i t s c o u r s e by 3000 s t e p s ( s e e a l s o t h e a p p r o p r i a t e c u r v e i n F i g . 7 ) . By c a r e f u l i n s p e c t i o n of F i g . 3 ( b ) one w i l l s e e t h a t s e v e r a l c r y s t a l l o g r a p h i c o r i e n t a t i o n s ( o r v a r i a n t s ) have formed i n t h e growth p r o c e s s and a m e t a s t a b l e i n t e r f a c e s e p a r a t e s t h e s e v a r i a n t s . S c a t t e r e d s m a l l s t r u c t u r a l d e f e c t s a r e a p p a r e n t a s w e l l . Some f l u c t u a t i o n o f t h e s e i m p e r f e c t i o n s c o n t i n u e s a s t h e s i m u l a t i o n is c o n t i n u e d , b u t t h e a v e r a g e d e f e c t c o n c e n t r a t i o n a p p e a r s t o have s t a b i l i z e d a b o u t a s shown. By c o n s i d e r i n g t h e " d a n g l i n g t a i l s " i n F i g . 3 ( b ) i t i s e a s y t o s e e t h a t d i f f e r e n t r e g i m e s o f c o o p e r a t i v e motion developed t o r e a c h t h e
" f i n a l c o n f i g u r a t i o n " . It may b e n o t e d t h a t i n t h e f i r s t s t a g e s of t h e t r a n s f o r m a - t i o n , p r o p a g a t i o n o f t h e t r i a n g u l a r phase o c c u r r e d outward from t h e vacancy i n
JOURNAL DE PHYSIQUE
F i g . 3 Vacancy a t 50°K f o r ( a ) s t e p 100, ( b ) s t e p 1800.
" l a t h s " moving i n a l l f o u r [ O l ] d i r e c t i o n s a t a p p r o x i m a t e l y t h e same r a t e . When f u l l y e x t e n d e d t h e s e l a t h s grew l a t e r a l l y t r a n s f o r m i n g t h e i n t e r v e n i n g m a t e r i a l . F i g u r e s 4 ( a ) a n d 4 ( b ) show t h e e v o l u t i o n o f t h e a r r a y a t 30K w i t h a s e c o n d t y p e o f n u c l e a t i n g d e f e c t . The l e f t h a l f o f t h e a r r a y was d i s p l a c e d upward u n i f o r m l y by one q u a r t e r of t h e a t o m i c r e p e a t d i s t a n c e i n t h i s d i r e c t i o n Ci.e.,
fi
a014 i f a o i s t h e l a t t i c e para- m e t e r o f t h e s q u a r e l a t t i c e ) . T h i s c o n f i g u r a t i o n was n o t s t a t i c a l l y r e l a x e d b e c a u s eit would s i m p l y r e t u r n t o t h e p e r f e c t s q u a r e l a t t i c e . The r a t i o n a l e t h a t prompted t h e u s e o f t h i s " s h e a r d e f e c t " was t h a t i t would i n i t i a l l y p l a c e t h e two c e n t r a l v e r t i c a l rows o f atoms i n a t r i a n g u l a r a r r a n g e m e n t a p p r o x i m a t i n g t h a t o f t h e e q u i l a t e r a l tri- a n g u l a r c o o r d i n a t i o n of t h e m a r t e n s i t i c p h a s e . A s s u c h , i t c o u l d be e x p e c t e d t o be a more p o t e n t n u c l e a t i n g d e f e c t t h a n t h e s i n g l e vacancy. The r e s u l t s seem t o b e a r t h i s o u t s i n c e n u c l e a t i o n ( F i g . 4 ( a ) ) and r e a s o n a b l y c o m p l e t e t r a n s f o r m a t i o n ( F i g . 4 ( b ) ) o c c u r f o r t h i s d e f e c t a t 30K, w h e r e a s o n l y a " s t r a i n embryo" d e v e l o p s from t h e vacancy
F i g . 4 S h e a r a t 30°K f o r ( a ) s t e p 200, ( b ) s t e p 3000.
a t t h i s same t e m p e r a t u r e . I n F i g . 4 ( a ) i t would a p p e a r t h a t n u c l e a t i o n C e n t e r s a r e p r e s e n t a t b o t h t h e t o p and bottom o f t h e a r r a y , b u t t h e p e r i o d i c boundary c o n d i t i o n s i n f a c t make t h i s j u s t one c o n n e c t e d r e g i o n .
F i g . 5 S h e a r a t 50°K f o r ( a ) s t e p 100, (b) s t e p 3000.
F i g . 6 No d e f e c t a t 5 0 K f o r ( a ) s t e p 600, ( b ) s t e p 1000.
An a r r a y w i t h o u t d e f e c t s o f any k i n d was a l s o r u n a t 30K and 50K t o d e t e r m i n e t h e s u s c e p t i b i l i t y o f t h e system t o homogeneous n u c l e a t i o n . By 3000 s t e p s no n u c l e a - t i o n had o c c u r r e d a t 30K ( n o t shown), b u t n u c l e a t i o n d i d o c c u r a t a b o u t 600 s t e p s a t 50K ( F i g . 6 ( a ) ) and grew o u t f a i r l y r a p i d l y w i t h a g a i n t h e development of competing v a r i a n t s i n d i c a t i n g m u l t i p l e n u c l e a t i o n s i t e s ( F i g . 6 ( b ) ) .
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O T I M E S T E P 3 0 0 2
F i g . 7 L a t t i c e e n e r g y v s . t i m e s t e p f o r s i m u l a t i o n s . Curve f o r v a c a n c y a t 30°K l a b e l e d 30V, s h e a r a t 3 0 K l a b e l e d 30S, e t c .
C o n c l u s i o n s a n d F u t u r e D i r e c t i o n s . - T h e s e two d i m e n s i o n a l s t u d i e s h a v e i m m e d i a t e r e l e v a n c e t o f i r s t o r d e r c r y s t a l l o g r a p h i c t r a n s f o r m a t i o n s t h a t a r e known t o o c c u r i n monomolecular a d s o r p t i o n l a y e r s , a n d a r e b e l i e v e d t o o c c u r a t i n t e r f a c e s s u c h a s g r a i n b o u n d a r i e s , a l t h o u g h a s u b s t r a t e p o t e n t i a l would h a v e t o b e added t o f u l l y s i m u l a t e t h e n a t u r e o f t h e s e s y s t e m s . Of w i d e r s i g n i f i c a n c e , however, t h e s e s t u d i e s c l e a r l y show that f i r s t o r d e r 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 c a n b e s u c c e s s f u l l y s i m u l a t e d , t h a t t h e p o t e n c y o f d i f f e r e n t d e f e c t s c a n b e e v a l u a t e d , a n d t h a t q u i t e d i f f e r e n t c o n f i g u r a - t i o n a l p a t h s d e v e l o p d u r i n g t h e t r a n s f o r m a t i o n d e p e n d i n g o n t h e n a t u r e o f t h e i n i t i a l d e f e c t . T h i s g i v e s u s r e a s o n t o hope t h a t w i t h s u f f i c i e n t e x p e r i e n c e i t w i l l b e p o s s i b l e t o i n f e r t h e o r i g i n a l n u c l e a t i n g d e f e c t s t r u c t u r e f r o m t h e s t r u c t u r e o f t h e l a t e r g r o w t h s t a g e s i n r e a l s y s t e m s w h e r e t h e i n t e r a t o m i c p o t e n t i a l i s w e l l d e t e r m i n e d .
I n t h e i m m e d i a t e f u t u r e we p l a n t o s t u d y t h e e v o l u t i o n o f t h e v i b r a t i o n a l s p e c - t r u m o f t h e s e a r r a y s b o t h p r i o r t o a n d d u r i n g t r a n s f o r m a t i o n t o d e t e r m i n e w h e t h e r l o c a l i z e d s o f t modes ( 5 , 6 ) p l a y a n i m p o r t a n t i d e n t i f i a b l e r o l e i n t h e e a r l y s t a g e dynamics. A l s o , t h e i n t r i g u i n g s u g g e s t i o n h a s b e e n made ( 7 ) t h a t " s o l i t o n s " m i g h t b e i m p o r t a n t i n d i s p l a c i v e t r a n s i t i o n s i n a n h a r m o n i c l a t t i c e s , a l t h o u g h v e r y l i t t l e e v i d e n c e e x i s t s t h a t t h e y c o u l d h a v e a f i n i t e l i f e t i m e i n a n e x p a n d i n g t r a n f o r m a t i o n f r o n t . Our s i m u l a t i o n s t u d i e s s h o u l d p r o v e a n e x c e l l e n t t e s t i n g g r o u n d f o r t h i s im- p o r t a n t q u e s t i o n .
Acknowledgements.- T h e s u p p o r t o f t h i s r e s e a r c h b y NSF-DMR u n d e r g r a n t DMR80-07031 i s v e r y much a p p r e c i a t e d .
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