NUCLEATION AND CRYSTAL GROWTH IN SOFT-SPHERE SYSTEMS

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NUCLEATION AND CRYSTAL GROWTH IN SOFT-SPHERE SYSTEMS

G. Bushnell-Wye, J. Finney

To cite this version:

G. Bushnell-Wye, J. Finney. NUCLEATION AND CRYSTAL GROWTH IN SOFT-SPHERE SYS- TEMS. Journal de Physique Colloques, 1985, 46 (C3), pp.C3-311-C3-321. �10.1051/jphyscol:1985325�.

�jpa-00224642�

JOURNAL DE PHYSIQUE

Colloque C3, suppl6ment au n03, Tome 46, mars 1985 page C3-311

NUCLEATION AND CRYSTAL GROWTH I N SOFT-SPHERE SYSTEMS

G. Bushnell-Wye and J.L. Finney

Department of CrystaZZography, Birkbeck College, MaZet S t r e e t , London WClE 7 H X , U.K.

Resum6 - P a r t a n t d l u n e c o n f i g u r a t i o n amorphe mod&lisGe, d e 4000 "sphbres-molles", une mdthode d e dynamique m o l 6 c u l a i r e permet d l i l l u s t r e r l a g e r m i n a t i o n s p o n t a n 6 e . L 1 a n a l y s e du p r o c e s s u s d e c r i s t a l l i s a t i o n , 5 t r o i s s t a p e s d e l a simu- l a t i o n , 5 l t a i d e d e s i n d i c e s d e s p o l y s d r e s de Voronoi montre que d e s germes bcc e t f c c c o n t i n u e n t 5 c r o 4 t r e . Des ambi- g u i t 6 s dans l a s t r u c t u r e , a p p a r a i s s a n t pour c e r t a i n s t y p e s d t i n d i c e , p e u v e n t S t r e l e v 6 e s en u t i l i s a n t d e s msthodes g r a p h i q u e s num6riques. C e r t a i n e s de c e s f i g u r e s donnent l t i m p r e s s i o n d l u n changement de p h a s e l o c a l i s 6 d t u n e s t r u c - t u r e bcc en f c c , c e c i demandant 2 S t r e confirm6 exp6rimen- t a l e m e n t .

A b s t r a c t - Spontaneous homogeneous n u c l e a t i o n can be de- m o n s t r a t e d i n a model of 4000 s o f t - s p h e r e s a n n e a l e d from a n amorphous s t a r t i n g c o n f i g u r a t i o n v i a a m o l e c u l a r dynamics method. A n a l y s i s of t h e c r y s t a l l i s a t i o n p r o c e s s a t t h r e e s t a g e s of t h e s i m u l a t i o n u s i n g Voronoi p o l y h e d r a i n d i c e s shows t h a t b o t h b c c and f c c n u c l e i c o n t i n u e t o grow. Methods of d e l i n e a t i n g t h e n u c l e i and t h e e f f e c t s on t h e i r r e s u l t i n g s h a p e and s i z e a r e d i s c u s s e d , and it i s shown how a m b i g u i t i e s i n s t r u c t u r e which a r i s e b o t h f o r p a r t i c u l a r index t y p e s and l o c a l a r r a n g e m e n t s can be r e s o l v e d u s i n g computer g r a p h i c s t e c h n i q u e s . Some of t h e drawings d i s p l a y e d g i v e t h e impres- s i o n of l o c a l i s e d phase change from b c c t o f c c s t r u c t u r e though t h e s e e v e n t s have y e t t o b e c o n f i r m e d .

1 . INTRODUCTION

I n computer s i m u l a t i o n of s u p e r c o o l e d l i q u i d s i t i s r e l a t i v e l y e a s y t o e n s u r e t h a t f o r e i g n b o d i e s a r e excluded and ( u s i n g s u i t a b l e p e r i o d i c boundary c o n d i t i o n s ) t h a t t h e r e a r e no c o n t a i n e r w a l l s , t h e r e b y p r o v i - d i n g an environment conducive t o homogeneous n u c l e a t i o n r a t h e r t h a n h e t e r o g e n e o u s n u c l e a t i o n which i s more p r o b a b l e i n n a t u r e . Even s o , it w i l l n o t o c c u r w i t h o u t f u r t h e r e n s u r i n g t h a t t h e t e m p e r a t u r e and d e n s i t y o f t h e ensemble p r e s e n t no b a r r i e r s t o n u c l e a t i o n . I n t h e a b s e n c e of h e t e r o g e n e o u s s u b s t r a t e s , l o c a l i n h o m o g e n e i t i e s o c c u r by chance f l u c t u - a t i o n s i n s t r u c t u r e and t h e s e c a n form t h e b a s e s o f n u c l e a t i o n . I n t h e s m a l l s y s t e m s f i r s t s t u d i e d , c r y s t a l l i n e s t r u c t u r e s formed o n l y under t h e i n f l u e n c e of boundary c o n d i t i o n s . More r e c e n t l y i t h a s been de- m o n s t r a t e d t h a t t h e phenomenom of s p o n t a n e o u s homogeneous n u c l e a t i o n c a n o c c u r i n a s u p e r c o o l e d Lennard-Jones l i q u i d p r o v i d e d t h e r e a r e no b a r - r i e r s t o n u c l e a t i o n (Mandell, McTague and Rahman (1976) ) .

M o l e c u l a r dynamics c a l c u l a t i o n s on a system of 4000 s o f t - s p h e r e s (Cape,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985325

C3-312 JOURNAL DE PHYSIQUE

Finney and Woodcock(l981)) i n t e r a c t i n g by t h e p o t e n t i a l

have a l s o been r e p o r t e d w h e r e o n c e a g a i n t h e r e was no dependence of t h e c r y s t a l l i t e a x e s upon t h e s i m u l a t i o n b o u n d a r i e s and where b o t h bcc and f c c s t r u c t u r e s c o e x i s t e d . A r a p i d l y quenched l i q u i d (Cape and Woodcock

(1980)) formed t h e g l a s s y s t a r t i n g c o n f i g u r a t i o n which was a n n e a l e d a t c o n s t a n t t e m p e r a t u r e f o r 3230 t i m e s t e p s a t A t = 0.005

.When

u s i n g t h e s o f t - s p h e r e p o t e n t i a l t h e e q u a t i o n s of s t a t e a r e f u n c t i o n s o f a s i n g l e reduced v a r i a b l e

t h e reduced d e n s i t y , and a s s o c i a t e d thermodynamic v a r i a b l e s , such a s P * c a n a l s o be e x p r e s s e d i n d i m e n s i o n l e s s u n i t s , t h u s

P * ( o C )

= - PV o*

(where P i s p r e s s u r e , V i s t h e volume NkT o f t h e s y s t e m , N i s t h e number of p a r t i c l e s , k i s Boltzmann's c o n s t a n t and T i s t h e t e m p e r a t u r e ) . T h i s s i m u l a t i o n a l l o w e d t h e system t o r e l a x t o a s t a t e of lower p o t e n t i a l e n e r g y by m a i n t a i n i n g a c o n s t a n t k i n e t i c e n e r g y (and t e m p e r a t u r e ) and a c o n s t a n t r e d u c e d d e n s i t y o f y* ⁼ 1.831, by r e s c a l i n g t h e v e l o c i t i e s a t e a c h t i m e s t e p . The approach t o e q u i l i b r i u m u n d e r t h e s e c o n d i t i o n s c a n be m o n i t o r e d by t h e d e c r e a s e i n t o t a l i n t e r n a l e n e r g y o r reduced p r e s s u r e , which s h o u l d e v e n t u a l l y become t h a t o f t h e most s t a b l e s t r u c - t u r e f o r t h e s y s t e m , t h e f c c c r y s t a l .

I n a p r e v i o u s p u b l i c a t i o n , Cape e t a 1 (1981) r e p o r t e d t h e p r e s e n c e of b o t h bcc and f c c n u c l e i w i t h i n t h e system a f t e r 3230 t i m e s t e p s ( r e f e r - r e d t o a s s t a g e I below). F u r t h e r m o l e c u l a r dynamics i t e r a t i o n s have been completed f o r t h i s model t o s e e i f t h e n u c l e i would d i s p e r s e o r c o n t i n u e t o grow, and whether f c c n u c l e i would d e v e l o p from b c c n u c l e i , s i n c e a t s t a g e I it was c l e a r t h a t t h e reduced p r e s s u r e was f a r from t h e e q u i l i b r i u m v a l u e ( f i g . 1 ) . ( C l a s s i c a l n u c l e a t i o n t h e o r y p r e d i c t s t h a t t h e b a r r i e r t o n u c l e a t i o n i s minimal f o r t h i s system and t h e nu- c l e i s h o u l d grow - Cape e t a 1 ( 1 9 8 1 ) ) . The s t r u c t u r e was f u r t h e r ana- l y s e d a f t e r 4430 and 8330 t i m e s t e p s ( s t a g e s I 1 and 1 1 1 ) . The v a r i - a t i o n of P * i s p l o t t e d i n f i g u r e 1 and i s summarized i n t a b l e 1. C l e a r - l y t h e reduced p r e s s u r e f e l l from i t s s t a r t i n g v a l u e and a p p e a r s t o have l e v e l l e d o u t b u t i s m a i n t a i n i n g a v a l u e above t h a t which i t would be e x p e c t e d t o a t t a i n i f t h e e q u i l i b r i u m s t r u c t u r e e x i s t e d t h r o u g h o u t t h e system.

*

Table 1

Values o f reduced pressure P a t d i f f e r e n t s t a g e s o f the simulation

Stage

s t a r t I

111 equilibrium

*

Reduced p r e s s u r e , P

1 3 7 . 3 N u m b e r o f time s t e p s

Structure amorphous -r f c c c r y s t a l

c r y s t a l l i t e s

Figure 1. The v a r i a t i o n of the reduced pressure, P * during t h e course of the molecular dynamics simulation. The average values f o r each 50 time s t e p s a r e p l o t t e d .

8 5 , -

1 5 4 -

1 5 3 -

252-

151-

350-

1 4 3 -

3.6- 0

2 . STRUCTURE CHARACTERISATION

The problem o f i d e n t i f y i n g o r d e r w i t h i n t h e system i s e s s e n t i a l l y t o l o c a t e n u c l e i w i t h i n a f l u i d of much t h e same d e n s i t y , which i s d i f - f i c u l t u n l e s s a n u c l e u s i s l a r g e and w e l l d e f i n e d . I t c a n b e done by c a l c u l a t i n g a s t r u c t u r e f a c t o r f o r s p e c i f i c l a t t i c e v e c t o r s ( s e e f o r example Mandell e t a 1 (1976)) b u t t h e r e s u l t s c a n be m i s l e a d i n g and may n o t d i s t i n g u i s h between bcc and f c c s t r u c t u r e (Hsu and Rahman ( 1 9 7 9 ) ) . The method a d o p t e d i n t h e s e i n v e s t i g a t i o n s was t h a t employing t h e ' i n - d i c e s ' of t h e Voronoi p o l y h e d r a which c a n be c o n s t r u c t e d t o e n c l o s e a l l s p a c e c l o s e s t t o e a c h atom w i t h i n t h e system p r o v i d i n g a complete s t a t - i s t i c a l c h a r a c t e r i s a t i o n ( F i n n e y ( l 9 7 0 ) ) . Each i n d e x ( n g , n 4 , n , n , . . ^.)

l i s t s t h e number of f a c e s , n . having a t o t a l of j e d g e s , ( t r i i l g n g z e r o s b e i n g s u p p r e s s e d a f t e r J j = 6), t h u s a p e r f e c t bcc s t r u c t u r e would g e n e r a t e i n d i c e s of (0 6 0 8 ) o n l y ( f i g u r e 2 ) .

A fundamental d i f f i c u l t y a r i s e s , however, when a t t e m p t i n g t o uncover t h e p r e s e n c e of c r y s t a l l i n e c l o s e - p a c k e d s t r u c t u r e s . Under i d e a l c o n d i t i o n s , having no t h e r m a l f l u c t u a t i o n s , t h e s e would g i v e r i s e t o i n d i c e s (0 12 0 0) f o r b o t h bcc and hcp a r r a n g e m e n t s ( s e e f i g u r e 2 ) . With even t h e s l i g h t e s t d i s t u r b a n c e t h e d e g e n e r a t e v e r t i c e s formed a t t h e i n t e r s e c t - i o n s of f o u r edges i n t h e s e p o l y h e d r a ( s o c a l l e d b e c a u s e any t h r e e o f t h e f o u r edges d e f i n e t h e v e r t e x - f i g u r e 2) s e p a r a t e i n t o two o r more v e r t i c e s a t - j u n c t i o n s o f t h r e e e d g e s and a d d i t i o n a l f a c e s may a r i s e . Consequently t h e i n d i c e s a r e e a s i l y d i s t o r t e d and c o n f u s i o n r e s u l t s .

time s t e p s

s t a g e 1 s t a g e 1 1 s t a g e 111

::::\

I I I

1000 2000 3000

I

,000

,

5000 6 0 0 0 7000 BOO0

JOURNAL DE PHYSIQUE

Figure 2. Polyhedra f o r p e r f e c t (a) bcc, (b) f c c , and ( c ) hcp s t r u c t u r e s . Some degenerate v e r t i c e s ( i l i t e r s e c t i o n s o f four edges) are in'dicated i n (b) and ( c ) .

The u s e o f t i m e a v e r a g e d c o o r d i n a t e s ( a v e r a g e d o v e r 100 t i m e s t e p s ) d o e s n o t improve t h e s i t u a t i o n (Cape(1979)). I t i s p o s s i b l e t o r e - v e a l t h e u n d e r l y i n g c l o s e - p a c k e d c r y s t a l s t r u c t u r e by merging v e r t i c e s c l o s e r t h a n a d i s t a n c e 5 ( i . e . removing s h o r t edges) s o c r e a t i n g t h e i d e a l p o l y h e d r a from t h e d i s t o r t e d v e r s i o n s . For t h i s p r o c e d u r e t o be u s e f u l a v a l u e of 5 i s chosen t o be commensurate 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 TI and t h e i n t e r - n e i g h b o u r d i s t a n c e d . Hsu and Rahmann

(1979) have d e m o n s t r a t e d t h e e f f e c t i v e n e s s o f t h i s method f o r Lennard- J o n e s systems and models of s u p e r c o o l e d m e t a l s .

A s i m i l a r e s t i m a t i o n of t h e p a r a m e t e r s a p p r o p r i a t e f o r t h i s s o f t - s p h e r e system gave n/d = 0.09, s u g g e s t i n g / = 3 s h o u l d be u s e d when modi- f y i n g t h e p o l y h e d r a t o i n d i c a t e t h e p r e s e n t of f c c ( o r hcp) n u c l e i .

( d

0.077% ; q i s e s t i m a t e d from t h e mean s q u a r e d i s p l a c e m e n t o f t h e atoms and i s , 0.007R , g i v i n g < ⁼ 0.02% where R = - ) T h i s

was i n agreement w i t h a s e r i e s o f t e s t s t o d e t e r m i n e t h e v a l u e o f 5

e m p i r i c a l l y f o r a r e g i o n a t s t a g e I p r e v i o u s l y i d e n t i f i e d a s b e i n g t h e

l a r g e s t f c c n u c l e u s (Cape e t a 1 ( 1 9 8 1 ) ) . These t e s t s i n d i c a t e d maxi-

mum r e s o l u t i o n of f c c s t r u c t u r e o c c u r r e d f o r 5 = 0.025% . I t was ap-

p a t e n t t h a t some p o l y h e d r a f o r atoms i n a bcc n u c l e u s were a l s o modi-

f i e d t o become (0 1 2 0 0) - t y p e when u s i n g t h i s p r o c e d u r e . T h i s emphasized a t an e a r l y s t a g e t h a t a m b i g u i t i e s c o u l d a r i s e owing t o min- o r d i f f e r e n c e s between d i s t o r t e d bcc and d i s t o r t e d f c c p o l y h e d r a , and t h a t i t would be n e c e s s a r y t o draw some o f t h e p o l y h e d r a t o r e s o l v e t h e problem.

The c e n t r a l r e g i o n o f t h e model, some 2200 atoms, was i n v e s t i g a t e d a t each of t h e s t a g e s d e f i n e d above u s i n g b o t h m o d i f i e d and unmodified i n - d i c e s . The p r o p o r t i o n s of bcc and f c c p o l y h e d r a a t e a c h s t a g e a r e sho.wn i n t a b l e 2 . I t a p p e a r s t h a t t h e number of l o c a l b c c a r r a n g e m e n t s i n c r e a s e s b u t remains t h e same between s t a g e s I 1 and I 1 1 whereas t h e f c c ( a n d / o r hcp) c l u s t e r s i n c r e a s e s t h r o u g h o u t t h e s i m u l a t i o n .

Table 2

Percentage of ( 0 6 0 8 ) polyhedra found before m o d i f i c a t i o n and t h e p e r c e n t a g e of ( 0 12 0 0 ) polyhedra found a f t e r m o d i f i c a t i o n ( e x p r e s s e d a s a p e r c e n t a g e of t h e t o t a l number o f polyhedra)

Stage

( 0 6 0 8)

unrnodi f i e d polyhedra

%

( 0 12 0 0 ) modified polyhedra

3. DELINEATING THE NUCLEI

I t i s p o s s i b l e t o i n v e s t i g a t e t h e e x t e n t (and p e r f e c t i o n ) of bcc n u c l e i by l i s t i n g a l l unmodified p o l y h e d r a of t y p e (0 6 0 8) and c h e c k i n g which of t h e s e have n e i g h b o u r s a t o r a b o u t t h e r i g h t d i s t a n c e and i n t h e r i g h t d i r e c t i o n s t o form bcc a g g r e g a t e s (Cape e t a 1 ( 1 9 8 1 ) ) , though t h i s c a n be a l a b o r i o u s t a s k . S i m i l a r l y , t h e f o u r i n t e r p e n e t r a t i n g s i m p l e c u b i c l a t t i c e s o f f c c c r y s t a l l i t e s c o u l d be mapped u s i n g t h e m o d i f i e d p o l y - h e d r a of t y p e (0 1 2 0 0) a s a g u i d e . However, s i n c e a computer c a n a c - c e s s a l l t h e d a t a r e q u i r e d , i t i s a l e s s a r d u o u s job t o w r i t e computer programs t o do t h i s : n e i g h b o u r i n g p o l y h e d r a o f s i m i l a r t y p e , i r r e s p e c - t i v e of d i s t a n c e and d i r e c t i o n c r i t e r i a , can be r e a d i l y t a b u l a t e d and

t h e n l i s t s c a n be compiled of t h o s e forming s e p a r a t e g r o u p s o r n u c l e i (Bushnell-Wye ( 1 9 8 3 ) ) . I t i s i m p o r t a n t t o r e c o g n i s e t h a t some atoms on t h e s u r f a c e of t h e n u c l e i may w e l l b e i n e x a c t r e g i s t e r w i t h t h e n u c l e u s b u t , b e c a u s e o f t h e i r d i s o r g a n i s e d n e i g h b o u r s away from t h e nu- c l e u s , t h e y w i l l n o t have t h e r e q u i s i t i n d i c e s (even when m o d i f i e d ) . Such atoms s h o u l d be i n c l u d e d i n t h e d e s c r i p t i o n of t h e e x t e n t o f t h e n u c l e i b u t n o t t o c o n n e c t n e i g h b o u r i n g n u c l e i , o t h e r w i s e t h e n u c l e i be- come v e r y d e n d r i t i c .

Using d i s t a n c e and d i r e c t i o n c r i t e r i a t o s p e c i f y n u c l e u s s i z e l i m i t s n u c l e i t o t h e n e a r l y p e r f e c t l y c r y s t a l l i n e r e g i o n s and p r o b a b l y u n d e r - e s t i m a t e s t h e i r s i z e though t h e i r s h a p e s t e n d t o be f a i r l y g l o b u l a r . D i s r e g a r d i n g s u c h c r i t e r i a r e s u l t s i n l a r g e r n u c l e i ( i n g e n e r a l ) which have r a m i f i c a t i o n s i n t o l e s s p e r f e c t l y o r g a n i s e d r e g i o n s of t h e model.

4 . RESULTS

T a b l e s 3 and 4 show t h e s i z e s of t h e l a r g e r n u c l e i found a t e a c h o f t h e

s t a g e s a n a l y s e d and emphasize how t h e a s s i g n e d s i z e i s i n f l u e n c e d by

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Table 3 . The s i z e s of the larger bcc nuclei i d e n t i f i e d a t each stage both with and without surface contributions.

Table 4 . The s i z e s o f the larger f c c nuclei i d e n t i f i e d a t each stage both with and without surface contributions.

Stage I11 (0 6 0

8 )

surface atoms

43 216

34 195

30 138

17 97

11 76

11 66

11 6' .

.

t h e i n c l u s i o n of s u r f a c e atoms. The t r u e s i z e a l m o s t c e r t a i n l y l i e s s o w w h e r e between t h e two t o t a l s g i v e n f o r e a c h n u c l e u s . Comparing t h e t o t a l s f o r e a c h s t a g e s u g g e s t s t h a t t h e bcc n u c l e i grow t o b e g i n w i t h b u t t h e n remain much t h e same s i z e whereas t h e f c c c l u s t e r s c o n t i n u - o u s l y i n c r e a s e i n s i z e p e r h a p s by merging w i t h n e i g h b o u r i n g c l u s t e r s . I t i s o b v i o u s t h a t some a m b i g u i t i e s a r i s e i n d e f i n i n g b o t h l o c a l sym- m e t r y and t h e e x t e n t of o r d e r e d g r o u p s , and t h a t t h e o n l y way t o r e -

s o l v e t h e problem i s t o examine some g r o u p s i n d e t a i l . An i n t e r a c t i v e computer g r a p h i c s system h a s been u s e d t o draw b o t h i n d i v i d u a l s and s m a l l c l u s t e r s of p o l y h e d r a ; some drawings a r e p r e s e n t e d h e r e t o de- m o n s t r a t e t h e r e s u l t s o f t h e a n a l y s i s s o f a r . F i g u r e 3 shows p o l y - h e d r a f o r two (0 6 0 8 ) atoms b e f o r e and a f t e r m o d i f i c a t i o n t o remove s h o r t e d g e s . C l e a r l y , some (0 6 0 8 ) environments a r e v e r y c l o s e t o b e i n g i d e a l f c c s t r u c t u r e s w h i l e o t h e r s a r e n o t which s u g g e s t s i n c o r - r e c t a s s i g n m e n t s c o u l d be made u s i n g unmodified p o l y h e d r a i n d i c e s a l o n e . Extended r e g i o n s of bcc and f c c c h a r a c t e r a l s o e x i s t a s d e p i c t e d i n t h e c l u s t e r s o f f i g u r e 4 . H i g h l i g h t e d i n f i g u r e 5 i s t h e t y p e of f e a - t u r e which c a n o c c u r b o t h i n r e a l s t r u c t u r e s and i n t h i s s o f t - s p h e r e model. These c a n o n l y be a l l o w e d t o e x i s t i n t h e l a t t e r i f s t r i c t d i s - t a n c e and d i r e c t i o n c r i t e r i a a r e n o t u s e d i n d e f i n i n g t h e e x t e n t of c r y s t a l l i n e c l o s e - p a c k e d r e g i o n s : s t a c k i n g f a u l t s i n t r o d u c e l a t t i c e t r a n s l a t i o n s and t h e r e f o r e form a boundary t o a n u c l e u s i f i t i s o n l y

Stage I1 (0 6 0 8) (0 6 0 8) +

surface atans

45 239

4 1 192

35 180

28 141

Stage I

P:bnber o f a t m s in nuclei of 10 or rare a t m s

nucleus 1 2 3 4

5

:.knkr o f a t m s in nuclei of 10 or enre a t m s

nucleus 1 2

3

4 5

(0 6 0 8 ) (0 6 0 8 )

+

19 97

14 8 1

11 61

Stage I1 (0 12 0 0 ) (0 12 0 0)

surface atoms

324

61 280

22 104

Stage I (0 12 0 0 ) (0 12 0 0 ) +

surface atans

151 598

32 139

30 171

10 59

10 56

: I

Stage I11 (0 12 0 0) (0 12 0 0)

*

surface atcms

405 1519

121 469

before ( 0 6 0 8 ) after (0 12 0 0)

before (0 6 0 8 ) after ( 3 7 1 0)

Figure 3. Two polyhedra having unmodified index (0 6 0 8) but clearly different environments. (a) The modified index (0 12 0 0) shows that the symmetry of the atoms' neighbours is very nearly fcc. (b) The modified polyhedron index (3 7 1 0) does not express a high symmetry whereas the drawing of the unmodified form has distinct bcc symmetry.

Figure 4 . (a) A group of mostly (0608)

polyhedra with strong bcc charac-

ter. (b) A group containing some

(0608) polyhedra having strong

fcc character.

JOURNAL DE PHYSIQUE

c u r v a t u r e

F i g u r e 5. A s t a c k i n g f a u l t w i t h i n a f a i r l y w e l l o r d e r e d p a r t o f t h e model.

The polyhedra can be compared with t h o s e of t h e p e r f e c t f c c and hcp s t r u c t u r e s .

F i g u r e 6.

arrangement of polyhedra s u g g e s t i n g p l a n e c u r v a t u r e . The sense of c u r v a t u r e i s i n d i c a t e d . An a l t e r n a t i v e i n t e r p r e t a t i o n i s t h a t a g r a i n boundary i s b e i n g observed.

allowed to have (by definition of the technique used) near perfectstruc- ture. Similarly, gross plane curvature, figure 6 , and grain boundaries, figure 7, can also be identified in this model.

5 . PHASE TRANSFORMATION

When drawing polyhedra, as described above, to examine the structure

associated with particular indices there were often polyhedra which

exhibited both fcc and bcc character. Sometimes this was due to the

small size of the six quadrilateral faces of a (0 6 0 8) polyhedron

and in other cases one 'half1 of the polyhedron could be recognised as

p l a n e d i r e c t i o n s

F i g u r e 7. A n arrangement of polyhedra s u g g e s t i n g t h e e x i s t e n c e of a g r a i n boundary.

having s t r o n g bcc c h a r a c t e r i s t i c s and t h e o t h e r was v e r y n e a r l y p e r f e c t f c c - t y p e . Once a g a i n , t h e s e f e a t u r e s c o u l d n o t be i d e n t i f i e d from i n - d i c e s a l o n e , making t h e u s e of t h e g r a p h i c s e s s e n t i a l . The o v e r a l l i m - p r e s s i o n s u g g e s t e d a change of p h a s e from bcc t o f c c was o c c u r r i n g . Re- f e r e n c e t o t a b l e 2 and maps d e t a i l i n g t h e d i s t r i b u t i o n of i n d e x t y p e s t h r o u g h o u t t h e system ( n o t shown.here) a t e a c h of t h e t h r e e s t a g e s a n a l y s e d d o e s n o t s u p p o r t t h i s however, b u t r a t h e r s u g g e s t s t h a t n u c l e i

grow by expancling i n t o r e g i o n s having i n d i s t i n c t c r y s t a l l i n e c h a r a c t e r . F u r t h e r a n a l y s i s a t d i f f e r e n t s t a g e s i n t h e s i m u l a t i o n must be u n d e r - t a k e n t o a s c e r t a i n e x a c t l y what t r a n s f o r m a t i o n s a r e t a k i n g p l a c e w i t h - i n t h e ensemble.

I t i s p o s s i b l e t o c o n s t r u c t h y p o t h e t i c a l i n t e r m e d i a t e s t a g e s o f t r a n s - f o r m a t i o n from a polyhedron of t y p e (0 6 0 8) t o one of t y p e (0 12 0 O), and some o f t h e f e a t u r e s of t h e i n t e r m e d i a t e s t a g e s c o u l d be s e e n i n p o l y h e d r a d i s p l a y e d u s i n g t h e g r a p h i c s programs. An example of t h i s i s t h e unmodified (0 4 4 4 ) polyhedron i n f i g u r e 8. Even though o n l y s l i g h t movement of n e i g h b o u r s would be r e q u i r e d t o make t h i s i n t o a

(0 12 0 0 ) - t y p e p o l y h e d r o n , i t i s e a s y t o s e e t h a t l a r g e c o o p e r a t i v e movements would be r e q u i r e d t o e f f e c t such a t r a n s f o r m a t i o n o v e r a

l a r g e number o f n e i g h b o u r s : t h e c o o p e r a t i v e movements shown i n f i g . 8 c o u l d n o t p o s s i b l y o c c u r owing t o t h e c o n s t r a i n t s of t h e p e r i o d i c boundary c o n d i t i o n s . T h i s type-of t r a n s f o r m a t i o n c o u l d t h e r e f o r e o n l y t a k e p l a c e w i t h d i f f e r e n t o r i e n t a t i o n s t h r o u g h o u t t h e model and e x t e n - s i v e g r a i n b o u n d a r i e s would r e s u l t . F u r t h e r o r g a n i s a t i o n might p r o - ceed u n d e r t h e r m a l f l u c t u a t i o n s t o a l l o w some g r a i n s t o grow.

6. CONCLUSIONS

S u i t a b l e c o n d i t i o n s have been p r o v i d e d f o r t h e a n n e a l i n g p r o c e s s i n t h e

s o f t - s p h e r e system t o p e r m i t s p o n t a n e o u s homogeneous n u c l e a t i o n of b o t h

bcc and c r y s t a l l i n e c l o s e - p a c k e d s t r u c t u r e s . The n u c l e i grow i n t h e

JOURNAL DE PHYSIQUE

s i m i l a r l y f o r lower h a l f , b u t

may be r o t a t e d by 90° 1

( a ) Successive a l t e r a t i o n s can be made t o t r a n s f o r m a polyhedron of index ( 0 6 0 8 ) t o one of t h e ( 0 1 2 0 0 ) - t y p e .

(b) An unmodified (0 4 4 4) polyhedron t a k e n from t h e model. Some f e a t u r e s of t h i s polyhedron can b e s e e n i n t h e s t r u c t u r e s i n ( a ) . Only minor rearrangements of neighbours would be r e q u i r e d t o form a f c c polyhedron.

1 ^+, c o o p e r a t i v e

-

^motion

/r r e q u i r e d

1

( c ) The r e l a t i o n s h i p between t h e bcc and f c c l a t t i c e s i s shown t o g e t h e r w i t h an i n d i c a t i o n o f t h e c o o p e r a t i v e movements n e c e s s a r y t o e f f e c t a phase change. A widespread motion of t h i s n a t u r e i s d e n i e d by t h e p e r i o d i c boundaries.

F i g u r e 8. The geometry of t h e bcc t o f c c phase t r a n s f o r m a t i o n .

continued simulation as judged by an increased number of Voronoi poly- hedra having indices typical of a crystalline environment. However, the technique used has limitations and it is essential to be aware of the ambiguities which arise. These difficulties can best be sorted out with the aid of computer graphics.

The (0 6 0 8) index can represent strong bcc character but additional- ly may indicate a distortion of crystalline close-packed symmetry.

Modified indices of type ( 0 12 0 0) occur for either fcc or hcp stack- ing sequences and can only be distinguished by resorting to geometri- cal considerations or by drawing the polyhedron concerned.

The technique used to delineate the nuclei curtails their size if geo-

metrical constraints are specified; if not, the nuclei can be more ne-

bulous and features such as stacking faults and grain boundaries can

be i n c l u d e d i n t h e c r y s t a l l i t e s . The l a t t e r approach seems r e a s o n a b l e b e c a u s e t h e s e f e a t u r e s would b e e x p e c t e d when c r y s t a l l i s i n g r a p i d l y from u n d e r c o o l e d l i q u i d s . Once a g a i n , t h e u n c e r t a i n t i e s r e l a t e d t o t h e e x t e n t of i n d i v i d u a l n u c l e i c a n o n l y be r e s o l v e d q u i c k l y w i t h t h e a i d of computer g r a p h i c s .

I t i s p o s s i b l e t h a t a p h a s e change from bcc t o f c c i s o c c u r r i n g l o c a l l y i n t h e model though i n s u f f i c i e n t e v i d e n c e h a s been found t o s u p p o r t t h i s . I t would n o t be p o s s i b l e f o r t h e whole system t o undergo a p h a s e change i n t h e same s e n s e owing t o t h e c o n s t r a i n t s of t h e p e r i o d i c b o u n d a r i e s . F u r t h e r a n a l y s i s i s r e q u i r e d (of s m a l l r e g i o n s of t h e model) o v e r s h o r t - e r t i m e i n t e r v a l s t o i n v e s t i g a t e t h e h y p o t h e s i s .

REFERENCES

J . N. Cape, J . L . F i n n e y and L . V. Woodcock ( 1 9 8 1 ) , J.Chem.Phys.75, 2366 J . N. Cape and L . V. Woodcock ( 1 9 8 0 ) , J . Chem.Phys.72, 976

J. L . Finney (1970), Proc.Roy.Soc. London, E , 479 J. N. Cape (1979), PhD T h e s i s , U n i v e r s i t y of Cambridge C . S. Hsu and A. Rahman (1979) , ^{J .} Chem.Phys. 1_2, 4974

M. J . M a n d e l l , J . P. McTague and A. Rahman (1976), J . C h e m . P h y s . E , 3699

G. Bushnell-Wye (1983), PhD T h e s i s , U n i v e r s i t y of London.