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SURFACE ENERGY AND EQUILIBRIUM SHAPE OF L12-TYPE A3B ORDERING ALLOYS
M. Yamamoto, T. Fukuda, S. Nenno
To cite this version:
M. Yamamoto, T. Fukuda, S. Nenno. SURFACE ENERGY AND EQUILIBRIUM SHAPE OF L12- TYPE A3B ORDERING ALLOYS. Journal de Physique Colloques, 1987, 48 (C6), pp.C6-323-C6-328.
�10.1051/jphyscol:1987653�. �jpa-00226858�
JOURNAL DE PHYSIQUE
Colloque C6, suppldment au n O 1 l , Tome 48, novembre 1987
SURFACE ENERGY AND EQUILIBRIUM SHAPE OF LIZ-TYPE A3B ORDERING ALLOYS
M. Yamarnoto, T. Fukuda and S. Nenno
Department of Materials Science and Engineering, Osaka University, Suita, Osaka 565, Japan
Abstract - Calculation of surface energy of LIZ-type A3B ordering alloy was made on the basis of a broken bond model using Bragg-Williams approximation. Prior to the calculation of surface energy, calculation of broken bond density in the same alloy was made. Numerical results for Cu3Au alloy were presented as functions of surface orientation and degree of order. Applying ~ u l f f ' s theorem to calculated y-plots of the alloy, their external equilibrium shapes for the alloy with various degree of order ( n ) were predicted.
I - INTRODUCTION
Surface energy (y) is an important parameter to understand the nature and character of the surface and the interface in metals and alloys.
The equilibrium shape of isolated metallic solids like emitters and particles is determined by the anisotropy of their specific surface energy. Study o n surface energy and the equilibrium shape is few for ordered alloys and intermetallic compounds.
Recently we have observed the shape of a thermally faceted emitter of a Dla-type ordering alloy by FIM /I ,2/. Moreover, we have calculated theoretically surface energy for the same alloy and have predicted the equilibrium shape 131. Then we have compared the above two each other.
In the present study we selected an L12-type A3B ordering alloy because it is a typical ordering alloy, and calculated surface energy theoretically on the basis of a broken bond model using Bragg-Williams approximation. Prior to the calculation of surface energy, w e made calculation of broken bond density in the same alloy. Numerical results for Cu3Au alloy are presented as functions of surface orientation and degree of order. The change of surface energy is demonstrated in a polar plot of y as a function of orientation (the y- plot). Applying Wulff's theorem to calculated y-plots of the alloy, their external equilibrium shapes for the alloy with various degree of order ( 0 ) are predicted. The Miller indices of the facet planes in the equilibrium shapes are obtained.
I1 - A CALCULATION METHOD
General framework of calculation method is schematically shown in Fig.1 and is explained in the previous paper 131. Here, the method is briefly described below. In the present calculation it is assumed
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987653
JOURNAL DE PHYSIQUE
Density Surf ace Equilibrium Energy Shape
Bragg- Williams
Fig.1 - S c h e m a t i c r e p r e - s e n t a t i o n of a c a l c u l a - t i o n m e t h o d o f s u r f a c e e n e r g y a n d e q u i l i b r i u m s h a p e .
t h a t c o m p o s i t i o n a n d d e g r e e o f o r d e r a t / n e a r t h e s u r f a c e a r e t h e same a s t h o s e i n t h e b u l k .
S p e c i f i c s u r f a c e e n e r g y c s i s g i v e n by t h e m u l t i p l i c a t i o n o f b r o k e n bond d e n s i t y a n d i n t e r a c t i o n e n e r g y , a n d i s e x p r e s s e d by t h e f o l l o w i n g e q u a t i o n ,
E S = - ( 1 / 2 ) C C C vIJ(') uIJ(') ( 1 )
R I J
w h e r e v ~ ~i s t h e number o f b r o k e n bonds b e t w e e n ( ~ ) I a n d J a t o m s ( I = A o r B a n d J = A o r B ) i n t h e R t h n e a r n e i g h b o r p e r u n i t a r e a o f t h e s u r f a c e f o r t h e s t a t e s w i t h v a r i o u s d e g r e e o f o r d e r a n d uIJ(') ( U A B ( & )
= uBA(R)) i s t h e i n t e r a c t i o n e n e r g y b e t w e e n t h e R t h n e a r n e i g h b o u r I-J atoms.
T h e n u m b e r o f b r o k e n b o n d s V I J ( R ) i s g i v e n , u s i n g B r a g g - W i l l i a m s a p p r o x i m a t i o n , by
w h e r e v i j ( & ) i s t h e n u m b e r o f b r o k e n b o n d s b e t w e e n i a n d j s u b l a t t i c e s i n t h e R t h n e a r n e i g h b o r p e r u n i t a r e a o f t h e s u r f a c e f o r t h e p e r f e c t l y o r d e r e d s t a t e , Pi1(!?.) i s p r o b a b i l i t y t h a t a n I - a t o m o c c u p i e s a n i - s u b l a t t i c e p o i n t , i o r j i s a- o r P - s u b l a t t i c e and q i s d e g r e e o f o r d e r . S o t h e n u m b e r o f b r o k e n b o n d s i s e v a l u a t e d f r o m t h e g e o m e t r i c a l c o n s i d e r a t i o n .
V a l u e s o f t h e i n t e r a c t i o n e n e r g y i J A A ( % ) a n d u ~ ~f o r Cu3Au a r e ( ~ ) o b t a i n e d u s i n g Morse p o t e n t i a l e n e r g y and t h a t o f u ~ ~i s ( o b t a i n e d ~ ) u t i l i z i n g t h e r e l a t i v e v a l u e s b e t w e e n a t o m s i n t h e Rth n e a r n e i q h b o r s g i v e n b y C l a p a n d M o s s 1 4 1 , t h a t i s ~ 2 1 ~ 1 = ( U A A ( ~ ) + U B - B ( ~ ) -
~ u ~ B ( ~ ) ) / ( u ~ ~ ( ) + u B B ( ' ) - 2 u A B ( ' ) ) a n d v3/v1.
I11 - RESULTS AND DISCUSSION
1 . Broken Bond D e n s i t y
The c r y s t a l p l a n e s o f t h e L I Z - t y p e o r d e r e d a l l o y a r e c l a s s i f i e d i n t o t w o t y p e s : f u n d a m e n t a l a n d s u p e r l a t t i c e p l a n e s . I n f u n d a m e n t a l p l a n e ,
F i g . 2 - Maps o f b r o k e n bond d e n s i t y , [O -1 I ] z o n e , f u n d a m e n t a l p l a n e o n l y .
e a c h s u c c e s s i v e l a y e r i s c r y s t a l l o g r a p h i c a l l y i d e n t i c a l a n d h a s t h e s t o i c h i o m e t r i c c o m p o s i t i o n of 75 a t % A a n d 25 a t % B. I n s u p e r l a t t i c e p l a n e , i t a l t e r n a t e s p e r i o d i c a l l y f r o m 100at%A t o 50at%B-5Oat%A. The s u p e r l a t t i c e p l a n e h a v i n g a 1 0 0 a t % A p l a n e a t t h e t o p p l a n e i s named h e r e t y p e I , a n d t h a t h a v i n g a 50at%A-5Oat%B p l a n e a t t h e t o p p l a n e i s t y p e 11. B r o k e n b o n d d e n s i t y w a s c a l c u l a t e d f o r f u n d a m e n t a l p l a n e o n l y , s u p e r l a t t i c e p l a n e o f t y p e I o n l y , a n d o f t y p e I1 o n l y , s e p a r a t e l y .
F i g u r e 2 i s maps o f b r o k e n b o n d d e n s i t y ~ i j ( ~ ) , a l o n g [O -1 1 I z o n e . E a c h o f t h e s e maps i n c l u d e s o n l y f u n d a m e n t a l p l a n e s . The n u m b e r s a t
F i g . 3 - Maps o f b r o k e n bond d e n s i t y , [O -1 1 1 z o n e , s u p e r l a t t i c e p l a n e o f t y p e I o n l y .
C6-326 JOURNAL DE PHYSIQUE
t h e r i g h t u p p e r s i d e o f e a c h map mean m a g n i f i c a t i o n o n d r a w i n g . Broken bond d e n s i t y c h a n g e s s m o o t h l y w i t h c h a n g i n g s u r f a c e o r i e n t a t i o n f o r a l l o f t h e I s t , 2nd a n d 3 r d n e a r e s t n e i g h b o r s . F o r t h e 1 s t n.n.
t h e b r o k e n b o n d d e n s i t y i s t h e l o w e s t a t ( 1 1 1 ) p l a n e a n d i s l o w e r a t ( 1 0 0 ) p l a n e t h a n a t o t h e r p l a n e s . F o r t h e 2nd n.n. ( 1 0 0 ) a n d ( 0 1 1 ) p l a n e s a r e l o w e r t h a n t h e o t h e r s , a n d f o r t h e 3 r d n.n. ( 1 11 ) , ( 0 1 1 ) , ( 2 1 0 ) a n d (31 1 ) p l a n e s h a v e l o w e r b r o k e n bond d e n s i t y .
F o r s u p e r l a t t i c e p l a n e o f t y p e I o n l y , F i g . 3 , M i l l e r i n d i c e s o f t h e p l a n e s h a v i n g l o w e r b r o k e n b o n d d e n s i t y a r e s i m i l a r t o t h o s e o f t h e p l a n e s f o r f u n d a m e n t a l p l a n e o n l y . However b r o k e n bond d e n s i t i e s f o r B c i a n d a @ b o n d c h a n g e j a g g e d l y . T h i s i s d u e t o t h e a l t e r n a t i v e s t a c k i n g o f a t o m i c l a y e r i n t h e s u p e r l a t t i c e p l a n e d e s c r i b e d a b o v e . On t h e c o n t r a r y f o r @ @ and cia t h e y c h a n g e s m o o t h l y . F o r s u p e r l a t t i c e p l a n e o f t y p e I1 o n l y , i t h a s t h e s a m e f e a t u r e s a s t h o s e f o r t y p e I d e s c r i b e d a b o v e , a s shown i n Fig.3.
2. S u r f a c e Energy
W h i l e b r o k e n bond d e n s i t y i s d e t e r m i n e d o n l y by g e o m e t r i c a l f a c t o r s , s u r f a c e e n e r g y n e e d s t h e i n t e r a c t i o n e n e r g y on i t s d e t e r m i n a t i o n a n d t h u s a n a l l o y h a s t o b e sampled. H e r e we s e l e c t e d Cu3Au a l l o y b e c a u s e i t i s o n e o f t y p i c a l L I Z - t y p e o r d e r i n g a l l o y a n d i t s i n t e r a c t i o n e n e r g y h a s b e e n a l r e a d y known 141.
F i g u r e s 4 ( a ) , ( b ) a n d ( c ) s h o w t h r e e c a l c u l a t e d y - p l o t s o f a L I Z - t y p e Cu3Au a l l o y , a l o n g [O -1 I ] z o n e f o r f u n d a m e n t a l p l a n e o n l y , s u p e r l a t t i c e p l a n e o f t y p e 1 o n l y a n d s u p e r l a t t i c e p l a n e o f t y p e I1 o n l y , r e s p e c t i v e l y . T h e y a r e f o r t h e c a s e o f q=0.346, w h i c h i s t h e v a l u e a t t h e c r i t i c a l t e m p e r a t u r e i n t h e o r d e r - d i s o r d e r t r a n s f o r m a t i o n of t h i s a l l o y . S u r f a c e e n e r g y c h a n g e s s m o o t h l y w i t h c h a n g i n g s u r f a c e o r i e n t a t i o n b o t h a t f u n d a m e n t a l a n d s u p e r l a t t i c e p l a n e s . I t i s t h e l o w e s t a t ( 1 1 1 ) p l a n e , a n d a t ( l o o ) , ( I I O ) , ( 2 1 0 ) , a n d ( 3 1 1 ) p l a n e s i n o r d e r o f i n c r e a s i n g . S u r f a c e e n e r g y i s e x a c t l y t h e s a m e f o r s u p e r l a t t i c e p l a n e o f t y p e I a n d 11, a n d i s n o t much c h a n g e d b e t w e e n f u n d a m e n t a l a n d s u p e r l a t t i c e p l a n e s .
When y - p l o t s f o r f u n d a m e n t a l p l a n e , F i g . Q ( a ) a n d s u p e r l a t t i c e p l a n e o f t y p e I , F i g . 4 ( b ) ( o r t y p e 11, F i g . 4 ( c ) ) a r e s u p e r p o s e d , F i g . S ( c ) i s o b t a i n e d , w h i c h i s f o r q=0.346. E v e n a f t e r s u p e r p o s i t i o n o f t w o y - p l o t s o f F i g . $ ( a ) a n d 4 ( b ) ( o r F i g . 4 ( a ) a n d 4 ( c ) ) , s u r f a c e e n e r g y c h a n g e s s m o o t h l y , a n d i t i s l o w e r a t ( 1 11 ) , ( 1 O O ) , ( 1 l o ) , ( 2 1 0 ) a n d
( 3 1 1 ) p l a n e s t h a n a t o t h e r s .
01 1
F. Plane S,( I 1 Plane S.( It ) Plane
F i g . 4 - C a l c u l a t e d y - p l o t s o f a L I Z - t y p e Cu3Au a l l o y w i t h q = 0 . 3 4 6 , a l o n g [ 0 -1 1 ] z o n e .
Figures 5(a)-(f) are calculated y-plots of a LIZ-type Cu3Au alloy with various degree of order: Fig.5(a) and 5(b) for q=1, Fig.5(c) and 5(d) for r)=0.346 and Fig.5(e) and 5(f) for n=0. Figures 5(a), ( c ) and (e) are drawn along [O -1 1 ] zone and Fig.S(b), 5(d) and 5(f) are along
[OOI] zone. Planes having the lowest and lower surface energy are the same as those in Fig.4. The feature of y-plots in Fig.5 does not change much, nevertheless degree of order changes largely.
3. Equilibrium Shape
Applying ~ u l f f ' s theorem to the calculated y-plots of the alloy, equilibrium external shapes for the alloy with various degree of order
Fig.5 - Calculated y-plots and equilFbrium shape of a LIZ-type Cu3Au alloy with various degree of order.
C6-328 JOURNAL DE PHYSIQUE
w e r e p r e d i c t e d . I n F i g . 5 ( a ) - ( f ) t h e p r e d i c t e d e q u i l i b r i u m s h a p e s a r e s h o w n b y s o l i d l i n e s . ( I l l ) , ( I O O ) , ( I I O ) , ( 2 1 0 ) a n d ( 3 1 1 ) p l a n e s h a v i n g low s u r f a c e e n e r g y a r e f a c e t t e d . The e q u i l i b r i u m s h a p e k e e p s a l m o s t t h e s a m e e v e n w h e n d e g r e e o f o r d e r c h a n g e s f r o m 1 t o 0.
4. Dependence of S u r f a c e E n e r g y o n Degree of O r d e r
F i g u r e 6 shows how s u r f a c e e n e r g y c h a n g e A&, c h a n g e s a g a i n s t d e g r e e o f o r d e r . (751 ) i s a n e x a m p l e o f h i g h i n d e x p l a n e . I n Fig.6, s u r f a c e e n e r g y i n c r e a s e s m o n o t o n i c a l l y w i t h i n c r e a s i n g d e g r e e o f o r d e r (q).
The i n c r e a s i n g r a t e was t h e l o w e s t a t ( 1 0 0 ) p l a n e .
Fig.6 - R e l a t i o n s b e t w e e n s u r f a c e e n e r g y c h a n g e AE, a n d d e g r e e o f o r d e r 11.
0 0.2 0.4 0.6 0.8 1.0
IV - SUMMARY ?
1 ) B r o k e n b o n d d e n s i t y i s l o w e r a t ( 1 11 ) , ( 1 0 0 ) a n d ( 1 1 0 ) p l a n e s t h a n a t o t h e r p l a n e s f o r t h e 1 s t n e a r e s t n e i g h b o r . F o r 2nd n.n. ( 1 0 0 ) a n d ( 1 1 0 ) p l a n e s a r e l o w . ( I I O ) , ( I l l ) , ( 2 1 0 ) a n d ( 3 1 1 ) p l a n e s h a v e low b r o k e n bond d e n s i t y f o r 3 r d n.n.
2 ) s u r f a c e e n e r g y i s t h e l o w e s t a t ( I 11 ) p l a n e , and a t ( l o o ) , ( 1 l o ) , ( 2 1 0 ) a n d ( 3 1 1 ) p l a n e s i n o r d e r o f i n c r e a s i n g . T h i s f e a t u r e i s s t r o n g l y i n f l u e n c e d b y t h e b r o k e n bond d e n s i t y .
3 ) E q u i l i b r i u m e x t e r n a l s h a p e i s d e t e r m i n e d by t h e a b o v e p l a n e s h a v i n g low s u r f a c e e n e r g y .
4 ) S u r f a c e e n e r g y i n c r e a s e s w i t h i n c r e a s i n g d e g r e e o f o r d e r Q b u t t h e i n c r e a s e o f 17 d o e s n o t g i v e a n y s i g n i f i c a n t i n f l u e n c e o n e q u i l i b - r i u m shape.
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