COMPUTER SIMULATION OF THE THERMODYNAMIC PROPERTIES OF GRAIN BOUNDARIES

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COMPUTER SIMULATION OF THE

THERMODYNAMIC PROPERTIES OF GRAIN BOUNDARIES

P. Deymier, G. Kalonji

To cite this version:

P. Deymier, G. Kalonji. COMPUTER SIMULATION OF THE THERMODYNAMIC PROPER- TIES OF GRAIN BOUNDARIES. Journal de Physique Colloques, 1985, 46 (C4), pp.C4-213-C4-224.

�10.1051/jphyscol:1985423�. �jpa-00224673�

JOURNAL DE PHYSIQUE

Colloque Cf, supplément au n"4, Tome 46, avril 1985 page C4-213

COMPUTER SIMULATION OF THE THERMODYNAMIC PROPERTIES OF GRAIN BOUNDARIES

P . Deymier and G. K a l o n j i

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02159, U.S.A.

Résume - On présente ici un calcul explicite des grandeurs thermodynamiques d'excès caractéristiques des joints de grain à partir de simulation de dynami- que moléculaire. On s'intéresse particulièrement au calcul de l'énergie libre interfaciale en fonction de la désorientation du joint. On observe en outre des transitions de phase dans la structure du joint qui se traduisent par une ré- orientation spontanée des deux cristaux et/ou l'apparition d'un film quasi- liquide au niveau du joint.

Abstract - Methods are discussed for the explicit calculation of the excess thermodynamic properties of grain boundaries using molecular dynamics simula- tions in the NPT ensemble. Specific attention is focussed on techniques for the calculation of excess free energies. Results on the variation of interfacial free energy with misorientation are reported. In addition, observations of pha- se transitions among grain boundary structures are discussed, including grain boundary melting transitions and reorientation phenomena.

1. INTRODUCTION

Interest in the structure and physical properties of internal interfaces in solids has spawned the development of a variety of tools, both theoretical and experimental, for their study. One of the approaches that has been employed in a considerable number of studies is atomistic computer simulation, employing almost exclusively molecular statics techniques, using central pair-wise interatomic potentials. The results of such studies are rather difficult to quantitatively evaluate because of the current limitations on resolution of the corresponding experimental methods of probing structure, but some qualitative agreement has, in certain cases, been attained. Several rather severe limitations are inherent in molecular statics simulations in general, and additional limitations arise through the methods with which they have been applied to grain boundaries in particular.

The most major intrinsic limitation of the molecular statics method is the obvious one: one accesses, at best, the T=0 structure of the system, and those properties which are functions of instantaneous coordinates only, such as the internal energy of the system. The vast majority of the interfacial properties and processes of interest to materials scientists, such as diffusion, migration and deformation, occur at finite temperatures. To answer even such comparatively simple questions as the relative stability of different interfacial structures becomes impossible in the regimes of interest. In addition to its intrinsic limitations, the

molecular statics calculations of grain boundaries have been limited by additional constraints. Often the atomic relaxations that are allowed in searching for local minima in the energy of the system are a severely constrained subset of the conceivable relaxations for an actual grain boundary. Sometimes the excess volume of the grain boundary is constrained to be zero because the interatomic potential functions employed in the simulation are derived for constant electron density.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985423

JOURNAL DE PHYSIQUE

Many o f t h e i n t r i n s i c l i m i t a t i o n s of m o l e c u l a r s t a t i c s s i m u l a t i o n s a r e r e l i e v e d b y u s e of dynamical s i m u l a t i o n t e c h n i q u e s , s u c h a s m o l e c u l a r dynamics o r Monte C a r l o . I n t h i s p a p e r we d i s c u s s methods o f c a l c u l a t i n g thermodynamic p r o p e r t i e s o f g r a i n b o u n d a r i e s u s i n g m o l e c u l a r dynamics. As w i l l b e shown, a number of p r o p e r t i e s a r e o b t a i n e d w i t h r e l a t i v e e a s e , w h i l e o t h e r s , s u c h a s e n t r o p y , r e q u i r e c o n s i d e r a b l e work. T e c h n i q u e s f o r c a l c u l a t i n g i n t e r f a c i a l e x c e s s e n t r o p y w i l l b e d e s c r i b e d and r e s u l t s p r e s e n t e d . Our g o a l i s t o s y s t e m a t i c a l l y i n v e s t i g a t e t h e p h a s e d i a g r a m o f a boundary, c o n s i d e r e d a s an i n d e p e n d e n t thermodynamic e n t i t y . Because thermodynamic p r o p e r t i e s o f g r a i n b o u n d a r i e s depend on a number of g e o m e t r i c a l p a r a m e t e r s , i n a d d i t i o n t o t h e more t y p i c a l o n e s c h a r a c t e r i s t i c of b u l k p h a s e s , t h e t y p e s of p h a s e t r a n s i t i o n s t h a t a g r a i n boundary may undergo a r e r i c h and v a r i e d . I n t h i s p a p e r we d e s c r i b e two t y p e s of g r a i n boundary p h a s e t r a n s i t i o n t h a t we have o b s e r v e d i n o u r m o l e c u l a r dynamics s i m u l a t i o n s of two-dimensional Lennard-Jones b i c r y s t a l s . The f i r s t i s a m e l t i n g t r a n s i t i o n i n which a g r a i n boundary which i s a w e l l - d e f i n e d c r y s t a l l i n e e n t i t y a t low t e m p e r a t u r e i s r e p l a c e d a t a p p r o x i m a t e l y 80% o f t h e b u l k m e l t i n g p o i n t w i t h a d i s o r d e r e d l i q u i d - 1 i k e l a y e r o f h i g h m o b i l i t y . Tne second phenomena we have o b s e r v e d i s t h e r e l a t i v e r e o r i e n t a t i o n of a b i c r y s t a l from one w e l l - d e f i n e d g r a i n boundary o r i e n t a t i o n i n t o a n o t h e r a s t h e t e m p e r a t u r e i s changed. B o t h t r a n s i t i o n s a r e r e f l e c t e d i n d i s c o n t i n u i t i e s i n t h e c a l c u l a t e d e x c e s s p r o p e r t i e s of t h e system.

2. GRAIN BOUNDARY PHASE EQUILIBRIA

B e f o r e d i s c u s s i n g how m o l e c u l a r dynamics t e c h n i q u e s a r e used t o c a l c u l a t e e x c e s s thermodynamic p r o p e r t i e s o f g r a i n b o u n d a r i e s i t i s u s e f u l t o r e c a l l how t h e y a r e d e f i n e d . F o l l o w i n g G i b b s ' c l a s s i c a l development of s u r f a c e

thermodynamics (1) we c a n d e f i n e t h e e x c e s s thermodynamic p r o p e r t i e s of a n

i n t e r f a c e , o r a n y o t h e r d e f e c t f o r t h a t m a t t e r . The e x c e s s v a l u e of any e x t e n s i v e thermodynamic p r o p e r t y o f a d e f e c t i s s i m p l y d e f i n e d a s t h e d i f f e r e n c e between t h e v a l u e o f t h e p r o p e r t y i n a system c o n t a i n i n g t h e d e f e c t , i n o u r c a s e a . b i c r y s t a 1 ,

and i t s v a l u e i n a r e f e r e n c e p e r f e c t c r y s t a l system. A p h a s e t r a n s i t i o n of a d e f e c t i s t h e n d e f i n e d i n t e r m s o f d i s c o n t i n u i t i e s i n t h e c a l c u l a t e d , o r measured.

e x c e s s p r o p e r t i e s , i n r e g i o n s where t h e b u l k p h a s e s a r e s t a b l e .

The s t u d y of p h a s e e q u i l i b r i a of c r y s t a l l i n e d e f e c t s i s c o n s i d e r a b l y e n l i v e n e d b y t h e f u n c t i o n a l dependence o f t h e i r p r o p e r t i e s on v a r i o u s g e o m e t r i c a l

p a r a m e t e r s . The f a c t t h a t t h e thermodynamic p r o p e r t i e s o f g r a i n b o u n d a r i e s depend on t h e s e g e o m e t r i c a l p a r a m e t e r s r a i s e s t h e p o s s i b i l i t y of i n t e r e s t i n g c l a s s e s of p h a s e t r a n s i t i o n s t h a t have no d i r e c t a n a l o g y i n t h e c o r r e s p o n d i n g b u l k p h a s e s . These g e o m e t r i c a l p a r a m e t e r s a r e i n v o l v e d i n t h e i d e a l i z e d d e s c r i p t i o n o f t h e c r e a t i o n o f t h e d e f e c t . F o r t h e c a s e of a p l a n a r c r y s t a l l i n e i n t e r f a c e t h e y a r e 10 i n number. F o u r d e s c r i b e t h e t r a n s f o r m a t i o n , s u c h a s r o t a t i o n , r e l e c t i o n o r i n v e r s i o n which g e n e r a t e s one c r y s t a l from t h e o t h e r . T h r e e more d e s c r i b e any r e l a t i v e r i g i d body t r a n s l a t i o n between t h e two c r y s t a l s , and t h r e e a r e r e q u i r e d t o s p e c i f y t h e e x a c t l o c a t i o n of t h e i n t e r f a c e p l a n e . A l l of t h e s e p a r a m e t e r s , w i t h t h e e x c e p t i o n o f t h e p a r a m e t e r d e s c r i b i n g t h e c h i r a l i t y of t h e t r a n s f o r m a t i o n o p e r a t i o n , w i l l , i n g e n e r a l , undergo r e l a x a t i o n s . Experiment o r t h e o r y w i l l u s u a l l y f i x t h e v a l u e s of some of t h e p a r a m e t e r s , a l l o w i n g t h e o t h e r s t o r e l a x t o c o n s t r a i n e d e q u i l i b r i u m v a l u e s . Under c e r t a i n c o n d i t i o n s changes i n t h e

g e o m e t r i c a l p a r a m e t e r s w i l l mark t h e o c c u r e n c e i n i n t e r f a c i a l phase t r a n s i t i o n s . The t y p e s of t r a n s i t i o n s t h a t c a n , i n p r i n c i p l e , o c c u r a r e r i c h and v a r i e d . F a c e t t i n g t r a n s i t i o n s i n v o l v e changes i n t h e p a r a m e t e r s d e s c r i b i n g t h e l o c a t i o n of t h e boundary p l a n e ( 2 ) . A boundary p l a n e o r i e n t a t i o n t h a t i s s t a b l e u n d e r c e r t a i n c o n d i t i o n s may b e r e p l a c e d by two o r more d i f f e r i n g o r i e n t a t i o n s a s t h e c o n d i t i o n s a r e v a r i e d . Another phenomena t h a t c a n o c c u r i n v o l v e s c h a n g e s i n t h e p a r a m e t e r s which d e s c r i b e t h e r e l a t i v e m i s o r i e n t a t i o n betvieen t h e two c r y s t a l s . S i m i l a r changes c a n o c c u r i n t h e p a r a n e t e r s d e s c r i b i n g t h e r i g i d body t r a n s l a t i o n between

t h e two c r y s t a l s . Of c o u r s e t h e r e i s a l s o t h e p o s s i b i l i t y t h a t i n t e r f a c i a l phase t r a n s i t i o n s o c c u r which do n o t i n v o l v e changes i n t h e g e o m e t r i c a l p a r a m e t e r s , though t h i s w i l l b e t h e e x c e p t i o n r a t h e r t h a n t h e r u l e . Such t r a n s i t i o n s c a n i n c l u d e g r a i n boundary m e l t i n g t r a n s i t i o n s , which have b e e n o b s e r v e d i n a v a r i e t y o f s t u d i e s , o r d i s p l a c i v e i n t e r f a c i a l p h a s e t r a n s i t i o n s .

The dependence of t h e thermodynamic p r o p e r t i e s of g r a i n b o u n d a r i e s o n t h e g e o m e t r i c a l p a r a m e t e r s a l s o c a u s e s symmetry a r g u m e n t s t o become e x t r e m e l y powerful i n t h e s t u d y o f phase e q u i l i b r i a o f s u c h d e f e c t s . I n p a r t i c u l a r , t h r o u g h

knowledge o f how t h e symmetry changes w i t h t h e g e o m e t r i c a l p a r a m e t e r s one i s a b l e t o l o c a t e p o s t i o n s on t h e g r a i n boundary f r e e e n e r g y s u r f a c e which must, b y symmetry, b e extrema ( 3 ) . These symmetry d i c t a t e d e x t r e m a , t h e n , s e r v e a s l o g i c a l s t a r t i n g p o i n t s f o r computer s i m u l a t i o n s t u d i e s of g r a i n boundary p r o p e r t i e s .

The e v i d e n c e t h a t e x i s t s t o d a t e c o n c e r n i n g g r a i n boundary p h a s e t r a n s i t i o n s i s somewhat skimpy. There have'been a number o f t h e o r e t i c a l s t u d i e s which i n d i c a t e t h a t low 1 t i l t b o u n d a r i e s undergo a t r a n s i t i o n t o a m e l t e d l a y e r . These i n c l u d e a l a t t i c e g a s s t u d y (41, and s e v e r a l m o l e c u l a r dynamics s t u d i e s (5-7). On t h e e x p e r i m e n t a l s i d e , t h e r e i s t h e TEM s t u d y of Glicksman and Vold, i n which c e r t a i n b o u n d a r i e s a r e o b s e r v e d t o b e r e p l a c e d b y l i q u i d l a y e r s ( 8 ) . R e c e n t l y , Watanabe and co-workers have s e e n i n t e r e s t i n g i n d i c a t i o n s of a g r a i n boundary p h a s e t r a n s i t i o n t h r o u g h measurement o f t h e a c t i v a t i o n e n e r g y a s a f u n c t i o n of t e m p e r a t u r e f o r g r a i n boundary s l i d i n g f o r w e l l c h a r a c t e r i z e d b i c r y s t a l s (9). I n t h i s p a p e r we w i l l p r e s e n t r e s u l t s on m o l e c u l a r dynamics s t u d i e s i n which b o t h m e l t i n g and r e o r i e n t a t i o n t r a n s i t i o n s a r e o b s e r v e d .

3. MOLECULAR DYNAMICS CALCUIATIONS OF TEERFXCQYNAMIC PROPERTIES

I n t h i s s e c t i o n we b r i e f l y d e s c r i b e how m o l e c u l a r dynamics t e c h n i q u e s a r e used t o c a l c u l a t e e x c e s s thermodynan~ic p r o p e r t i e s o f d e f e c t s i n g e n e r a l , and o f i n t e r f a c e s i n p a r t i c u l a r . As was p o i n t e d o u t i n t h e p r e v i o u s s e c t i o n , e x c e s s thermodynamic p r o p e r t i e s o f a d e f e c t c a n b e unambiguously d e f i n e d t h r o u g h t h e u s e o f p e r f e c t c r y s t a l r e f e r e n c e s t a t e s . T h e r e f o r e it w i l l b e i m p o r t a n t t o e s t a b l i s h t h e thermodynamic p r o p e r t i e s of t h e b u l k .

We b e g i n b y r e c a l l i n g what m o l e c u l a r dynamics i s . M o l e c u l a r dynamics i s a c o m p u t a t i o n a l t e c h n i q u e whereby t h e c l a s s i c a l mechanical e q u a t i o n s of motion o f a s y s t e m of m o l e c u l e s o r atoms a r e s o l v e d n u m e r i c a l l y . Given i n i t i a l c o o r d i n a t e s and v e l o c i t i e s o f e a c h atom, and a n i n t e r a t o m i c p o t e n t i a l f u n c t i o n , t h e

t r a j e c t o r i e s of e a c h atom i n time a r e c a l c u l a t e d . The c o n s t r a i n t s imposed o n t h e system, s u c h a s p e r i o d i c o r f i x e d b o r d e r c o n d i t i o n s , c o n s t a n t p r e s s u r e o r c o n s t a n t volume v a r y from s i m u l a t i o n t o s i m u l a t i o n . The o u t p u t o f a m o l e c u l a r dynamics r u n i s t h e c o o r d i n a t e s and v e l o c i t i e s o f e a c h atom a t e a c h time s t e p . Our t a s k i s t o c a l c u l a t e t h e p r o p e r t i e s of t h e s y s t e m from t h e s e q u a n t i t i e s .

Many thermodynamic p r o p e r t i e s a r e d i r e c t l y o b t a i n a b l e from a MD s i m u l a t i o n . These a r e t h o s e p r o p e r t i e s which a r e d e f i n e d i n terms of i n s t a n t a n e o u s c o o r d i n a t e s a n d / o r v e l o c i t i e s , and t h u s a r e d e f i n e d a t e a c h t i m e s t e p . T h i s c a t e g o r y of p r o p e r t i e s i n c l u d e s i n t e r n a l e n e r g y , p r e s s u r e , e n t h a l p y , t e m p e r a t u r e , and d e n s i t y . The thermodynamic v a l u e of s u c h a p r o p e r t y , t h e n , i s s i m p l y t h e time a v e r a g e v a l u e . C l e a r l y one w i l l o n l y b e a b l e o p e r a t i o n a l l y t o c a l c u l a t e s u c h a n a v e r a g e i f t h e s y s t e m i s a t l e a s t m e t a s t a b l e , and remains i n t h e s t a t e o f i n t e r e s t f o r s u f f i c . i e n t l y long t i m e s .

C a l c u l a t i o n s of e n t r o p y , and hence s u c h q u a n t i t i e s a s f r e e e n e r g i e s , however, a r e n o t o b t a i n a b l e i n s o s i m p l e a f a s h i o n . T h i s i s b e c a u s e t h e y a r e n o t

e x p r e s s i b l e a s ensemble a v e r a g e s and r e q u i r e knowledge of t h e p a r t i t i o n f u n c t i o n o f t h e system. There a r e s e v e r a l t e c h n i q u e s b y which m o l e c u l a r dynamics methods

C4-216 JOURNAL DE PHYSIQUE

may be u s e d t o e s t i m a t e f r e e e n e r g i e s . The s i m p l e s t a p p r o a c h t h a t we have used t o e s t i m a t e f r e e e n e r g i e s i s due t o Hoover ( 1 0 ) . I t c o n s i s t s o f a d i r e c t e v a l u a t i o n o f t h e c a n o n i c a l p a r t i t i o n f u n c t i o n w i t h i n t h e quasi-harmonic a p p r o x i m a t i o n . Under t h i s a p p r o x i m a t i o n , t h e t o t a l p o t e n t i a l e n e r g y o f t h e s y s t e m c a n b e w r i t t e n a s a q u a d r a t i c form i n t h e p a r t i c l e d i s p l a c e m e n t s , u . :

C a r r y i n g o u t t h e i n t e g r a t i o n of t h e p a r t i t i o n f u n c t i o n one f i n d s t h e p a r t i t i o n f u n c t i o n t o b e p r o p o r t i o n a l t o t h e i n v e r s e s q u a r e r o o t of t h e d e t e r m i n a n t o f t h e m a t r i x of t h e f o r c e c o n s t a n t c o e f f i c i e n t s k

.

f u n c t i o n o f t h e s t a t i c l a t t i c e p o t e n t i a l uOi? t e m p e r a t u r e , volume and number o f p a r t i c l e s . F o r a system u n d e r p e r i o d i c boundary c o n d i t i o n s t h e d e t e r m i n a n t v a n i s h e s due t o t h e e x i s t e n c e o f d i s p l a c e m e n t s f o r which n o r e s t o r i n g f o r c e s a r e i n v o l v e d , s u c h a s t r a n s l a t i o n of t h e c r y s t a l . T h i s m o t i o n i s t h e n c o n s t r a i n e d by p i n n i n g one p a r t i c l e i n t h e s y s t e m . The new d e t e r m i n a n t w i l l be t h e d e t e r m i n a n t o f a m a t r i x b u i l t from t h e o r i g i n a l f o r c e c o n s t a n t m a t r i x by removing rows and columns c o r r e s p o n d i n g t o t h e f i x e d p a r t i c l e . The f o r c e c o n s t a n t m a t r i x i s c a l c u l a t e d from a n a v e r a g e s t r u c t u r e g e n e r a t e d i n a biD s i m u l a t i o n . C l e a r l y t h i s method h a s l i m i t a t i o n s i n t r i n s i c t o t h e quasi-harmonic approximat i o n , and i s n o t u s e f u l e x c e p t a t low t e m p e r a t u r e s .

We have u s e d a s e c o n d , more g e n e r a l l y a p p l i c a b l e method c a l l e d computer

c a l o r i m e t r y ( 1 1 ) t o c a l c u l a t e f r e e e n e r g i e s . The c r u x o f t h i s method i s t h a t one i s a b l e t o e x p r e s s r a t i o s o f p a r t i t i o n f u n c t i o n s , between t h e system o f i n t e r e s t and a r e f e r e n c e s y s t e m , a s a n ensemble a v e r a g e . T h i s a l l o w s u s t o compute d i f f e r e n c e s i n f r e e e n e r g i e s . The r e f e r e n c e s y s t e m d i f f e r s from t h e s y s t e m o f i n t e r e s t i n i t s i n t e r a t o m i c p o t e n t i a l f u n c t i o n . If we choose a r e f e r e n c e system o f known f r e e e n e r g y , we a r e t h e n a b l e t o c a l c u l a t e e x p l i c i t v a l u e s f o r t h e f r e e e n e r g y o f t h e s y s t e m o f i n t e r e s t .

I n o u r g r a i n b o u n d a r y s t u d i e s t h e system o f i n t e r e s t i s c h a r a c t e r i z e d by a Lennard-Jones p o t e n t i a l , U, , and t h e r e f e r e n c e s y s t e m i s c h a r a c t e r i z e d by a Hooke's law t y p e p o t e n t i a l , Uo

.

f u n c t i o n s h , ( x ) and h , ( x ) i n t h e i n d e p e n d e n t ensembles g e n e r a t e d b y U, and U, :

Here A . i s t h e i s o t h e r m a l - i s o b a r i c p a r t i t i o n f u n c t i o n of system i , and 6 i s t h e D i r a c d e l t a f u n c t i o n . These f u n c t i o n s , h i ( x ) , c a n be e s t i m a t e d i n a m o l e c u l a r dynamics s i m u l a t i o n by c o m p i l i n g a h i s t o g r a m o f t h e f r e q u e n c y o f o c c u r e n c e . o f s t a t e s w i t h p o t e n t i a l e n e r g y d i f f e r e n c e U, - U, between x - d x / 2 and x + dx/2.

The f r e e e n e r g y d i f f e r e n c e i s t h e n g i v e n b y :

When t h e two d i s t r i b u t i o n s o v e r l a p c o m p l e t e l y o r p a r t i a l l y , any p o i n t x i n t h e o v e r l a p p i n g r e g i o n c a n b e p i c k e d , and t h e f r e e e n e r g y d i f f e r e n c e o b t a i n e d from t h e above e x p e s s i o n . When n o o v e r l a p o c c u r s b u t t h e h i s t o g r a m s a r e s i g n i f i g a n t l y b r o a d e r t h a n kT and s e p a r a t e d by a gap n o t much b r o a d e r t h a n t h e h i s t o g r a m s

themselves', e x t r a p o l a t i o n o f t h e two f u n c t i o n s I n h , ( x ) + x/2kT and I n h , ( x )

-

x/2kT g i v e s good e s t i m a t e s of Go - G,

.

When t h e t e c h n i q u e s d e s c r i b e d above a r e a p p l i e d t o c a l c u l a t e t h e e x c e s s

p r o p e r t i e s o f c r y s t a l l i n e d e f e c t s a d d i t i o n a l work i s i n v o l v e d . Yfe must e s t a b l i s h a l s o t h e thermodynamic p r o p e r t i e s of t h e b u l k r e f e r e n c e system i n c l u d i n g any number and s h a p e dependence. Then t h e e x c e s s thermodynamic p r o p e r t i e s o f t h e d e f e c t a r e found by t a k i n g t h e d i f f e r e n c e i n t h o s e o f t h e d e f e c t e d system and t h e p e r f e c t c r y s t a l r e f e r e n c e s y s t e m .

4. RESULTS OF OUR MD SIMmATIONS

The m o l e c u l a r dynamics t e c h n i q u e we u s e d i s b a s e d on an e x t e n s i o n due t o P a r i n e l l o and Rahmann ( 1 2 ) o f a L a g r a n g i a n f o r m u l a t i o n due t o Andersen ( 1 3 ) . I n t h i s s o - c a l l e d f l e x i b l e b o r d e r s t e c h n i q u e , t h e s i m u l a t i o n c e l l e d g e s a r e dynamic v a r i a b l e s , a l l o w i n g t h e c e l l t o change b o t h s i z e and s h a p e i n t h e c o u r s e o f t h e s i m u l a t i o n . T h i s t e c h n i q u e i s e x t r e m e l y i m p o r t a n t f o r t h e s t u d y of p h a s e t r a n s i t i o n s , which c a n o f t e n be p r e v e n t e d from o c c u r i n g i f one h a s a f i x e d c e l l .

P e r i o d i c boundary c o n d i t i o n s were employed. P a r t i c l e s i n o u r system were i n t e r a c t i n g t h r o u g h 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 . Most o f t h e c a l c u l a t i o n s were done w i t h a Lennard-Jones p o t e n t i a l :

0 0

w i t h p a r a m e t e r s c h o s e n t o s i m u l a t e a r g o n ( E =119.79 K and a ~ 3 . 4 0 5 )A. However a Hooke's law t y p e p o t e n t i a l ,

w i t h w = ( 2 - 1 ) a was u t i l i z e d a s r e f e r e n c e s y s t e m f o r t h e f r e e e n e r g y c a l c u l a t i o n s . The e q u a t i o n s o f motion were s o l v e d by t h e f i n i t e d i f f e r e n c e method.

4.1. G r a i n Boundarv P h a s e T r a n s i t i o n s :

We i n v e s t i g a t e d a two-dimensional =13 ( 0 = 27.80') t i l t boundary w i t h 1 1 0 p a r t i c l e s u s i n g t h e f l e x i b l e b o r d e r s t e c h n i q u e w i t h Lennard-Jones i n t e r a c t i o n s between f i r s t n e i g h b o u r s . E n t h a l p y and d e n s i t y o f t h e s y s t e m were c a l c u l a t e d o v e r

* 2

a wide r a n g e o f t e m p e r a t u r e s and a t a reduced h y d r o s t a t i c p r e s s u r e P = Pa / 4 ~ o f 0.4936. I n o r d e r t o d e c r e a s e t h e e n e r g y o f t h e r i g i d s y s t e m , o n e p a r t i c l e i n t h e boundary o v e r l a p p i n g t o o c l o s e l y w i t h i t s n e i g h b o u r s was removed. The r e l a x e d s t r u c t u r e , shown i n F i g . 1, was o b t a i n e d by h o l d i n g t h e s y s t e m a t low r e d u c e d t e n i p e r a t u r e s T * = T / ~ E = 0 . 0 1 f o r long t i m e s and c a l c u l a t i n g t h e time-averaged a t o m i c c o o r d i n a t e s . H e a t i n g t h e b i c r y s t a l t o 8 0 1 o f t h e b u l k m e l t i n g p o i n t and c o o l i n g i t bacl: down t o T * = 0 . 0 1 gave a n i d e n t i c a l e n t h a l p y v a l u e c o n f i r m i n g t h a t t h i s r e l a x e d s t r u c t u r e i s a w e l l - d e f i n e d l o c a l minimum.

The r e s u l t s of t h e s t u d y o f t h e v a r i a t i o n o f b i c r y s t a l e n t h a l p y and volume w i t h t e m p e r a t u r e a r e p l o t t e d i n F i g . 2 . The low t e m p e r a t u r e g r a i n boundary i s a

*

-

w e l l - d e f i n e d p e r i o d i c e n t i t y . Above T = 0.16 t h e e n t i r e system m e l t s . I n t h e r a n g e o f t e m p e r a t u r e s T * = 0.14 - 0 . 1 5 , where a d i s c o n t i n u i t y i n b i c r y s t a l e n t h a l p y and d e n s i t y o c c u r s , two phenomena a r e o b s e r v e d .

JOURNAL DE PHYSIQUE

'. .

7 w

-

w ³

U C, U 8 s

0 L -1 .r C,

2 _LO

z *

> 3 w

5 .E 2

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W U O C ,

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- w w

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L . V ) . V)- 8

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? w .r U U C , L .r

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m m

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A w rn- r +r 0 I - a J L

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3 . w C ,

g ^"9- _{t a w 6}

-1 C,

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U 0 h ta

-r 3 7

.a ^C, m C , Y w

7 s .r I1 aJ 7

W r n l aJ -0

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s n s

C, aJ L .r 0- h

-

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5249-

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4 5 z -

LO 3 s

ENTHALPY AND VOLUME VS. TEMPERATUFE 0 liquid like reoriented FIG. 2. Reduced bicrystal enthalpy and volurne.per particle versus reduced temperature. The open circles represent the low T behaviour of a perfect C=13 bicrystal. The black circles represent the bulk liquid system. The open and black squares stand respectively for a liquid- like interface and a reoriented bicrystal.

C4-220 JOURNAL DE PHYSIQUE

The f i r s t i s t h e r e p l a c e m e n t o f t h e boundary by a l i q u i d - l i k e l a y e r , which i s v i s i b l e i n F i g l b . T h i s i s t h e phenomena t h a t we c a l l t h e m e l t i n g t r a n s i t i o n . It

i s s i m i l a r t o t h e m e l t i n g t r a n s i t i o n o b s e r v e d p r e v i o u s l y o n a C =7 g r a i n boundary ( 5 ) . From o u r d a t a we e s t i m a t e t h e g r a i n b o u n d a r y m e l t i n g t e m p e r a t u r e t o b e 80%

of t h e b u l k m e l t i n g p o i n t .

The second phenomena we o b s e r v e i n t h e h i g h t e m p e r a t u r e r e g i o n , b u t below t h e b u l k m e l t i n g , i s t h e r e o r i e n t a t i g n phenomena. The m i g o r i e n t a t i o n c h a r a c t e r i z i n g t h e b i c r y s t a l c h a n g e s from 27.80 t o a p p r o x i m a t e l y 44

.

4.2. F r e e Energy C a l c u l a t i o n s

The two t e c h n i q u e s t o c a l c u l a t e f r e e e n e r g i e s were t e s t e d on 2d c l o s e - p a c k e d p e r f e c t c r y s t a l s w i t h d i f f e r e n t s i z e s and g e o m e t r i e s . The d e t e r m i n a n t t e c h n i q u e was a p p l i e d on b o t h Lennard-Jones and E o o k e ' s law t y p e c r y s t a l s t o compute t h e i r Helmholtz f r e e e n e r g i e s . Gibbs f r e e e n e r g i e s o f t h e Lennard-Jones c r y s t a l s were e s t i m a t e d b y computer c a l o r i m e t r y u s i n g t h e p r e v i o u s Hooke's law t y p e c r y s t a l s a s r e f e r e n c e s y s t e m s .

Systems h a v i n g 52, 104 and 208 p a r t i c l e s w i t h . f i r s t n e i g h b o u r i n t e r a c t i o n s were i n v e s t i g a t e d a t r e d u c e d t e m p e r a t u r e s which were low b u t h i g h enough t o be above

*

..

t h e Debye t e m p e r a t u r e ( T = 0.04) s o t h a t t h e y a r e i n t h e c l a s s i c a l l i m i t . The f r e e e n e r g y and e n t r o p y r e s u l t s o b t a i n e d by t h e two t e c h n i q u e s a r e i n good a g r e e m e n t : w i t h i n 5% f o r t h e e n t r o p y and l e s s t h a n 3% f o r t h e f r e e e n e r g i e s . The r e s u l t s a r e t a b u l a t e d i n T a b l e 1.

G e n e r a l t r e n d s b e s i d e s t h e e x p e c t e d t e m p e r a t u r e dependence of f r e e e n e r g i e s and e n t r o p y c a n b e e x t r a c t e d . The l o o s e r t h e s y s t e m i s , t h e h i g h e r ( i n a b s o l u t e v a l u e ) t h e f r e e e n e r g i e s and e n t r o p y a r e , a n a r r o w w e l l p o t e n t i a l and b u l k s h a p e s y s t e m making a c r y s t a l s t i f f e r . The d a t a we g a t h e r e d have n o t e n a b l e d u s t o e s t a b l i s h q u a n t i t a t i v e l y t h e number dependence o f t h e thermodynamic p r o p e r t i e s . Meanwhile, q u a l i t a t i v e l y , w h a t e v e r t h e p o t e n t i a l i s , e n e r g i e s and e n t r o p i e s

i n c r e a s e ( i n a b s o l u t e v a l u e ) w i t h t h e number o f p a r t i c l e s sitriulated. We s t u d i e d t h e e f f e c t o f second n e i g h b o u r i n t e r a c t i o n s on thermodynamic p r o p e r t i e s o f t h e Lennard-Jones c r y s t a l s . Second n e i g h b o u r i n t e r a c t i o n s make t h e s y s t e m s s t i f f e r and t h e r e f o r e lower t h e e n t r o p y , w h i l e t h e e n e r g i e s a r e d e c r e a s e d due t o more p o t e n t i a l e n e r g y .

S i m i l a r r e s u l t s were o b s e r v e d o n C =7 and C =13 b i c r y s t a l s , and a r e g i v e n i n T a b l e 3 . D i r e c t e v a l u a t i o n o f t h e f o r c e c o n s t a n t m a t r i x was n o t p o s s i b l e due t o g r a i n b o u n d a r y m i g r a t i o n . We t h e n employed a b i a s e d method c o n s i s t i n g o f a r t i f i c i a l l y expanding t o match t h e volume a t t h e d e s i r e d t e m p e r a t u r e a n a v e r a g e s t r u c t u r e g e n e r a t e d a t lower t e m p e r a t u r e where m i g r a t i o n d o e s n o t o c c u r . T h i s p s e u d o d e t e r m i n a n t t e c h n i q u e u n f o r t u n a t e l y y i e l d s r e l a t i v e l y h i g h d i s c r e p a n c y w i t h computer c a l o r i m e t r y e n t r o p i e s , which a r e more r e l i a b l e .

E x c e s s thermodynamic p r o p e r t i e s p e r u n i t l e n g t h of g r a i n boundary were c a l c u l a t e d from t h e computer c a l o r i m e t r y and d e t e r m i n a n t t e c h n i q u e d a t a

r e s p e c t i v e l y i n t h e c a s e of f i r s t and second n e i g h b o u r i n t e r a c t i o n . The r e s u l t s a r e g i v e n i n T a b l e 4, which a l s o l i s t s t h e r e f e r e n c e p e r f e c t c r y s t a l s . The C=7 g r a i n boundary h a s a s l i g h t l y lower e x c e s s e n e r g y and h i g h e r e x c e s s volume t h e n t h e C =13 b o u n d a r y . The e n t r o p i e s , however, d i f f e r b y a f a c t o r o f 10 i n f a v o u r o f t h e C = 7 , a t t h e t e m p e r a t u r e s s t u d i e d , q u a l i t a t i v e l y due t o a l o o s e r boundary

structure. The free energy of the C =7 is loner than that of the C =13. We expect that in the series of close-packed 2d CSL boundaries entropy will decrease with the excess volume

.

DET C C

Table 1

Perfect crystal thermodynamic data calculated by

molecular dynamics, the determinant technique, and computer calorimetry, in the case of first neighbour interactions. GEO stands for the geometry of the system and is expressed in reduced units (r*= rlc. PC stands for potentiall LJ for

Lennard-Jones and h for Hookers law type potential.

Table 2. Perfect crystal thermodynamic data, for the case of second neighboar interactions.

JOURNAL DE PHYSIQUE

'DET'

DET+ 'DET' C C

N

112

112

110

110

Table 3. Bicrystal thermodynamic data. 'DET' means that a biased determinant technique was used.

/ ^A \

lSt neighbor l S t neighbor 2nd neighbor GEO

C=7 6x21 C=7 6x21 C=13 4x30 C=13 4x30

N

112

Table 4. Excess thermodynamic quantities of 2 = 7 and C=13 grain boundaries.

PO

h

LJ

h

LJ

GEO C =7 6x21

PO

LJ T*

0.055

0.005

0.055

0.055

r

S*

35.48

125.25

32.24

104.18

T*

0.055 V*

121.65

124.63

119.59

121.50

F*

-67.98

-75.63

-68.94

-74.12 H*

-5.99

-7.23

-8.14

-8.42

V*

122.80

\

G*

-7.94

-14.11

-9.91

-14.15

H*

-15.42

F

S*

-

137.94

-

117.70

S*

101.63

F*

-

-76.33

-

-74.86

F*

-81.62

1

G*

-

-14.81

-

-14.89

G *

-21.01

5. DISCUSSION

The r e s u l t s of o u r m o l e c u l a r dynamics s t u d i e s t o d a t e a r e i n t r i g u i n g , b u t t h e y d e s e r v e more i n t e n s i v e s t u d y i n t h e f u t u r e . The method of c a l c u l a t i n g e x c e s s thermodynamic p r o p e r t i e s e x p l i c i t l y t h r o u g h t h e u s e o f p e r f e c t c r y s t a l r e f e r e n c e s y s t e m s h a s t h e a d v a n t a g e t h a t i t a l l o w s u s t o p i n p o i n t an i n t e r f a c i a l phase t r a n s i t i o n unambiguously. However, i t o f f e r s i n i t s e l f l i t t l e knowledge of t h e n a t u r e o f t h e t r a n s i t i o n . C l e a r l y an i m p o r t a n t t a s k i s t o c h a r a c t e r i z e t h e s t r u c t u r e o f t h e l i q u i d - l i k e l a y e r a f t e r t h e t r a n s i t i o n and i t s v a r i a t i o n w i t h t e m p e r a t u r e more t h o r o u g h l y and q u a n t i t a t i v e l y . The q u a s i - e x p e r i m e n t a l problems o f a s s e s i n g t h e s t r u c t u r e o f t h i s l a y e r a r e n o n - t r i v i a l b e c a u s e o f i t s h i g h m o b i l i t y . The s i t u a t i o n i n f a c t mimics t h e e x p e r i m e n t a l s c e n a r i o when one t r i e s t o p r o b e atomic s t r u c t u r e of w e l l - p r e p a r e d b i c r y s t a l s a t e l e v a t e d t e m p e r a t u r e s and t h e boundary l e a v e s t h e b i c r y s t a l d u r i n g t h e c o u r s e o f t h e e x p e r i m e n t . Eiowever t h i s i s a c o m p u t a t i o n a l i n c o n v e n i e n c e t h a t c a n and w i l l b e overcome t h r o u g h t h e u s e o f more i m a g i n a t i v e though time-consuming c o m p u t a t i o n a l t e c h n i q u e s . To a l a r g e d e g r e e t h i s i s one of a l a r g e c l a s s of problems which a r i s e due t o t h e s m a l l s i z e s o f o u r s i m u l a t i o n c e l l s , a problem t h a t w i l l b e a l l e v i a t e d , b u t n o t

e l i m i n a t e d . i n t h e c o u r s e of time a s c o m p u t u a t i o n a l power i n c r e a s e s . I n t h e a b s e n c e of t h e p o s s i b i l i t y of macroscopic s c a l e s i m u l a t i o n c e l l s we must c o n t i n u e t o t r y t o s y s t e m a t i c a l l y e v a l u a t e t h e dependence of t h e p r o p e r t i e s of t h e system on t h e s i z e and geometry of t h e c e l l c h o s e n .

A l a r g e u n s e t t l e d q u e s t i o n i s t h e mechanism and k i n e t i c s of t h e r e o r i e n t a t i o n t r a n s i t i o n . A d e t a i l e d s t u d y t o of t h i s phenomenon i s p l a n n e d f o r t h e X =13 boundary and f o r o t h e r s .

I n t h e a r e a of f r e e e n e r g y c a l c u l a t i o n a l a r g e amount of work r e m a i n s t o b e done. We a r e now c a p a b l e of q u a n t i t a t i v e l y e v a l u a t i n g t h e r e l a t i v e s t a b i l i t y o f d i f f e r e n t i n t e r f a c e s a t f i n i t e t e m p e r a t u r e . Both t h e t e c h n i q u e s f o r c a l c u l a t i n g f r e e e n e r g i e s , however, have t h e i r annoying a s p e c t s . The d e t e r m i n a n t t e c h n i q u e i s l i m i t e d i n i t s v a l i d i t y t o low t e m p e r a t u r e s . The computer c a l o r i m e t r y t e c h n i q u e h a s i t s own i n t r i n s i c l i m i t a t i o n s . Both systenls have t o b e d e f i n e d on t h e same c o n f i g u r a t i o n s p a c e which means t h a t we c a n n o t u s e i t t o e v a l u a t e d i r e c t l y f r e e e n e r g y d i f f e r e n c e s between d i f f e r e n t boundary s t r u c t u r e s . We a r e t h e n o b l i g e d t o c a l c u l a t e e x p l i c i t l y t h e f r e e e n e r g y of e a c h s t r u c t u r e , t h r o u g h t h e u s e of a r e f e r e n c e system of known f r e e e n e r g y . I f o u r r e f e r e n c e s y s t e m h a s a l i m i t e d r a n g e of a p p l i c a b i l i t y , a s o u r quasi-harmonic r e f e r e n c e d o e s , t h e n we w i l l b e l i n i i t e d s e v e r e l y i n t h e r e g i m e s where we c a n e f f e c t i v e l y u s e computer

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

It i s o u r i n t e n t i o n now t o s y s t e m a t i c a l l y e v a l u a t e t h e e f f e c t o f g r a i n boundary p h a s e t r a n s i t i o n s on t h e p h y s i c a l p r o p e r t i e s of t h e i n t e r f a c e s . The p r o p e r t i e s we have c h o s e n f o r p a r t i c u l a r a t t e n t i o n a r e t h e g r a i n boundary m e c h a n i c a l

p r o p e r t i e s .

6. CONCLUSIONS

The c o n c l u s i o n s we c a n draw from o u r s t u d y a r e t h e f o l l o w i n g . A t r a n s i t i o n , which we c a l l a m e l t i n g t r a n s i t i o n , o c c u r s i n t h e 2d t i l t b o u n d a r i e s we have s t u d i e d . The t r a n s i t i o n o c c u r s a t a p p r o x i m a t e l y 80% o f t h e b u l k m e l t i n g p o i n t , and i s r e f l e c t e d i n d i s c o n t i n u i t i t e s i n t h e c a l c u l a t e d b i c r y s t a l e n t h a l p y and d e n s i t y . The t r a n s i t i o n i s c h a r a c t e r i z e d by t h e r e p l a c e m e n t of a well-ordered p e r i o d i c g r a i n boundary w i t h a l i q u i d - l i k e l a y e r o f I . i g h n o b i l i t y . I n one of o u r b o u n d a r i e g , t h e C =13, we o b s e r v e a n o t h e r phenomena. The boundary o r i e n t a t i o n of

0 =27.80 , which i s s t a b l e a t low t e n l p e r a t u r e s , i s o r e p l a c e d a t h i g h e r

t e m p e r a t u r e s w i t h an o r i e n t a t i o n of a p p r o x i m a t e l y 44

.

C4-224 JOURNAL DE PHYSIQUE

We have s u c c e s s f u l l y d e v e l o p e d t e c h n i q u e s f o r c a l c u l a t i n g f r e e e n e r g i e s of g r a i n b o u n d a r i e s . We a r e t h u s now a b l e t o a d d r e s s f o r t h e f i r s t t i m e t h e q u e s t i o n o f t h e r e l a t i v e s t a b i l i t y o f d i f f e r e n t g r a i n boundary p h a s e s a t f i n i t e

t e m p e r a t u r e . B o t h t e c h n i q u e s , t h e d e t e r m i n a n t t e c h n i q u e , which i n v o k e s t h e q u a s i h a r m o n i c a p p r o x i m a t i o n , and t h e computer c a l o r i n ~ e t r y t e c h n i q u e have b e e n a p p l i e d t o t h e s t u d y o f t h e C =7 and C =13 g r a i n b o u n d a r i e s and t o t h e i r p e r f e c t c r y s t a l r e f e r e n c e s y s t e m s . We found t h e C =7 boundary t o b e s t a b l e w i t h r e s p e c t t o t h e C =13 boundary a t t h e t e m p e r a t u r e s s t u d i e d due t o i t s s i g n i f i g a n t l y l a r g e r e n t r o p y . The t e c h n i q u e s f o r f r e e e n e r g y c a l c u l a t i o n need t o b e expanded and improved.

We would l i k e t o acknowledge t h e s u p p o r t o f t h i s r e s e a r c h t h r o u g h t h e NSF g r a n t

#DhR 81-1925 and t h r o u g h a P r e s i d e n t i a l Young I n v e s t i g a t o r Award. I n a d d i t i o n we a r e p a r t i c u l a r l y g r a t e f u l f o r t h e h o s p i t a l i t y o f t h e Max P l a n c k I n s t i t u t f u r l e t a l l f o r s c h u n g . I n s t i t u t f i i r W e r k s t o f f w i s s e n s c h a f t e n i n S t u t t g a r t , where a p o r t i o n o f t h i s work was a c c o m p l i s h e d , and i n p a r t i c u l a r t o D r . Manfred Riihle, o u r h o s t .

REFERENCES

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(1983) 5576.

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8. Glicksman, M.E. and Vold. C.L.. A c t a . Met.. 1 3 . ( 1 9 6 9 ) 1.

9 . Watanabe, T . , Kimura, S. and Karashima. S . , P h i l . Mae. A, 49 (1984) 845.

1 0 . Hoover, W.G.. Hindmarsh. A.C. and E o l i a n , B.L.. J . of Chen. P h v s . , 57 (1972) 1980.

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1 3 . Andersen, B.C., J . Chem. Phvs., 72 (1980) 2384.

DISCUSSION

D.R. C l a r k Do your r e s u l t s r e v e a l any evidence o f f a c e t t i n g ?

G. Kalonji: No. If one c o n s i d e r s , though, t h e geometrical c o n s t r a i n t s a r t i f i c i a l l y imposed on t h e system by t h e p e r i o d i c boundary c o n d i t i o n s , which may, i n f a c t , mimic n a t u r e i n t h e l i m i t o f small g r a i n s i z e , one can only conclude t h a t t h e p o s s i b i l i t y o f studying f a c e t t i n g depends d i r e c t l y on t h e computer budget. Wait a few months.