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THE STATISTICAL GEOMETRY OF THE
STRUCTURE OF THE MOLECULAR DYNAMIC
MODEL OF LIQUID AND AMORPHOUS
ALUMINIUM
V. Poluchin, M. Dzugutov, V. Uchov, R. Vatolin
To cite this version:
JOURNAL DE PHYSIQUE CoZZoque C8, suppZdment au n08, Tame 42, a o c t 2980, page C8-284
T H E S T A T I S T I C A L GEOMETRY OF T H E STRUCTURE OF T H E MCLECULAR DYNAMIC MODEL OF L I G U I D AND AMORPHOUS A L U M I N I U M
V.A. P o l u c h i n , M.M. Dzugutov, V.F. Uchov and X.A. V a t o l i n
I n s t i t u t e o f Metallurgy, UraZ Science Research Centre o f t h e USSR Acadew o f S c i e n c e s , S v e r d l o ~ v s k , USSR.
The p r e s e n t paper d e a l s with t h e t h e case of metals depends on t h e den- s t a b i l i t y of t h e l i q u i d s t a t e and with s i t y of t h e system).
t h e t r a n s i t i o n t o t h e metastable amorph- The system was cooled by i n s t a n t - ous s t a t e . An attempt i s made t o con- aneous v a r i a t i o n of t h e v e l o c i t i e s of s t r h c t a model of t h e m e t a l l i c amorphous a l l p a r t i c l e s (corresponding t o about phase which i s known t o possess a number 25
K)
a t some i n t e g r a t i o n s t e p . Thenof unique mechanical p r o p e r t i e s and t o be capable of conserving i t s s t r u c t u r e a t normal temperatures?
The r e a l i s t i c i t y of mathematical models is determined primaxily by t h e i r
a b i l i t y t o reproduce t h e experimentally measured r a d i a l d i s t r i b u t i o n f u n c t i o n
(RDF) o r t h e s t r u c t u r e f a c t o r . The mo- d e l s constructed using t h e method of random c l o s e packing of s o l i d spheres
/I/
which a r e then subjected t o a r e l a x a t i o n process with t h e use of various kinds of s o f t i n t e r a c t i o n p o t e n t i a l s / 2 ,3/
do .not s a t i s f y t h e above c r i t e r i o n , which ap- p a r e n t l y is due t o t h e packing procedure.The nolecul~r-dynamic model con- s i s t i n g of 2048 p a r t i c l e s with p a i r e d i n t e r a c t i o n , which reproduces i n d e t a i l t h e experimentally measured s t r u c t u r e f a c t o r of l i q u i d aluminium /4/, was sub- j e c t e d t o a s e r i e s of instantaneous i a o c h o r i c supercoolings ( t h e condition of i a o c h o r i c i t y permits conservation of t h e i n i t i a l p a i r e d p o t e n t i a l which i n
C . S . Hsu, A. Rahman (J.C.P.
70,
5234 (1979)).we observed t h e evo1ut;ion of t h e system u n t i l i t came t o equilibrium. I n t h e equilibrium s t a t e we c a l c u l a t e d the mean square o f t h e displacement of p a r t i c l e s . Using t h e well known asymptotic r e l a t i o n
for<^^>*
witht--=
<R~)*
'D t / 6
+ C ,where C is a constant it i s p o s s i b l e t o e s t i m a t e t h e temperature dependence of t h e s e l f - d i f f u s i o n c o e f f i c i e n t
D
.
When t h e temperature of t h e system
is 85 K t h e d i f f u s i o n p r a c t i c a l l y ceases t o e x i s t i n t h e system. Observa- t i o n s of t h e curve r i t h i r i t h e time i n t e r v a l 0 . 5 ' 1 0 ' ~ ~ s e c s showed t h a t t h e s e l f -dif f usion c o e f f i c i e n t
D
decseas- e s with decreasing temperature p a r t i c u l - a r l y sharply (almost by a f a c t o r of lo4) i n t h e region of 86K.
The s t r u c t u r e of t h e system was
analyzed on t h e b a s i s of t h e c a l c u l a t e d difference f u n c t i o h
~ x R ~ [ ~ ( R )
-&I.
where f, is t h e mean densiky of t h e system and
f(R)
is t h e mean d e n s i t y a ta d i s t a n c e from each p a r t i c l e ( f i g . 1).
I A
e
i
Q.
+'
3
%3
2
1
10
20
R , A.E.
Pig. '1 : RDF f o r l i q u i d (A) and amorphous (B) aluminium
Pig. 2 p r e s e n t s t h e s t r u c t u r e f a c t o r S(K)
which may be obtained by Fourier t r a n s - formation of t h e d i f f e r e n c e d e n s i t y func- t i o n
fl(R).
Comparing f i g s 1 and 2 with t h e mean-square displacement c a l c u l a t i o n r e s u l t s it can be e a s i l y seen t h a t t h e sudden disappearance of d i f f u s i o n i n t h e system a tT
= 8.5 K is not accompanied by any q u a l i t a t i v e changes i n t h e struo- t u r e . %%is i n d i c a t e s t h a t a tT
=85
K our- aodel system passes t o a s o l i d "amor- phous" s t a t e without any s i g n s of, t h e occurrence of a c r y s t a l s t r u c t u r e . TheFig. 2 :
S ( k )
f o r l i q u i d (A) and amorphous(B)
aluminium."shoulder" appearing i n t h e second peak
S(K)
is not transformed t o an a d d i t i - onal maximum whose presence is charac- t e r i s t i c of a number of amorphous metals /5, 6 ,7/.
It should be noted t h a t t h e so c a l l e d mean-f i e l d theopies/ 8 / based on t h e d e n s i t y functional. associated f h e q u a s i - c r i t i c a l behaviour of t h e t r a n s f e r c o e f f i c i e n t s with t h e anomalous behaviour, not confirmed ex- perimentally
/5,
6/, of t h e s t a t i c s t r u c t u r e f a c t o r i n t h e region of t h e wave v e c t o r corresponding t o t h e s t a b l e cryskal l a t t i c e of She substance under analyses.C8-286 JOURNAL DE PHYSIQUE noted e a r l i e r i n normal l i q u i d aluminium
/I/.
The occurrence of t h e s e peaks evid-ences t h a t t h r e e neighbowing p a r t i c l e s a r e most l i k e l y t o form angles m u l t i p l e of 60°, which i s i n agreement with our e a r l i e r assumption /LC/. Since t h e RDF gives no p e n e t r a t i n g i n s i g h t i n t o t h e l o c a l ordering i n t h e s t r u c t u r e i n v e s t i g a t e d w e have performed a s t a t i s t f i a l and geometric a n a l y s i s of t h e r e s u l t i n g amorphous and l i q u i d s t a - t e s . The a n a l y s i s i s based on t h e concept of Voronoy polyhedra /9/ ( f i g .
3 ) .
The l a t t e r were c l a s s i f i e d by a s e t of indi- c e sn,
,
n,
,
n,
,.
...
,
whereni
ist h e number of i hedral edges o f a given polyhedron. Attention should be given t o
t h e f a c t t h a t i n both s t a t e s t h e r e a r e present a considerable number of poly- hedra such a s 0365, 044.6, 0366, 0447 oc- c u r r i n g i n t h e r e a l f c c l a t t i c e /10/. A t t h e same time, 0608 t n e polyhedra typic- a l f o r t h e bcc s t r u c t u r e
/lo/
a r e absent i n t h e system under i n v e s t i g a t i o n . This f a c t may be a t t r i b u t e d t o t h e occurrence of fcc-type c r y s t a l l i n e n u c l e i due t o t h e c r y s t a l l i s a t i o n of aluminium.The s t a t i s t i c of t h e angles formed by d i f f e r e n t p a i r s of t h e n e a r e s t neigh- b o w s of a given ion i s shown is f i g . 4, f o r l i q u i d and amorphous s t a t e s . What i s t y p i c a l f o r t h e amorphous s t a t e i s t h e
presence of two sharp maxima correspond- Fig.
1
: R e l a t i v e n m b e r of xtost f r e - ing t o t h e angles 45' and 60'. F u r t h e r , A quent-
l i q u i d , B y occurring Voronoy polyhedra,-
amorphous s t a t e . t h e r e is a s e r i e s of small peaks f o r theevidently a s a r e s u l t of thermal f l u c - v a l u e s of t h e angles which a r e a l i n e a r
t u a t i o n s . B y comparing f i g s 1 and 4 one combination of t h e above two peaks. For
t h e l o c a l order occurring i n t h e system being considered. The maxima a t
R,
=5.3
atomic u n i t s i n f i g . 1 and with t h e angle 60' may be explained by t h e presence of e q u i l a t e r a l t r i a n g l e s with t h e s i d e
R,
i n t h e s t r u c t u r e . The quantity I?, may be regarded a s an e f f e c t i v e ion diameter.Fig. 4 : Angular d i s t r i b u t i o n of d i f f e r - e n t p a i r s of n e a r e s t neighbours, A
-
l i q u l d , B-
amocphous s t a t e .!l%en t o t h e angular d i s t r i b u t i o n peak a t
45' t h e r e should correspond t h e d i s t a n c e t o t h e n e a r e s t neighbour which is equal t o
R,P
.
I n f i g . I t h e r e is r e a l l y t h e corresponding peak a t t h e above-men- tioned d i s t a n c e f o r t h e amo-rphous s t a t e . Meanwhile, t h i s peak is absent i n t h eRLlF f o r l i q u i d aluminium, which agrees with t h e f a c t t h a t t h e s e p a r a t e peak f o r t h e angle 45' is smeared. f o r t h i s s t a t e . The a n a l y s i s performed permits some con- c l u s i o n s concerning t h e p o s s i b l e types of Delone t e t r a h e d r a of t h e s t r u c t u r e
C8-287 being i n v e s t i g a t e d where t h e v e r t i c e s a r e determined by s e t s of f o u r n e a r e s t neighbours
/9/.
The observed maxima i n t h e angular d i s t r i b u t i o n determine t h e dominating value of t h e i n t e r i o r angles of t h e t r i a n g l e s forming t h e edges of' t h e above t e t r a h e d r a (60' i n t h e l i q u i d , 4.5' and 60' i n an amorphous body). A s isknown, t h e r e s u l t s f o r
s(<)
obtained by d i f f r a c t i o n methods i n d i c a t e t h a t t h e s t r u c t u r e of the l i q u i d s t a t e is isotrop- i c i n t h e macroscopic s c a l e . However, i n t h e molecular-dynamic,models t h e l o c a l o r d e r l i n e s s manifests i t s e l f i n t h e an- g u l a r d i s t r i b u t i o n of n e a r e s t neighbours. To t h i s end we made an attempt t o inves- t i g a t e t h e anisotropy of l i q u i d and amor- phous aluminium within t h e framework of a molecular-dynamic model. The calcula- t i o n was performed only f o r t h e c e n t r a l group of atoms, t o e l i m i n a t e boundary e f f e c t s.
Table 1 c a r r i e s c a l c u l a t i o n r e s u l t s f o r S(G)
.
In s p i t e of t h e pres- ence of a well defined l o c a l anisotrdpy, t h e sequence of maximum-density planesI i s i r r e g u l a r i n t h e s c a l e of the e n t i r e model. The p r o j e c t i o n of t h e d e n s i t y of p a r t i c l e s t o t h e d i r e c t i o n of t h e main maxima &so p o i n t s t o t h e absence of well defined p e r i o d i c c o r r e l a t i o n s of global character. The r a t i o of t h e
max-
imum value of
s(<
) t o t h e value aver- aged with r e s p e c t t o a l l d i r e c t i o n sJOURNAL DE PHYSIQUE C8-288 f a c t o r of t h e system b e i n g considered and i t s s p a t i a l s c a l e a r e e v i d e n t l y due t o t h e presence of a s e t of independent s c a t t e r e r s , i.e. s e p a r a t e l o c a l s e c t i o n s w i t h ordered s t r u c t u r e . The s i z e of the- s e s e c t i o n s i s much l e s s t h a n t h a t of t h e model involved.
Table 1
Some peaks ~ ( f ? ) may be explained a s r e s u l t i n g from t h e random superposi- t i o n of s c a t t e r i n g from d i f f e r e n t inde- pendent small volumes w i t h a l o c a l p e r i - o d i c d e n s i t y c o r r e l a t i o n . /I/ B e r n a l , J. D., Nature 188 (1960), 910. rand. 7.5 6.7 7.1 7.6
/2/ Heimendahl, L. V., Journ. Phys. F , 5 (1 975) 141
-
/3/ Leung, P. K . , Quin, J. J . , Wright,
SmoxCG) 11.4 5.1 6.0 8.9 J. G . , P h i l . Mag.
-
38 (1978),
P a r t 2, 127. /4/ Uchov, V. Ph., D Z U ~ U ~ O V , M. M., Polu- c h i n , V. A . , S o v i e t Phys.-
F i z . Model 12 MD MD MD L Metallov i Metallovedenie 47 (1979) A 904../5/
C a r l s o n , D. G., Feder, J., Segmiiler, A ,,
Phys.
Rev.Ag
(1 974) 400./ 6 / B i z i d , A., Bosio, L., .Curien, H.,
Defrain, A . , Dupont, M . , P.hys. S f . S o l i d i ( a )
-
23 (1974) 135. Sk,) 2.93 2.78 2.78 2.78JV
890 277 527 1080 /7/ B e r n a l , J. D., I n : P w s i c s of Simple L i q u i d s , Amsterdam, 1968. /8/ S j o l a n d e r , A., T u r s k i i , L. A . , Journ. Phys.C
(1973) 1973./9/ Rogers, C
