ORIENTATIONAL ENVIRONMENTS AND HIGH ORDER CORRELATION FUNCTIONS IN LIQUIDS

HAL Id: jpa-00225265

https://hal.archives-ouvertes.fr/jpa-00225265

Submitted on 1 Jan 1985

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

ORIENTATIONAL ENVIRONMENTS AND HIGH

ORDER CORRELATION FUNCTIONS IN LIQUIDS

A. Haymet

To cite this version:

JOURNAL

DE

PHYSIQUE

Colloque C9, supplément au n012, Torne 46, décembre 1985 _{page c9-27}

ORIENTATIONAL ENVIRONMENTS AND HIGH ORDER CORRELATION FUNCTIONS I N L I Q U I D S

A.D.J. ~ a ~ m e t *

Department of Chemistry, University of California, Berkeley, California 94720, U.S.A.

Abstract

-

Recent experimental and t h e o r e t i c a l s t u d i e s o t amorphous s o l i d s and supercooled l i q u i d s have generated renewed i n t e r e s t i n t r i p l e t and higher-order c o r r e l a t i o n f u n c t i o n s i n n o n c r y s t a l l i n e m a t e r i h l s . Here we discuss t h e conhection between o r i e n t a t i o n a l l y ordered condensed phases and high-order c o r r e l a t i o n functions.

1. INTRODUCTION

Since t h e pioneering work o f Yvon, Kirkwood, and Born and Green /1,2/, p a i r and t r i p l e t c o r r e l a t i o n f u n c t i o n s have been s t u d i e d i n t e n s e l y t o t r y - t o o b t a i n a workable, f i r s t p r i n c i p l e s theory o f l i q u i d s . We review t h e progress i n such a t h e o r y beyond t h e s u p e r p o s i t i o n approximation, two numeri c a l methods f o r c a l c u l a- t i n g t h e t r i p l e t c o r r e l a t i o n f u n c t i o n g3 and p o s s i b l y high-order functions, t h e reasons f o r c o n t i n u i n g t h e development o f t h i s approach, and t h e i n t e r e s t i n g p r o p e r t i e s o f l i q u i d s which can be obtained. We discuss t h e r o l e o f g3 i n thermodynamic p r o p e r t i e s , such as t h e entropy, and some simple connections w i t h experiments.

Ihe f a c t t h a t long-range o r i e n t a t i o n a l order can e x i s t i n t h e absence o f t r a n s l a t i onal ( p e r i o d i c ) order has been r e a l i zed r e l a t i v e l y r e c e n t l y . The d e s c r i p t i o n o f such o r d e r i n g by c o r r e l a t i o n t u n c t i o n s i s n a t u r a l and useful. For example, t h e s t r u c t u r e o f t h e so-called " h e x a t i c " phase which can e x i s t , i n p r i n c i ple, i n some two dimensional m a t e r i a l s i s d e s c r i bed by t h e t o u r t h - o r d e r c o r r e l a t i o n function.

High-order c o r r e l a t i o n f u n c t i o n s p l a y important r o l e s i n many aspects o f t h e t h e o r y o f condensed phases. A f i r s t p r i n c i p l e s t h e o r y o f l i q u i d s t r u c t u r e /3-13/, based on d i r e c t c a l c u l a t i o n o f t h e t r i p l e t c o r r e l a t i o n f u n c t i o n , seems t o be w i t h i n reach. Dynami c a l p r o p e r t i e s o f dense 1 iq u i d s a r e i n f l uenced by h i gh-order s t a t i c c o r r e l a t i o n s /14-15/. The connection between " c l a s s i c a l " i n t e g r a l equation

*u.s.

Presidential Young Investigator, 1985

-

1990

Cg-28 JOURNAL

DE

PHYSIQUE

methods and modern r e n o r m a l i z a t i o n group methods i n v o l v e s t r i p l e t and p o s s i b l y higher order c o r r e l a t i o n f u n c t i o n s 117-19/. Recent s t u d i e s o f molten s a l t s 120-21/, molecular l i q u i d s 122-23/, and m e t a l l i c glasses /24/ d i s p l a y c l e a r l y t h e c e n t r a l r o l e o f t r i p 1 e t c o r r e l a t i o n f u n c t i ons 1251.

I n a d d i t i o n , recent developments i n t h e t h e o r y o f long-range o r i e n t a t i o n a l order i n l i q u i d s and amorphous m a t e r i a l s /26-39/ may be understood i n terms o f high- order c o r r e l a t i o n f u n c t i o n s , although t h e t h e o r i e s a r e not always phrased i n t h e t r a d i t i onal 1 anguage o f c o r r e l a t i on f u n c t i ons.

I n Section 2 we review t h e c a l c u l a t i o n from f i r s t p r i n c i p l e s o f t h e t r i p l e t c o r r e l a t i o n f u n c t i o n f o r .simple 1 iq u i d s and some important a p p l i c a t i o n s . The importance o f a c l o s e l y r e l a t e d q u a n t i t y , t h e t r i p l e t s t r u c t u r e f a c t o r ~ ( ~ 1 , i s discussed i n Section 3, w f t h emphasis on modern d e n s i t y f u n c t i o n t h e o r i e s o f f r e e z i n g 140-49/. P r e l im i n a r y r e s u l t s from a molecular dynamics s i m u l a t i o n designed t o measure s ( ~ ) are presented. F i n a l l y , a more approximate " d e n s i t y f u n c t i o n a l " t h e o r y i s used t o discuss t h e o r d e r i n g i n c e r t a i n metal a l l o y s which appear t o have q u a s i p e r i o d i c ( r a t h e r than s t r i c t l y p e r i o d i c )

1

ong range order. This l e v e l o f theory has proved u s e f u l i n d e s c r i b i n g t h e f r e e z i n g o f simple models such as hard spheres and Lennard-Jones p a r t i c l e s i n t o p e r i o d i c c r y s t a l s , and i t may be a b l e t o p r e d i c t t h e s t a b i l i t y , o r otherwise, o f icosahedral o r pentagonal q u a s i p e r i o d i c s o l ids.

Before proceeding, 1 would l i k e t o thank Professors S.A. Rice and D.R. Nelson, who i n t r o d u c e d me t o l o c a l and long-range o r i e n t a t i o n a l order i n l i q u i d s , and

Professor W.G. Madden, w i t h whom much o f t h e work on t r i p l e t c o r r e l a t i o n f u n c t i o n s was done.

II. TRIPLET CORRELATION FUNCTIONS

The importance o f t r i p l e t and h i g h e r order c o r r e l a t i o n t u n c t i o n s i s seen even i n t h e simpl e s t 1 iq u i d s , which a r e discussed here. These c o r r e l a t i o n f u n c t i o n s p l a y an even more important r o l e i n two component systems; f o r example, i n molten s a l t s /20,21/ and i n i n t e r a c t i o n s i t e models (ISM) o f molecular l i q u i d s /22,23/. To i n t r o d u c e t h e c e n t r a l ideas we use t h e BBGKY h i e r a r c h y o f i n t e g r o - d i f f e r e n t i a l equations f o r t h e c o r r e l a t i o n f u n c t i o n s o f many-particle systems. As a simple example, consider a c l a s s i c a l

,

one-component t l u i d o f N i d e n t i c a l

,

s t r u c t u r e l e s s p a r t i c l e s w i t h p o s i t i o n s

{ r i )

enclosed i n a volume V a t a temperature T. Each p a i r o t p a r t i c l e s i n t e r a c t s w i t h a p o t e n t i a l u ( r . .), where r - .=Ici -r.

1.

The p a i r

1 J 1 J Y

d where k ~ , = N/V, t =

I L - ~ I

and u l ( r ) = fl(r).

The d i f f i c u l t y w i t h t h i s equation i s , o f course, t h a t t h e t r i p l e t c o r r e l a t i o n f u n c t i o n g ( 3 ) ( r , s , t ) i s an unknown, complicated q u a n t i t y . For many years an exact r e l a t i o n has been known which expresses g ( 3 ) as a f u n c t i o n a l o f g(2), a l b e i t one which an i n f i n i t e number o f terms 131. Recently, t h e numerical accuracy o f t h i s expansion, t r u n c a t e d a f t e r a f i n i t e number o t terms, has been i n v e s t i g a t e d 14-81. For a s i n g l e component t h e expansion i s

The approximation

r

= 1 i s c a l l e d t h e Kirkwood s u p e r p o s i t i o n approximation and i s known t o be inadequate t o r l i q u i d s o f one /4/ (and two /9,10/) components. A number o f workers, most completely S t e l l 131, have d e r i v e d t h e expansion f o r t h e c o r r e c t i o n s t o s u p e r p o s i t i o n r ( r , s , t ) , i n terms o f "h-bonds," w h e ~ e h ( r ) = g ( r )

-

1. By c a l c u l a t i n o t h e f i r s t two terms i n t h i s s e r i e s we have generated accurate t r i p l e t c o r r e l a t i o n f u n c t i o n s and an attempt a t a f i r s t - p r i n c i p l e s theory o f t h e l i q u i d s t a t e f o r t h e Lennard-Jones l i q u i d 141 and f o r a mode1 o f l i q u i d sodium 161. This theory compliments t h e very successful p e r t u r b a t i o n t h e o r i e s o f t h e

1 i q u i d s t a t e : t h e above t h e o r y addresses t h e expl i c i t l y excl uded v o l ume p r o b l em, u n l i k e those t h e o r i e s which r e l y upon a computer-simulated hard sphere reference s t a t e , and which are i n t h i s sense semiempirical 151.

The c o r r e c t i o n s t o t h e s u p e r p o s i t i o n approximation a r e contained i n t h e expression

For a s i n g l e component system t h e f i r s t term i n t h i s s e r i e s i s

where h . . = g( l

r. -r.

1 )

-

1

.

I he second order term i s t h e sum o t seven separaTe

1 J -1 -J

c o e t t i c i e n t s , each i n v o l v i n g a six-dimensional i n t e g r a t i o n . A convenient diagram- m a t i c r e p r e s e n t a t i o n o f these terms has been published 141. l h e f i r s t two terms o f t h e s e r i e s (2.3) have been evaluated by expanding one o r more h-bonds i n

Legendre polynomials /4,6/, i n t e g r a t i n g over some degrees o f freedom a n a l y t i c a l l y and completing t h e i n t e g r a t i o n s numerically. F u l l d e t a i l s may be found i n references 4 and 6.

C9-30

JOURNAL

DE PHYSIQUE

c a l c u l a t e d t r i p l e t c o r r e l a t i o n f u n c t i o n s w i t h computer s i m u l a t i o n "experiments" 151. The t r i p l e t c o r r e l a t i o n f u n c t i o n s f o r e q u i l a t e r a l c o n f i g u r a t i o n s of t h e Lennard-Jones f l u i d are shown i n Figures 1 and 2 f o r two thermodynamic states.

F i g u r e 1 Figure 2

T & i p l e t c o r r e l a g i o n s f o r e q u i l a t e r a l congigurations of,the Lennard-Jones f l u i d a t

p = 0.80 and T = 2.74 (Figure 1) and p = 0.85 and T = 0.73, from Reference

5.

The open c i r c l e s a r e molecular dynamics r e s u l t s ; t h e s o l i d - l i n e s g i v e t h e r e s u l t s o f t h e second-order h-bond theory. The e r r o r bars a r e t h e 95% confidence 1 i g j t s described i n t h e reference. (a) The t r i p l e t c o r r e l a t i o n f u n c t i o n

g r , r , r ) . (b) The r a t i o r ( r , r , r ) of t h e t r i p l e t c o r r e l a t i o n f u n c t i o n t o i t s s u p e r p o s i t i o n estimate. C e r t a i n i s o s c e l e s c o n f i g u r a t i o n s have a l s o been c a l c u l a - t e d and measured by molecular dynamics, and a r e contained i n Reference 5.

At h i g h d e n s i t y and low temperature, near t h e t r i p l e p o i n t , t h e Legendre expansion i s n o t i n complete agreement w i t h computer s i m u l a t i o n r e s u l t s , although i t i s much b e t t e r than t h e Kirkwood approximation

r

= 1.

To check t h e convergence o f t h e s e r i e s (2.3), we are c a l c u l a t i n g a number o f " l o o s e l y connected" t h i r d order diagrams. This f o l l o w s t h e suggestion /5/ t h a t , s i n c e t h e highly-connected diagrams c o n t r i bute l i t t l e t o equation (2.3), t h e diagrammatic s e r i e s be reorganized according t o =order r a t h e r than nodal order. The new diagrams c a l c u l a t e d are shown i n F i g u r e 3, and i n c l u d e t h e s i n g l e s i x node/six bond diagram (a), and t h e f o u r u n l a b e f l e d diagrams (b)-(e) which r e s u l t from t h e 18 t o p o l o g i c a l l y d i s t i n c t s i x nodelseven bond diagrams. Using t h e d i r e c t numerical i n t e g r a t i o n method described above, w i t h i n Our present numerical accuracy, these diagrams appear t o be small even a t t h e t r i p l e p o i n t density. However, f u r t h e r refinement o f t h i s c a l c u l a t i o n i s required.

F i g u r e 3. New h i g h order diagrams which a r e being evaluated (W.G. Madden and A.D.J. Haymet, unpublished), f o r t h e Lennard-Jones f l u i d a t t h e t r i p l e p o i n t , i n o r d e r t o check t h e convergence ( o r otherwise) o f t h e s e r i e s (2.3).

JOURNAL

DE

PHYSIQUE

It should be noted t h a t r e c e n t l y t h e r e has been great progress i n a f i r s t - p r i n c i p l e s theory o f l i q u i d s t r u c t u r e based on t h e Rosenfeld-Ashcroft

/50/

m o d i f i e d hypernetted c h a i n scheme f o r modeling t h e s o - c a l l e d "bridge" f u n c t i o n , which i s o m i t t e d from t h e usual HNC equation. Lado and others /51-54/ have generated extremely accurate p a i r c o r r e l a t i o n functions. Uf course, t h e b r i d g e f u n c t i o n which i s modelled i n t h i s approach contains information, i n i n t e g r a t e d form, about t r i p l e t and higher-order c o r r e l a t i o n functions.

A number o f q u a n t i t i e s , such as t h e d e n s i t y d e r i v a t i v e o f t h e p a i r c o r r e l a t i o n function, are s e n s i t i v e t o c o r r e c t i o n s t o t h e s u p e r p o s i t i o n approximation /55/. Haymet and Rice /7/ have compared c a l c u l a t e d values o t t h i s d e r i v a t i v e w i t h r e l a t i v e l y o l d experimental data f o r Neon /56/. The r e s u l t s a r e displayed i n F i g u r e 4. Very r e c e n t l y , accurate s c a t t e r i n g data have been obtained f o r Argon /57/, and i t w i l l be i n t e r e s t i n g t o see how t h e present t h e o r y compares t o these new experimental r e s u l t s . Haymet and Rice a l s o examined t h e i n v e r s i o n o f e x p e r i - mental s c a t t e r i n g data t o o b t a i n e f f e c t i v e p a i r p o t e n t i a l s , and concluded t h a t t h e r e are important e f f e c t s due t o c o r r e c t i o n s t o superposition.

Distance

r/a

The e n t r o p y o f a l i q u i d can be c a l c u l a t e d from knowledge o f t h e l i q u i d c o r r e l a t i o n f u n c t i o n s . It i s w e l l known t h a t f o r t h e s i m p l e l i q u i d discussed above t h e p r e s s u r e p, excess i n t e r n a l energy U and i s o t h e r m a l c o m p r e s s i b i l i t y K~ may be o b t a i n e d t r o m t h e p a i r c o r r e l a t i o n t u n c t i o n u s i n g t h e w e l l known e q u a t i o n s

I n c o n t r a s t , c a l c u l a t i o n o t t h e excess entropy, SE, i s n o t o r i o u s l y d i f f i c u l t , d e s p i t e t h e f a c t t h a t one can e a s i l y d e r i v e t h e f o l l o w i n g " c l u s t e r expansion" e x p r e s s i o n /11/,

-SE/Nk = ~ / Z P d_r g ( r ) I n g ( r )

where t =

1 ~ - 2 1 .

l h e r e a r e a t l e a s t two m a j o r problems i n u s i n g e q u a t i o n (2.5) i n p r a c t i c a l problems. The f i r s t i s t h a t t h e i n t e g r a l i n t h e f i r s t t e r m i s v e r y l o n g ranged. I n F i g u r e 5 (on t h e n e x t page) i s d i s p l a y e d t h e i n t e g r a l

f o r t h e Lennard-Jones l i q u i d a t t h e t r i p l e p o i n t . Under t h e s u p e r p o s i t i o n 1 im

approximation, p R+,I(R) = -SE/Nk. The p a i r c o r r e l a t i o n f u n c t i o n i s o b t a i n e d f r o m

YBb t h e o r y /4/, which f o r t u n a t e l y p r o v i d e s g ( r ) o u t t o l a r g e d i s t a n c e s . I n t y p i c a l computer s i m u l a t i o n s , g ( r ) i s a v a i l a b l e u s u a l l y o n l y f o r d i s t a n c e s l e s s t h a n a p p r o x i m a t e l y 3 reduced u n i t s . Given a l o n g range p a i r c o r r e l a t i o n f u n c t i o n , t h i s f i r s t d i f f i c u l t y can be overcome.

The second and more s e r i o u s d i f f i c u l t y i s t h a t t h e second t e r m i s

not

s m a l l : 1 2

approximate i n i t i a l c a l c u l a t i o n s f o r t h e second t e r m y i e l d ~ p x 5, o r -20% o f t h e f i r s t term. Note t h a t under t h e s u p e r p o s i t i o n approximation, a l l terms b u t t h e f i r s t vanish. Hence, a c c u r a t e c a l c u l a t i o n o f t h e e n t r o p y r e q u i r e s knowledge o f g ( 3 ) a t l e a s t . The goal here i s t o p r e d i c t t h e e n t r o p y and hence t h e f r e e energy o f s i m p l e l i q u i d s f r o m t h e i n t e g r a l e q u a t i o n a t a g i v e n temperature and

JOURNAL

DE

PHYSIQUE

F i g u r e 5. The i n t e g r a l d e f i n e d i n equation (2.6) as a f u n c t i o n o f d i s t a n c e R i n t h e Lennard-Jones l i q u i d near t h e t r i p l e p o i n t .

The next question i s , o f course, t h e magnitude o f s t i l l h i g h e r order terms i n equation (2.5). The p r e l i m i n a r y evidence, from comparing t h e sum o f t h e f i r s t two terms w i t h t h e r e s u l t s trom i n t e g r a t i n g data from computer simulations, i s t h a t h i g h e r o r d e r c o n t r i b u t i o n s a r e small. This would imply t h a t g(4) i s w e l l approxi- mated by t h e a p p r o p r i a t e l y g e n e r a l i z e d s u p e r p o s i t i o n approximation 1121. As discussed by Ziman /13/, and Haymet and Rice 171, t h e r e i s some evidence t h a t t h i s i s t h e case, and i t may be a b l e t o be checked by c a r e f u l s c a t t e r i n g experiments on r e a l 1 iq u i d s .

i n v a r i e n t under s i x - f o l d r o t a t i o n o f a neighbor about a p a r t i c l e . I n t h r e e dimen- sions, f o r example i n t h e i c o s a h e d r a l l y q u a s i p e r i o d i c a l l o y made by Shechtman e t a l . /28/, t h e r e should a l s o be h i g h symmetry i n g(4). Note t h a t t h i s k i n d o f order can e x i s t i n p r i n c i p l e , and apparently now i n p r a c t i c e , i n t h e absence o f t r a n s l a t i o n a l ( p e r i o d i c ) order. However, l i t t l e i s known about g(4) i n l i q u i d s o r amorphous m a t e r i a l s , although s e r i e s expansions analogous t o (2.3) can be used t o estimate it from lower order c o r r e l a t i o n functions.

III. TRIPLET STRUCTURE FACTOH

The t r i p l e t s t r u c t u r e f a c t o r s ( ~ ) ( ? ~ , k 2 ) i s another important s t r u c t u r a l q u a n t i t y , which i s r e l a t e d t o t h e t r i p l e t c o r r e l a t i o n f u n c t i o n g(3) by F o u r i e r transformation. I n t h i s s e c t i o n we p o i n t out t h e importance o f t h i s q u a n t i t y i n t h e modern d e n s i t y f u n c t i o n a l theory o f freezing, and r e p o r t some p r e l i m i n a r y data from a computer s i m u l a t i o n designed t o measure s ( ~ ) .

A successful theory o f f i r s t order phase t r a n s i t i o n s has been constructed by expanding c o r r e l a t i o n f u n c t i o n s o f t h e s o l i d i n terms o f c o r r e l a t i o n f u n c t i o n s o f t h e c o e x i s t i n g l i q u i d /40,41/. Some d e t a i l s o f t h i s theory a r e o u t l i n e d i n Section 4. When such expansions a r e t r u n c a t e d a f t e r t h e f i r s t term, t h e o n l y i n p u t t o t h e f r e e z i n g theory i s t h e s t r u c t u r e f a c t o r s ( ~ ) o f t h e l i q u i d , y e t t h e t h e o r y appears t o be reasonably accurate. For a more complete understanding o f t h i s theory, i t i s necessary t o understand t h e h i g h e r order terms i n t h e d e n s i t y f u n c t i o n a l expansion, which i n v o l v e s ( ~ ) and h i g h e r order s t r u c t u r e f a c t o r s .

The q u a n t i t y o f i n t e r e s t i n t h e f r e e z i n g theory i s c(3)($.,, $,,) which i s r e l a t e d d i r e c t l y t o t h e t r i p l e t f a c t o r s ( ~ ) ( $ , $,,) by

where S(k) i s t h e usual ( p a i r ) s t r u c t u r e f a c t o r . For t h e purpose o t c a l c u l a t i o n from a s i m u l a t i o n i n v o l v i n g N p a r t i c l e s , t h e p a i r and t r i p l e t s t r u c t u r e f a c t o r s a r e expressed conveniently as and N

ikn-cj

N i g , * ~ ~ N

-i(bn

+

r )*k, s ( ~ ) (bn,

B)

=

1

e C e Z e

-a)

(3.3) j = l k = l

a = l

JOURNAL

DE

PHYSIQUE

Apart from c o n t r i b u t i o n s due t o d e n s i t y d e r i v a t i v e (zero wavevector) terms, t h e most important second order c o n t r i b u t i o n s a r i s e from s e t s o f t h r e e r e c i p r o c a l l a t t i c e vectors which form an e q u i l a t e r a l t r i a n g l e w i t h s i d e l e n g t h approximately equal t o t h e magnitude o f t h e wavevector a t t h e peak o f t h e l i q u i d s t r u c t u r e f a c t o r . It was Landau who f i r s t p o i n t e d o u t 145,461 t h a t such t r i a n g l e s e x i s t i n a bcc l a t t i c e b u t n o t i n an f c c l a t t i c e . I n t h e d e n s i t y f u n c t i o n a l theory o f f r e e z i n g /40,41/, t h e r e a r e f i r s t order terms which already i n f l u e n c e t h e r e l a t i v e s t a b i l i t y o f f c c versus bcc c r y s t a l s . I n a d d i t i o n , t h e t h r e e body c o n t r i b u t i o n d i s f a v o r s t h e bcc s o l i d i f t h e c o e f f i c i e n t c i i A f o r such t r i a n g l e s i s p o s i t i v e , whereas i t f u r t h e r d i s f a v o r s t h e f c c s o l i d i f i t i s negative. Unfortunately, t h e value o f t h i s c o e f f i c i e n t i s unknown.

Al though t h e c o e f f i c i e n t s c l i i have n o t y e t been measured experimental l y , they may be c a l c u l a t e d from computer simulations. The p r e l i m i n a r y r e s u l t s o f such a c a l c u l a t i o n are presented here (Haymet, Rahman and Oxtoby, unpubl ished), although f i r m conclusions cannot be drawn due t o s t a t i s t i c a l f l u c t u a t i o n s i n t h e data. We have c a l c u l a t e d t h e t r i p l e t d i r e c t c o r r e l a t i o n f u n c t i o n f o r a number o f

(approximately) e q u i l a t e r a l t r i a n g l e s near t h e peak o f t h e p a i r s t r u c t u r e f a c t o r from molecular dynamics s i m u l a t i o n s o f two simple l i q u i d s near t h e i r t r i p l e points. The f i r s t i s a mode1 o f l i q u i d rubidium which has been s t u d i e d extensive- l y by A. Rahman and h i s c o l l a b o r a t o r s /47-48/. This "substance" i s known t o freeze t o a bcc s o l i d , and i n f a c t t h e s t a b i l i t y o f t h i s phase w i t h respect t o an f c c s o l i d i s s u r p r i s i n g l y w e l l e s t a b l i s h e d /49/. The second c a l c u l a t i o n i s a s i m u l a t i o n o f t h e Lennard-Jones system, which freezes t o an f c c s o l i d . Both s i m u l a t i o n s c o n t a i n 500 p a r t i c l e s and are performed a t a d e n s i t y o f .90 i n t h e customary reduced u n i t s ; t h i s i m p l i e s t h a t t h e s i d e l e n g t h o f t h e s i m u l a t i o n cube i s 8.2207 u n i t s .

Three separate s i m u l a t i o n s were performed, each run f o r -5000 t i m e steps w i t h averages taken over every f i f t h step. The f i r s t s i m u l a t i o n used t h e rubidium p o t e n t i a l and a t i m e step o f -0075 u n i t s . The temperature, a c a l c u l a t e d q u a n t i t y i n Molecular Dynamics simulations, was .91 u n i t s and hence t h e thermodynamic c o n d i t i o n s o f t h e s i m u l a t i o n represent a s t a t e near t h e t r i p l e p o i n t . The o t h e r two s i m u l a t i o n s were f o r t h e Lennard-Jones system w i t h t i m e steps o f .O075 and

.O050 u n i t s , w i t h temperatures o f .69 and .73 u n i t s , r e s p e c t i v e l y ; these condi- t i o n s are a l s o c l o s e t o t h e t r i p l e p o i n t o f t h i s "substance.' The s i m u l a t i o n s a r e known t o be long enough t o p r o v i d e a good s t a t i s t i c a l r e p r e s e n t a t i o n o f t h e p a i r c o r r e l a t i o n function.

p e r i o d i c boundary c o n d i t i o n s suppress a l 1 f l u c t u a t i o n s w i t h wavelength l a r g e r than t h e cube l e n g t h L. More p r e c i s e l y , t h e o n l y allowed f l u c t u a t i o n s are those w i t h wavevector 2 r ~ - l ( i , j , k ) , where i, j, and k are integers. I n u n i t s o f 2 , ~t h e ~ ~ magnitude squared o f an allowed wavevector i s l2 = i2

+

j 2

+

k2, and t h i s q u a n t i t y i s c l o s e t o 78 near t h e peak o f t h e p a i r s t r u c t u r e f a c t o r s .

For t h e p r e l i m i n a r y s i m u l a t i o n s reported here, both t h e p a i r and t r i p l e t s t r u c t u r e f a c t o r s show s t a t i s t i c a l f l u c t u a t i o n s . The f i r s t peak o f t h e p a i r s t r u c t u r e f a c t o r s f o r " a l lowedt1 wavevectors shows s i g n i f i c a n t s t a t i s t i c a l f l u c t u a t i o n s , i n d i c a t i n g t h a t l o n g e r s i m u l a t i o n s are required. Note t h a t these a r e c a l c u l a t e d d i r e c t l y from equation (3.2), not by F o u r i e r transformation o f t h e p a i r c o r r e l a - t i o n function.

The t r i p l e t s t r u c t u r e f a c t o r was c a l c u l a t e d f o r a l 1 t r i a n g l e s w i t h i n t h e s i m u l a t i o n cube w i t h square s i d e l e n g t h s between 76 and 82 i n t h e above u n i t s . For 500 p a r t i c l e s t h e r e are 183 such d i s t i n c t t r i a n g l e s , y e t w i t h e f f i c i e n t r o u t i n e s f o r c a l c u l a t i n g sines and cosines the c a l c u l a t i o n s are by no means burdensome. Results f o r t h e t r i p l e t s t r u c t u r e f a c t o r and t r i p l e t d i r e c t

c o r r e l a t i o n f u n c t i o n a r e presented i n Table 1. It i s c l e a r t h a t t h e s t a t i s t i c a l TABLE 1. Calcul ated t r i p l e t s t r u c t u r e f a c t o r s

-

p r e l i m i n a r y data.

R b ( ~ t = .0075) L J ( h t = .0075) L J ( A ~ = .0050) 1: 1; 1; s(3) c ( 3 ) s ( 3 ) c ( 3 ) ~ ( 3 ) ,(3)

Cg-38 JOURNAL

DE

PHYSlQUE

metastable (supercooled) l i q u i d . ( I t i s known t o be a l i q u i d because t h e c a l c u l a - t e d d i f f u s i o n constant i s .O4 i n t h e usual reduced units.) Although t h e comple- t i o n o f these c a l c u l a t i o n s w i l l t a k e time and computational resources, we b e l i e v e t h a t t h e r e s u l t s presented i n Table 1 e s t a b l i s h e s t h i s new, comparatively simple, c a l c u l a t i o n as a method t o determine t h e c r u c i a l parameters i n t h e t h e o r y o f freezing, and t h a t t h e method i s e s p e c i a l l y v a l u a b l e s i n c e t h i s i n f o r m a t i o n has y e t t o be obtained experimentally.

I V . DENSITY FUNCTIONAL THEORY OF QUASIPERIODIC ORDER

The study o f long-range o r i e n t a t i o n a l order i n condensed phases has i n t e n s i f i e d r e c e n t l y due t o t h e observation by Shechtman e t a l 1281, o f an apparently meta- s t a b l e , condensed phase o f Al-Mn a l l o y which d i f f r a c t s e l e c t r o n s c o n s i s t e n t w i t h a s t r u c t u r e o f icosahedral p o i n t group symmetry. A number o f workers 129-361 have addressed t h e p r o p e r t i e s and s t a b i l i t y o f t h i s quasi p e r i o d i c icosahedral phase, which has been c a l l e d a " q u a s i c r y s t a l " by Levine and S t e i n h a r d t 1291, although t h i s k i n d o f s t r u c t u r e has been understood by mathematicians e a r l i e r 158,591.

I n t e r e s t i n icosahedral o r d e r i n g was r e v i v e d by t h e computer s i m u l a t i o n s o f S t e i n h a r d t e t a l 1371, who observed long-range o r i e n t a t i o n a l order i n a system o f supercool ed Lennard-Jones p a r t i c l es. Mean-f i e l d t h e o r i es were devel oped 138,391 t o d e s c r i be t h e onset o f o r i e n t a t i o n a l order. Simultaneously, d e n s i t y f u n c t i o n a l t h e o r i e s f o r f r e e z i n g i n t o p e r i o d i c s o l i d s 140,441 were developed. Very r e c e n t l y two independent groups, Haymet 1351 and Sachdev and Nelson 1361, have used t h e d e n s i t y f u n c t i o n a l t h e o r y t o examine t h e s t a b i l i t y ( o r otherwise) o f q u a s i p e r i o d i c phases.

The d e n s i t y f u n c t i o n a l theory o f q u a s i p e r i o d i c order assumes t h a t t h e s t r u c t u r e o f t h e i c o s a h e d r a l l y ordered phase may be w r i t t e n

where t h e v e c t o r s

5,

a r e constructed from 12 b a s i s v e c t o r s described below, and where pL i s t h e d e n s i t y o f t h e l i q u i d phase,

n

= uo i s t h e f r a c t i o n a l d e n s i t y

increase on o r d e r i n g and {un) i s t h e set o f order parameters which describe quasi- p e r i o d i c order.

For an icosahedral q u a s i c r y s t a l , t h e vectors kn i n t h e upper h a l f plane ( p o s i t i v e 6 .

K

,

where {jna) a r e z) o f t h e ordered system may be w r i t t e n

Sn

= _{La,1 ,J,}

_,,

K = A ( s i n y cosbn, s i n y siribn, cosy), n = 2, 6

-n

where 6, = 2.(n-2)/5, cos^ = 5-112, and A i s an amplitude t o be determined. S i m i l a r b a s i s vectors e x i s t f o r pentagonal symmetry i n two dimensions.

Icosahedral phases are determined by sets o f order parameters {un) w i t h n o t a l 1 un equal t o zero. The d e n s i t y f u n c t i o n a l theory o f icosahedral o r d e r i n g proceeds by analogy w i t h t h e successful d e n s i t y f u n c t i o n a l t h e o r y o f c r y s t a l l i z a t i o n

140-441. The o n l y d i f f e r e n c e i s t h a t t h e " d e n s i t y waves" which c h a r a c t e r i z e t h e ordered phase are incommensurate i n t h e present t h e o r y (equation 4.1), whereas t h e y are commensurate i n t h e e a r l i e r t h e o r i e s 1411. Note t h a t a completely d i f f e r e n t approach t o condensation i n t o a p e r i o d i c s t r u c t u r e s has been developed by Wolynes and coworkers 1421.

From t h e d e n s i t y f u n c t i o n a l a n a l y s i s /35/, t h e s t a b i l i t y o f t h e icosahedral phase i s determined expl i c i t l y by n o n t r i v i a l s o l u t i o n s o f t h e impl i c i t equation

where c ( r ) i s t h e Ornstein Zernike d i r e c t c o r r e l a t i o n f u n c t i o n o f t h e l i q u i d o r supercooled l i q u i d . S u b s t i t u t i o n o f equation (4.1) i n t o (4.3) provides expl i c i t equations f o r t h e f r a c t i o n a l d e n s i t y change on o r d e r i n g ri and t h e order parameters

pn. Due t o t h e approximations o f t h e theory, m u l t i p l e s o l u t i o n s o f equation (4.3) may e x i s t over a range o f d e n s i t i e s ( o r e q u i v a l e n t l y , a range o f chemical

p o t e n t i a l s ) a t f i x e d temperature. The chemical p o t e n t i a l a t which t h e t r a n s i t i o n occurs i s uniquely determined, t o t h e same order i n p e r t u r b a t i o n t h e o r y as t h e above c a l c u l a t i o n , by t h e e q u a l i t y o f t h e grand thermodynamic p o t e n t i a l s (i .e., t h e Maxwell c o n s t r u c t i o n ) ,

The amplitude A o f t h e b a s i s vectors tcn which d e s c r i b e t h e icosahedral phase i s determined by t h e average d e n s i t y pI = p L ( l + q ) of t h e ordered phase. I n t h e case o f f r e e z i n g i n t o a p e r i o d i c s t r u c t u r e /41/, t h i s 5s r e a d i l y determined by i n t e g r a - t i o n o f equation (4.3) over a ( t h r e e dimensional). u n i t ce11 o f t h e s o l i d . I n t h e present case, i t i s n o t so t r i v i a l : i n t e g r a t i o n s must be performed over a volume l a r g e enough t o sample t h e t r u e average d e n s i t y o f t h e m a t e r i a l . This leads t o a n u m e r i c a l l y somewhat d e l i c a t e s e l f - c o n s i s t e n c y problem f o r t h e b a s i s v e c t o r ampli- t u d e A. Sachdev and Nelson 1361 have use an elegant mapping i n t o a s i x

dimensional v e c t o r space t o overcome t h i s d i f f i c u l t y . I n t h i s space, each o f t h e v e c t o r s (4.2) i s represented by an orthogonal d i r e c t i o n , t h e r e a r e no

JOURNAL

DE

PHYSIQUE

s i x dimensions.

Under t h e approximation 0 = O, Sachdev and Nelson have used t h e "relaxed dense random packing' mode1 o f Ichikawa 1601 and t h e s t r u c t u r e f a c t o r o f amorphous c o b a l t /61/ as i n p u t t o t h e theory, and f i n d metastable q u a s i p e r i o d i c phases! These icosahedral phases are l o c a l l y s t a b l e and have a f r e e energy lower than t h e amorphous ( a p e r i o d i c ) m a t e r i a l , b u t s t i l l higher than t h e p e r i o d i c f c c c r y s t a l

.

It now seems c l e a r t h a t even some s i n g l e component systems can be made i n t o q u a s i p e r i o d i c m a t e r i a l s provided they can be supercooled q u i c k l y enough. The t i m e s c a l e on which such m a t e r i a l s remain q u a s i p e r i o d i c depends on t h e separate q u e s t i o n o f t h e n u c l e a t i o n r a t e t o t h e ground s t a t e p e r i o d i c c r y s t a l .

This l e v e l o f t h e o r y p r e d i c t s o n l y t h e s i n g l e - p a r t i c l e q u a n t i t i e s (e.g., t h e d e n s i t y p(t-)) i n t h e ordered phase, since t h i s i s a l 1 t h a t i s needed t o c a l c u l a t e t h e thermodynamic p r o p e r t i e s . However, t h e p a i r d i r e c t c o r r e l a t i o n f u n c t i o n c ( ~ ) ( [ ~ , r ) i n t h e ordered phase can be found from t h e t r i p l e t c o r r e l a t i o n f u n c t i o n i f 3 ) o f t h e l i q u i d using t h e equation analogous t o (4.3). To

nv

knowl edge, t h i s r e l a t i o n has n o t been expl ored i n d e t a i 1.

There are e x c i t i n g developments i n t h e theory o f l i q u i d s , supercooled l i q u i d s , and amorphous m a t e r i a l s . There a r e a number o f d i f f e r e n t kinds o f long-range order, having no connection w i t h t r a n s l a t i o n a l p e r i o d i c order, which

may

e x i s t i n so- c a l l e d " d i s o r d e r e d ' m a t e r i a l s . The challenges a r e ( i ) t o d e t e c t t h e presence o f such order experimentally, and ( i i ) p r e d i c t which kinds o f order can e x i s t i n a given m a t e r i a l . High order c o r r e l a t i o n functions w i l l p l a y a r o l e i n b o t h ctial 1 enges.

REFERENCES

/1/ J. Yvon, Actual i t i e s S c i e n t i f i q u e s e t I n d u s t r i a l (Herman, Paris, 1935), Volume 203; J.G. Kirkwood, J. Chem. Phys. 3, 300 (1935).

/2/ M. Born and H.S. Green, Proc. Roy. Soc. London, Ser. A 188, 10 (1946); see a l s o N.N. Bogoliubov, J. Phys. (USSR) 10, 256 and 265 m 6 ) ; Engl i s h t r a n s l a t i o n by E.K. Gora i n Studies i n S t a t i s t i c a 1 Mechanics, Volume 1, P a r t A, e d i t e d by J. de Boer and G.E. Uhlenbeck North-Holland, Pmsterdam, 1962). /3/ G. S t e l l , Physica 29, 517 (1963); R. Abe, P!og. Theor. Phys. 22, 213 (1959);

L. Verlet, Nuovo ~ X e n t o

18,

77 (1960); F.A. Blood, J. Math PGs.

1,

1613 (1966).

/4/ A.D.J. Haymet, S.A. Rice and W.G. Madden, J. Chem. Phys.

75,

4696 (1981). /5/ W.J. McNeil, W.G. Madden, A.D.J. Haymet and S.A. Rice, J. Chem. Phys.

78,

388 (1983).

/6/ A.D.J. Haymet, S.A. Rice and W.G. Madden, J. Chem. Phys. 74, 3033 (1981). /7/ A.D.J. Haymet and S.A. Rice, J. Chem. Phys. 76, 661 ( 1 9 8 2 T

/8/ A.D.J. Haymet, J. Chem. Phys.

80,

3801 ( 1 9 8 4 T

I.Z. F i s h e r and B.L. Kopeliovich, Sov. Phys. Dokl. 5, 761 (1960). J.M. Ziman, Mode1 s o f Disorder (Cambridge uni v e r s i t y Press, 1974). J. Bosse, E. Leutheusser and S. Yip, Phys. Rev. A27,

-

1696 (1983). E. Leutheusser, J. Phys. C g , 2801 (1982).

M.E. F i s h e r and S. Fishman, J. Chem. Phys.

2,

4227 (1983) and Phys. Rev. L e t t . 47 421 (1981).

S. F i s K a n , Physica 109A, 382 (1981). M. Alexanian, Phys.

Rev.

A 25, 572 (1982).

W. K l e i n and A. D. J. HaymeK Phys. Rev. B 30, 1387 (1984).

L. Schafer and A. Klemm, Z. Naturforsch 34aT993 (1979); 36a, 584 (1981). See a l s o S. Gupta, J.M. H a i l e and W.A. S'Eele, Chemical W s i c s 72, 425 (1982), and Mol. Phys. 51, 675 (1984); H. Breitenfelder-Manske,

El.

Phys. 48, 209 (1983); M. L u c k z , K. Lucas, H. Manske and F. Kohler, Mol. Phys. 48,

n

(1983).

-

F. Hirata, P.J. Rossky and B.M. P e t t i t t , J. Chem. Phys. 78, 4133 (1983). D. Chandler, R. Si1 bey and B. Ladanyi, Mol. Phys. 46, 1335 (1982).

P. Jacobaeus, J.U. Madsen, F. Kragh and R.M.J . C o t s r i l l , P h i l . Mag. 841, 11 (1980); M. Favre-Bonte and P.J. Desre, Physics L e t t e r s 75A, 415 ( 1 9 8 0 ) F C. Hoheisel, Phys. Rev. A23,

-

1998 (1981); W. S c h o m m e r s , ~ y s . Rev. A g , 2855 (1980).

D.R. Nelson and B.I. Halperin, Phys. Rev. B E , 2547 (1979). A.P. Young, Phys. Rev. B19, 1855 (1979).

D. Shechtman, 1. Blech, Gratias and J.W. Cahn, Phys. Rev. L e t t .

53,

1951 (1984).

D. Levine and P.J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).

D. Levine, T.C. Lubensky, S. Ostlund, S. Ramaswa~, P.J. Steinhardt and J. Toner, Phys. Rev. Lett. 54, 1520 (1985).

P. Bak, Phys. Rev. Lett.-54, 1517 (1985).

N.D. Mermin and S.M. T r o i x , Phys. Rev. L e t t .

54,

1524 (1985).

D.R. Ne1 son and S. Sachdev,

"

Incommensurate Icosahedral Density Waves i n Rapidly Cooled Metals", Phys. Rev., t o be published.

D.R. Nelson and B. 1. Halperin, ''Pentagonal and Icosahedral Order i n Rapidly Cooled Metals," Science, t o be publ ished.

A.D.J. Haymet, "Density Functional Theory f o r t h e S t a b i l i t y o f Icosahedral Q u a s i c r y s t a l s , " submitted f o r p u b l i c a t i o n .

S. Sachdev and D.R. Ne1 son, "Order i n Meta1 1 i c Glasses and Icosahedral Crystals," submitted f o r publ i c a t i o n .

P.J. Steinhardt, D.R. Nelson and M. Ronchetti, Phys. Rev. L e t t .

47,

1297 (1981).

P.J. Steinhardt, D.R. Nelson and M. Ronchetti, Phys. Rev. 828, 784 (1983). A.D.J. Haymet, Phys. Rev. 827, 1725 (1983).

T.V. Ramakrishnan and M. Y u c o u f f , Phys. Rev. B19, 2775 (1979).

A.D.J. Haymet, J. Phys. Chem. 89, 887 (1985); A3.J. Haymet and D.W. Oxtoby, J. Chem. Phys. 74, 2559 ( 1 9 8 1 ) T

Y. Singh, J.P. SfToessel and P.G. Wolynes, Phys. Rev. L e t t e r s ,

54,

1059 (1985).

S. ~ m i t h l i n e and A.D.J. Haymet, J. Chem. Phys., t o be published.

C. Marshall, B.B. L a i r d and A.D.J. Haymet, Chem. Phys. L e t t e r s , submitted f o r p u b l i c a t i o n .

L.D. Landau, Phys. Z. Sowjetunion 11,

26

(1937); The C o l l e c t e d Papers o f L.D.

_-

_..

Landau, D. t e r Haar ( E d i t o r ) , (Gordon and Breach, New York, 1965), p.

193.

S. Alexander and J.P. McTague, Phys. Rev. L e t t . 41, 702 (1978).

M.J. Mandel, J.P. McTague and A. Rahman, J. ChemF~hys. 64: 3699 (1976);

s,

3070 (1977). 7

C.S. Hsu and A. Rahman, J. Chem. Phys. 70, 5234 (1979); 71, 4974 (1979). M. P a r r i n e l l o and A. Rahman, Phys. Rev.Tett. 45, 1196 m 8 0 ) .

Y. Rosenfeld and N.W. Ashcroft, Phys. Rev. ~ f l F 1 2 0 8 (1979). F. Lado, Phys. Lett. 89A, 196 (1982).

C9-42 JOURNAL DE PHYSIQUE

F.J. Rodgers, D.A. Young, H.E. DeWitt and M. Ross, Phys. Rev.

-

A28, 2990 (1983).

Y. Rosenfeld, Phys. Rev. A, i n press.

P.A. E g e l s t a f f ,

Ann.

Rev. Phys. Chem. 24, 159 (1973). L.A. deGraaf and B. Mozer, J. Chem. P h z .

55,

4967 (1971).

A.A. van Well

,

P. Verkerk, L.A. deGraaf, J.-B. Suck and J.R.D. Copley, Phys. Rev. A 31, 3391 (1985).

R. P e n r z e , B u l l . I n s t . Maths and I t s . Appl.

-

10, 266 (1974); see a l s o M. Gardner, Sci. Am. 236, 110 (1977).

A.L. MacKay, P h y s i Z 1 1 4 A , 609 (1982).