CENTRAL RESEARCH INSTITUTE FORPHYSICSBUDAPEST

F, IGLÓI J, KOLLÁR

CLUSTER PERTURBATION THEORY FOR CLASSICAL FLUIDS II.

APPLICATIONS FOR A HARD SPHERE REFERENCE SYSTEM

^Hungarian ^Academy o f Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

CLUSTER PERTURBATION THEORY FOR CLASSICAL FLUIDS II.

APPLICATIONS FOR A HARD SPHERE REFERENCE SYSTEM

F. Iglói, J. Kollár

Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary

HU ISSN 0368 5330 ISBN 963 371 767 1

The thermodynamic properties and the radial distribution functions for systems with inverse power potential and for a Lennard-Jones fluid at high temperature are calculated using a new perturbation method applied for a hard sphere reference system. The results are compared to those obtained from the Andersen-Weeks-Chandler theory and to the Monte Carlo results.

АННОТАЦИЯ

Термодинамические свойства и парные корреляционные функции для системы частиц, отталкивающихся по закону 1/гп , и для жидкости Ленард-Джонса были вычислены с помощью нового метода возмущения для системы референции твердых сфер. Результаты были сравнены с соответствующими величинами теории Андерсен- -Викс-Чендлера и с Монте-Карло "машинными" экспериментами.

KIVONAT

Az l/г11 taszitó potenciállal kölcsönható rendszerek és a magas hőmérsék­

letű Lennard-Jones folyadék termodinamikai tulajdonságait és radiális elosz­

lásfüggvényét számítottuk ki egy uj perturbációs módszer segítségével, ke­

mény-gömb referencia rendszert használva. Az eredményeket összevetettük az Andersen-Weeks-Chandler módszerrel számoltakkal és Monte Carlo számitógépes kísérletek eredményeivel.

In the first p a r t of this work.'*' (hereafter r e f e r r e d to as p a p e r I) a p e r t u r b a t i o n a l m ethod w a s d e v e l o p e d for the d e s c r i p t i o n of the t h e r m o d y n a m i c and s t r u c t u r a l p r o p e r t i e s of real fluids in t e r m s of a reference system, the p r o p e r t i e s

»

of w h i c h are a s s u m e d to be known. In t h i s paper we a p p l y thi s m e t h o d for a h a r d sphere reference system, for w h i c h c o n v e n i e n t a n a l y t i c f o r m u l a e are a v a i l a b l e bot h for the equa t i o n of state2 and the pair c o r r e l a t i o n function 3 4' .

D u r i n g the last y e ars the r e l a t i o n s h i p b e t w e e n the p r o ­ p e r t i e s of this i d e a l i z e d model s y s t e m and those of real fluids wit h smoothly v a r y i n g repulsive forces has been d i s c u s s e d in several papers. Rowlinson~* c o n s i d e r e d fluids w i t h an i n v e r s e nth p ower p o t e n t i a l and e x p a n d e d the t h e r m o d y n a m i c p r o p e r t i e s in p o w e r s of 1/n. B a r k e r and H e n d e r s o n ^ g e n e r a l i z e d the R o w - linson m e t h o d by a p p l y i n g it to a w i d e c l a s s of r e p u l s i v e

potentials. One of the m o s t s u c c e s s f u l m e t h o d w a s p r o p o s e d by Andersen, W e e k s and C h a n d l e r / A W C / ^ . In this m e t h o d a s e r i e s e x p a n s i o n is o b t a i n e d for the free e n e r g y in powers of a

"softness p a r a m e t e r " íj by w r i t i n g d o w n its functional T a y l o r e x p a n s i o n in p o w e r s of the B o l t z m a n n f a c t o r differ e n c e s . T h e hard sphere d i a m e t e r is chosen in s u c h a w a y w h i c h c a u s e s the first order t e r m of the free e n e r g y to vanish, at the s a m e time r e d u c i n g the m a g n i t u d e of the h i g h e r o r d e r terms as well.

However, if the i n t e r a c t i o n p o t e n t i a l is not steep e n o u g h , the a c c u r a c y of t h e s e m e t h o d s is not s a t i s f a c t o r y in m a n y c a s e s , e s p e c i a l l y for the radial d i s t r i b u t i o n f u n c t i o n of the system.

The i n a c c u r a c y of the t r e a t m e n t of s o f t e n i n g the core r e f l e c t s in t h e r e s u l t s of the p e r t u r b a t i o n c a l c u l a t i o n s for r e a l i s t i c p o t e n t i a l s , w h e n an a t t r a c t i v e tail is a d d e d to a s m o o t h l y v a r y i n g short range r e p u l s i v e potential. A l t h o u g h the d i f f e r e n t t h e o r e t i c a l m e t h o d s for t r e a t i n g s l o w l y varying a t t r a c t i v e p e r t u r b a t i o n s can be t e s t e d by r u l i n g o u t this

Q

i n a c c u r a c y , t h e a c c u r a t e t r e a t m e n t of soft cores h a s still r e m a i n e d an u n s o l v e d p r o b l e m in t h e t h e o r y of c l a s s i c a l fluids.

In this p a p e r we c a c u l a t e the t h e r m o d y n a m i c p r o p e r t i e s and radial d i s t r i b u t i o n f u n c t i o n s for systems w i t h i nverse n t h p o wer p o t e n t i a l s (n= 6, 9, 12) and for a L e n n a r d - J o n e s

fluid at h i g h t e m p e r a t u r e u s i n g the m e t h o d d e v e l o p e d in I.

T h i s m e t h o d e n a b l e s us to take i n t o a ccount an i m p o r t a n t part of the h i g h e r o r d e r t e r m s of the u s u a l g - d e p e n d e n t form of the t h e r m o d y n a m i c q u a n t i t i e s u s i n g onl y the 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 of the har d s p h e r e r e f e r e n c e s y s t e m (those p a r t s that c a n be e x p r e s s e d in t e r m s of the d e r i v a t i v e s of t h i s function w i t h r e s p e c t to the d e n s i t y ) . T h i s w i l l be d e m o n s t r a t e d on the

s y s t e m w i t h an i nverse t w e l f t h p o w e r potential. We d i s c u s s the p r o b l e m s r e l a t e d to the o p t i m a l c h o i c e of the har d sphere

r e f e r e n c e s y s t e m and the c o n d i t i o n g = § 0 p r o p o s e d in I is c o m p a r e d to t h a t g i v e n b y A n d e r s e n et al^. The t h e r m o d y n a m i c c o n s i s t e n c y w i l l also b e i n v e s t i g a t e d : the e q u a t i o n of state is c a l c u l a t e d b o t h from the v i r i a l e q u a t i o n and from the d e r i v a ­ ti v e of the f r e e e n e r g y w i t h r e s p e c t to the density. The results are c o m p a r e d t o those o b t a i n e d f r o m the AWC method. In the last

pa r t o f the p a per the c a l c u l a t e d 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 for the s y s t e m w i t h inv e r s e tw e l f t h p o w e r p o t e n t i a l is c o m p a r e d to the M o n t e C a r l o r e s u l t s and to that o b t a i n e d fro m the A W C theory.

D e s c r i p t i o n of the m e t h o d for a hard sphere r e f e r e n c e s y s t e m

In thi s s ection we r e f o r m u l a t e the m e t h o d d e s c r i b e d in p a p e r I for a har d sphere r e f e r e n c e system, a s s u m i n g that we k n o w o n l y the ha r d sphere p a i r c o r r e l a t i o n function. In this case t h e b a s i c e q u a t i o n s o f the m e t h o d ( e q u a t i o n s (lla-b) in p a p e r I) w i l l have the fo r m

(la)

and

(lb)

where, t o simp l i f y the f o r m ulas, we i n t r o d u c e d the q u a n t i t y

'ft* 3

"ß" S ^ ins t e a d of t h e d e n s i t y ( O ’ s t a n d s for the c h a r a c t e r i s t i c length o f the p o t e n t i a l U ( ^ ) ) a n d the p a c k i n g

7Г i3

f r a c t i o n n£0s . U g o a for the har d sphere r e f e r e n c e s y s t e m (d is the ha r d sphere d i a m e t e r ) .

In these e q u a t i o n s t h e prime d e n o t e s a d e r i v a t i v e w i t h

j|£

r e s p e c t to and d = ol (б" . The i n t e g r a l

J

is d e f i n e d as

oo

1(^ 0 , с 1%Л2.«£,> x ) [ e ( d x ) - e c t x ) ] x zőlj< (2)

О

H e r e в о ( * ) у £ ( ^ в < * ) stands' for the har d sphere 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 (which c o i n c i d e s w i t h for nr > d ),

в ( ^ ) and 6 0 ( ^ ) are the B o l t z m a n n f a c t o r s for the

s y s t e m u n d e r c o n s i d e r a t i o n and for the r e f e r e n c e system, r e s p e c ­ tively. T h e e x c e s s c h e m i c a l p o t e n t i a l of the s y s t e m can be

o b t a i n e d f r o m the e q u a t i o n (19) of p a per Is

) л f3jLCел

<Лг11' 7

T h e e x c e s s free e n e r g y p e r p a r t i c l e n o w f r o m the t h e r m o d y n a m i c r e l a t i o n

n . £ Д е х can be d e t e r m i n e d

у

a(^,ol ) я |3/^ex - 1

4

T h e c a l c u l a t i o n can be c a r r i e d o u t in t w o steps. First, one has to d e t e r m i n e the p a c k i n g f r a c t i o n о o f the har d sphere s y s t e m for a g i v e n v a l u e of *1 from the e q u a t i o n (lb). A f t e r t h i s c a l c u l a t i o n the c o r r e s p o n d i n g t h e r m o d y n a m i c q u a n t i t i e s can be o b t a i n e d fr o m the e q u a t i o n s (la), (3) o r (4) u s i n g this v a l u e for . W e can, h o w e v e r , p r o c e e d in an o t h e r way, starting f r o m the u s u a l d e n s i t y - d e p e n d e n t form for t h e t h e r m o d y n a m i c q u a n t i t i e s ; e x p r e s s i o n s of this k i n d for the e x c e s s chemical p o t e n t i a l and e x c e s s free e n e r g y can be o b t a i n e d from the

e q u a t i o n s (20) and (21) in p a p e r I, r e s p e c t i v e l y (first o r der in 5 ):

^ I

p / ^ k í ^ - ^ (5a)

a (*7

^{a 0} C 'f ) - I ( ^ » ol )

(5b)

T o s i m p l i f y the formulae, he r e w e i n t r o d u c e d the n o t a t i o n

•if 3

я <y£oL . The two k i n d s of d e s c r i p t i o n , in gene r a l , are n o t e q u i v a l e n t wit h e a c h other: the r e s u l t s of a c a l c u l a t i o n b a s e d on the e q u a t i o n s (lb) and (3) or (4) d i f f e r f r o m those o b t a i n e d f r o m the e q u a t i o n s (5a-b) in the sum of an infinite subseries. T h e y are e q u i v a l e n t o n l y if the h a r d s p h e r e r e f e r e n c e s y s t e m is d e t e r m i n e d f r o m the c o n d i t i o n £ = S ° p r o p o s e d in p a p e r I, w h e n the sum of this s u b s e r i e s e q u a l s to zero. The e q u i v a l e n c y o f the two d e s c r i p t i o n s can be se e n by c o m p a r i n g the e q u a t i o n s (5a-b) to (3) and (4) and t a k i n g into ac c o u n t the c o n d i t i o n ~ S ° • w h i c h n o w has the form

[•7 Г (ч ,с 1 * )]’= о (

)

Th i s e q u a t i o n s p e c ifies the r e f e r e n c e s y s t e m for a g i v e n value ^ t h r o u g h d e t e r m i n i n g the f u n c t i o n c L \ ) . T o i l l u s t r a t e the

d i f f e r e n c e b e t w e e n the t w o descriptions, we c o n s i d e r a system w i t h an i n v e r s e twelfth p o w e r p o t e n t i a l and c a l c u l a t e the e x c e s s

free e n e r g y a n d the e x c e s s c h e m i c a l p o t e n t i a l as a f u n c t i o n of

the har d sphere d i a m e t e r oL at d i f f e r e n t v a l u e s for oj u s i n g bot h k i n d s of d e s c r i p t i o n . A s it is w e l l known, the t h e r m o ­ d y n a m i c f u n c t i o n s of a s y s t e m wit h an inv e r s e p o w e r p o t e n t i a l of the fo r m

a ( r ) ~ £. ( — *]

n

(7) J/n

d e p e n d s o n l y on the q u a n t i t y *£ (£ P ) . In the p r e s e n t m e t h o d this s c a l i n g p r o p e r t y is p r e s e r v e d as w e can see from the d e f i n i ­ tion (2) for the func t i o n I, w h i c h d o e s not d e p e n d on p, £ , and

O' s e p a r a t e l y , but o n l y on a " t e m p e r a t u r e d e p e n d e n t

c h a r a c t e r i s t i c l e n g t h " Cf [£.{$) . R e p l a c i n g C7 by thi s n e w c h a r a c t e r i s t i c l e n g t h (i.e. t a k i n g s ^ £ p ^ and

ßl , r V „

Cl — ~ r v £ p Í ) , o u r f o r m u l a e b e c o m e a p p r o p r i a t e for t r e a t i n g systems w i t h a p o t e n t i a l o f the type (7).

To p e r f o r m t h i s c a l c u l a t i o n , one n e e d s the k n o w l e d g e o f the hard s p h e r e 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 as a f u n c t i o n of the p a c k i n g f r a c t i o n and the d i s t ance, and the e q u a t i o n of state

o

for ha r d sphere fluids. F o r o u t s i d e the c o r e we us e d the a n a l y t i c e x p r e s s i o n s o b t a i n e d e m p i r i c a l l y by V e r l e t and Weis'*

b a s e d on the s o l u t i o n of the P e r c u s - Y e v i c k equation^, w h i l e in the r 4. d r e g i o n w e took the T h i e l e - W e r t h e i m c u b i c p o l i n o m i a l form in r/d, w i t h c o e f f i c i e n t s , w h i c h a s s u r e the c o n t i n u i t y of the p a i r d i s t r i b u t i o n f u n c t i o n and its first a n d s econd d e r i v a ­ tives at r=d^. V e r l e t and W e i s state t h a t the f u n c t i o n y ^ o b t a i n e d

in this w a y d i f f e r s from t h e i r M o nte C a r l o r e s u l t s by at m o s t 3 %. For the e q u a t i o n of s t a t e of har d sphere f luids we u s e d the

e x p r e s s i o n s u g g e s t e d by C a r n a h a m and S t a r l i n g , w h i c h s u m m a r i z e s the a v a i l a b l e m o l e c u l a r - d y n a m i c s and M o n t e C a r l o r e s u l t s w i t h i n the s t a t i s t i c a l a c c u r a c y of these c o m p u t e r c a l c u l a t i o n s . F o r the e x c e s s free e n e r g y and for the e x c e s s c h e m i c a l p o t e n t i a l this g i ves

2

o - o i ä )

Í 4 -

8*j

H - n])3 (8b)

The c a l c u l a t e d v a l u e s for the e x c e s s free e n e r g y and for the e x c e s s c h e m i c a l p o t e n t i a l of the s y s t e m w i t h i nverse t w e l f t h p o w e r p o t e n t i a l vs. ol ■ * are shown in Fig. la and b at t h ree d i f f e r e n t v a l u e s for *1 ( **[ = 0.3, 0.4, 0.5). The solid c u r v e s w e r e o b t a i n e d f r o m the e q u a t i o n s ( l a ) , ( l b ) , (4) or (3) by e l i m i n a t i n g • w h i l e the r e s u l t s o b t a i n e d fro m the e q u a t i o n

(5b) o r (5a) for

CL

s t a r t i n g from a n y v a lue of ol . Thu s the e x t e n s i o n of the r e g i o n in

ol

of the n e g l e c t e d h i g h e r o r d e r terms. T h e r e f o r e this f igure

c l e a r l y shows t h e a d v a n t a g e s of a d e s c r i p t i o n w h i c h s t a r t s from the e q u a t i o n s (lb) and (4) or (3). In the figure we a l s o indi- c a t e d the v a l u e s c(I* d e t e r m i n e d f r o m the e q u a t i o n (6) and t h ose o b t a i n e d f r o m the AWC c o n d i t i o n ( K ^ i d ) = 0) .

The two m e t h o d s r e s u l t in s i m i l a r v a l u e s for сл » and therefore, I*

for this system, t h e y are n e a r l y e q u i v a l e n t fr o m the p o i n t of v i e w of the t h e r m o d y n a m i c q u a n t i t i e s (but not fro m the p o i n t of v i e w of the p a i r d i s t r i b u t i o n f u n c tions, as we w i l l see later).

F u r t h e r m o r e , it is easy to sho w t h a t at the v a l u e of w h ere ф ~ • n o t on l y the v a l u e s o f the free e n e r g i e s o b t a i n e d fro m the two k i n d s of d e s c r i p t i o n e q u a l to ea c h other, b u t the s l o p e s of the c u r v e s also c o i n c i d e at this p o i n t ^ :

(9)

On the o t h e r h a n d - since in z e r o t h o r d e r у г ( 1 ' £ ) = у » И 1 5 )

as w e can see f r o m the e q u a t i o n 23 o f p a p e r I - the v i r i a l p r e s s u r e , д J 4

Pv is r e l a t e d to the d e r i v a t i v e ( J ~ b y the e q u a t i o n

3(^ - иИ 1 г Ы ч 91

T h u s we can see t h a t p = p ^ w h e n ( A k e q u a l s to zero.

F r o m Fig. la w e c a n see that this c o n d i t i o n is a p p r o x i m a t e l y s a t i s f i e d for a l l the three v a l u e s of ^ at the p o i n t w h e r e

9 e ' w h i c h p r e d i c t s a g o o d z e r o t h o r d e r a p p r o x i m a t i o n for the 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 b y u s i n g the ?-$*<> condition.

Results

A. E q u a t i o n of state

In o r d e r to test the p r e s e n t met h o d , in t h i s s ection we c a l c u l a t e the e q u a t i o n of s t ate for several s y s t e m s from the c o n d i t i o n 5^= ' anc^ c o m p a r e the m to those o b t a i n e d from the AWC m e t h o d and to the r e s u l t s of c o m p u t e r s i m u l a t i o n s . The v a l u e s

for the free e n e r g y and the p r e s s u r e for a s y s t e m w i t h inverse t w e l f t h p o w e r p o t e n t i a l are shown in T a b l e 1. A s w a s alr e a d y

i n d i c a t e d in Fig. 1, there are no r e a s o n a b l e d i f f e r e n c e s b e t w e e n the r e s u l t s o b t a i n e d from the c o n d i t i o n = g 0 (equation (6)) and t h o s e o b t a i n e d from the A W C c o n d i t i o n (1=0) w h e n the e q u a t i o n of s t ate is d e t e r m i n e d by n u m e r i c a l d i f f e r e n t i a t i o n of the free e n e r g y w i t h r e s p e c t to the dens i t y . The v i r i a l e q u a t i o n of state, however, is m o r e accurate in the case of the c o n d i t i o n r as we c a n e x p e c t on the b a s i s of the d i s c u s s i o n in c o n n e c t i o n w i t h Fig. 1. The v alues for t h e p r e s s u r e o b t a i n e d from the e q u a t i o n (la) u s i n g the c o n d i t i o n are in g o o d a g r e e m e n t wit h t h o s e o b t a i n e d by n u m e r i c a l d i f f e r e n t i a t i o n , d e m o n s t r a t i n g again h o w the t h e r m o d y n a m i c c o n s i s t e n c y is f u l f i l l e d in the p r e s e n t method.

O t h e r a p p l i c a t i o n s o f the m e t h o d are s h o w n in T a b l e 2 and in Fig. 2, w h e r e the e q u a t i o n of s t ate is g i v e n for the systems w i t h i n v e r s e n i n t h and s i x t h p o w e r p o t e n t i a l s , respectively. In Fig. 2, the r e s u l t s o b t a i n e d fro m the c o n d i t i o n g = ^>0 (solid line) are c o m p a r e d to t h o s e of the A W C m e t h o d (dashed l i n e ) .

r

T h e d o t t e d lines s h o w the v i r i a l e q u a t i o n s o f state and the full c i r c l e s i n d i c a t e the M o n t e C a r l o r e s u l t s 12. W e can c o n c l u d e

a g a i n t h a t at low d e n s i t i e s t h e r e are no e s s e n t i a l d i f f e r e n c e s b e t w e e n the r e s u l t s of the t w o method, b u t for larger d e n s i t i e s the c o n d i t i o n $ ~ r e p r o d u c e s m o r e a c c u r a t e l y the M o n t e C a r l o values. The a d v a n t a g e of the c o n d i t i o n g = g a is even m o r e e v i d e n t if one c o m p a r e s the v i r i a l p r e s s u r e p ^ to that o b t a i n e d b y n u m e r i c a l d i f f e r e n t i a t i o n ( p ) . B o t h the d a t a in T a b l e 2 and the c u r v e s p l o t t e d in Fig. 2 c l e a r l y sh o w t h a t p <v» p ^ for the c o n d i t i o n = g 0 , w h i l e u s i n g the AWC m e t h o d p ^ is e s s e n t i a l l y l a r g e r tha n p at large d e n s i t i e s .

F i n a l l y in Fig. 3 the c a l c u l a t e d e q u a t i o n of state for a

*

L e n n a r d - J o n e s fluid is p l o t t e d at hig h t e m p e r a t u r e (T =5), to- g e t h e r w i t h the M o n t e C a r l o r e s u l t s 13. The r e s u l t s show t h a t in this t e m p e r a t u r e r a nge the m e t h o d can be a p p l i e d d i r e c t l y for a L e n n a r d - J o n e s s y s t e m as w e l l , but at low t e m p e r a t u r e s it fails to w o r k for the r e a s o n s d i s c u s s e d in p a p e r I (at low t e m p e r a t u r e s our z e r o t h o r d e r p a i r d i s t r i b u t i o n f u n c t i o n r^zC'r ) is n o t a g o o d a p p r o x i m a t i o n a n y more, and one s h o u l d t r eat s e p a ­ r a t e l y the short r a nge r e p u l s i v e and long r a n g e a t t r a c t i v e parts of the p o t e n t i a l ) .

B. 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

T h e e q u a t i o n (23) in p a p e r I g i ves an e x p r e s s i o n (first o r d e r in í= ) for the r a d i a l d i s t r i b u t i o n function, a n d we s u g g e s t e d two d i f f e r e n t r e c i p e s for the c h o i c e of the r e f e r e n c e s y s t e m in the

z e r o t h and the first o r d e r calculation. In z e r o t h o r d e r (when bi/t' <r ) = У*- (Я ' S ’ ) ) the r e f e r e n c e s y s t e m can be d e t e r ­ m i n e d from the c o n d i t i o n (equation (6)). The res u l t s o f s u c h a z eroth o r d e r c a l c u l a t i o n are s h own in the Fig. 4 and are c o m p a r e d to the re s u l t s of a M o n t e C a r l o c o m p u t e r c a l c u l a t i o n14

and to the p a i r d i s t r i b u t i o n function o b t a i n e d f r o m the A W C m e t h o d for the s y s t e m w i t h i n v e r s e t w e l f t h p o w e r p o t e n t i a l . In c o n t r a s t to the case of the t h e r m o d y n a m i c q u a n t i t i e s , in this case there is an e s s e n t i a l d i f f e r e n c e b e t w e e n the r e s u l t s o b t a i n e d from the A W C m e t h o d and from the c o n d i t i o n $ ~ • W h i l e the first p e a k o f the p a i r d i s t r i b u t i o n func t i o n is t o o large in the AWC m e t h o d , the c o n d i t i o n j? = j>0 a l m o s t e x a c t l y r e p r o d u c e s the r e s u l t s of the c o m p u t e r s i m u l a t i o n s in this r e g i o n (here we u s e d a m o d e r a t e l y large v a l u e of , i n d i c a t e d in the f i g u r e ) . In the r e g i o n of the first m i n i m u m , however, b o t h the A W C and the p r e s e n t m e t h o d give too s mall v a l u e s for the f u n c t i o n g 2 (r).

T h e s e r e s u l t s can be i m p r o v e d by t a k i n g into a c c o u n t the n e x t t e r m in the e x p a n s i o n of the pair d i s t r i b u t i o n function.

In t h i s case the r e f e r e n c e s y s t e m can be d e t e r m i n e d from the c o n s i s t e n c y c r i t e r i o n g i v e n by the e q u a t i o n (27) in p a p e r I, w h i c h n o w has the form (for a hard sphere r e f e r e n c e system)

(ID

To p e r f o r m the c a l c u l a t i o n , one s hould use some a p p r o x i m a t i o n for t h e three p a r t i c l e d i s t r i b u t i o n f u n c t i o n o f the ha r d sphere s ystem, w h i c h a p p e a r s in the f i rst o r d e r t e r m in the e q u a t i o n (23)

in p a p e r I. U s i n g the K i r k w o o d s u p e r p o s i t i o n a p p r o x i m a t i o n for this function, w e can o b t a i n an a p p r o x i m a t e e x p r e s s i o n for y 2 fro m the e q u a t i o n (23) o f p a p e r I:

w h e r e the i n t e g r a l J is d e f i n e d by

x + x

О ч

m a x (I x - x ' l , A

(13)

T h i s e x p r e s s i o n for J(x) was o b t a i n e d b y i n t r o d u c i n g a b i p o l a r c o o r d i n a t e system.

T h e r e s u l t of the f i r s t o r d e r c a l c u l a t i o n (solid line) is p l o t t e d in Fig. 5 for a s y s t e m w i t h in v e r s e t w e l f t h p o w e r p o t e n -

14 tial, t o g e t h e r w i t h the M o n t e C a r l o r e s u l t s (full circles)

T h e z e r o t h o r d e r curve (da s h e d line) is als o shown. One can see fr o m t h e figure that the r e s u l t of the first o r d e r c a l c u l a t i o n a g r e e s r e m a r k a b l y wel l w i t h the M o n t e C a r l o d a t a n o t on l y in the n e i g h b o u r h o o d o f the f i r s t peak, b u t the a g r e e m e n t is e s s e n t i a l l y

i m p r o v e d in t h e region o f the first m i n i m u m as w e l l c o m p a r e d to the z e r o t h o r d e r case.

C o n c l u s i o n

To s u m m a r i z e we c a n say that the a p p l i c a t i o n of the m e t h o d d e v e l o p e d in I for the systems w i t h in v e r s e p o w e r p o t e n t i a l u s i n g a hard sphere r e f e r e n c e system, v e r i f i e s our e x p e c t a t i o n s t h a t using this m e t h o d a m o r e a c c u r a t e and m o r e c o n s i s t e n t d e s ­ c r i p t i o n can be a c h i e v e d c o m p a r e d to the c a l c u l a t i o n s w h i c h start f r o m the usual d e n s i t y - d e p e n d e n t form for the t h e r m o d y n a m i c

q u a ntities. T h i s r e s u l t can be a s c r i b e d to the fact that the p r e s e n t m e t h o d m a k e s a m o r e c o m p l e t e use of the info r m a t i o n c o n t e n t of the r a d i a l d i s t r i b u t i o n function. The a d v a n t a g e of t h e m e t h o d can be seen c l e a r l y fr o m the r e m a r k a b l e a g r e e m e n t b e t w e e n the v a l u e s for the p r e s s u r e c a l c u l a t e d from the virial e q u a t i o n and t h o s e o b t a i n e d fr o m the d e r i v a t i v e of the free e n e r g y with r e s p e c t to the density, and f r o m the r e a s o n a b l e a g r e e m e n t b e t w e e n the c a l c u l a t e d r a d i a l d i s t r i b u t i o n functions a n d the Monte C a r l o results.

A c k n o w l e d g e m e n t

The a uthors are g r a t e f u l to Drs. P. F a z e k a s and A. SUto for v a l u a b l e d i scussions.

R e f e r e n c e s

1. See the p r e v i o u s p a p e r in this volume.

2. N.F. Carnahan, K.E. Starling, J. Chem. Phys. _51, 635 (1969) 3. L. Vériét, J.J. W e is, Phys. Rev. A 5, 939 (1972)

4. T h e e x act solu t i o n of the P e r c u s - Y e v i c k e q u a t i o n for the r a d i a l d i s t r i b u t i o n func t i o n of the h a r d s p h e r e s y s t e m was o b t a i n e d by F. Thiele, J. Chem. Phys. 3_9, 474 (1963) and M.S. W e r t h e i m , Phys. Rev. Lett. 10, 321 (1963). F o r this

f u n c t i o n a c o n v e n i e n t a n a l y t i c fo r m w a s g i v e n by W.R. Smith and D. Hen d e r s o n , Mol. Phys. 19^, 411 (1970), w h i l e the n u m e ­ r i c a l v a l u e s w e r e t a b u l a t e d b y G.J. T h r o o p and R.J. Bearman, J. Chem. Phys. 42_, 2408 (1965).

5. J.S. Rowlinson, Mol. Phys. 8, 107 (1964)

6. J.A. Barker, D. H e n d e r s o n , J. Chem. Phys. 4_7, 4714 (1967)

7. H.C. A n d e rsen, J.D. Weeks, D. C h a n dler, Phys. Rev. A 4, 1597 (1971) and

J.D. Weeks, D. C h a n dler, H.C. Andersen, J. Chem. Phys. 5 4 , 5237 (1971)

8. G. Stell, J.J. Weis, Phys. Rev. A 2_1, 645 (1980)

9. T h i s p r o c e d u r e is c l e a r l y arbitrary; we can expect, however, t h a t the w a y of e x t r a p o l a t i o n into the co r e has little e f fect

7 on the results, as it w a s p o i n t e d out by A n d e r s e n et al.

T h e r e a s o n for c h a n g i n g the o r i g i n a l p r o c e d u r e by V e r l e t and 3

W e i s was the fact that at large d e n s i t i e s it g i v e s large n e g a t i v e v a l u e s for i n side the co r e a l r e a d y at a d i s t a n c e as large as (0.8-0.9)d.

10. J.P. Hansen, Phys. Rev. A 2, 221 (1970)

11. In the c a l c u l a t i o n of this d e r i v a t i v e from the e q u a t i o n s (4) and (lb) we used the r e l a t i o n I =s О This

d"[o lg

rela t i o n c a n be v e r i f i e d e a s i l y s t a r t i n g from the e q u a t i o n s (4), (3) and (la) and u s ing the c o n d i t i o n (6). On the b a s i s of this r e s u l t w e can say that the r e f e r e n c e s y s t e m is d e t e r m i n e d f r o m the c o n d i t i o n that the free e n e r g y of the s y s t e m as a func t i o n of the d e n s i t y o f the r e f e r e n c e s y s t e m s hould have an extremum, w h i c h is e q u i v a l e n t w i t h the

c o n d i t i o n £ = g 0 .

12. W.G. Hoover, S.G. Gray, K.W. Johnson, J. Chem. Phys. 5 5 , 1128 (1971)

13. J.J. Nicolas, K.E. Gubbins, W.B. Stre e t t , D.J. Tildesley, Mol. Phys. 37_, 1429 (1979)

14. J.P. Hansen, J.J. Weis, Mol. Phys. 23, 853 (1972)

F igure C a p t i o n s F i g u r e 1.

T h e e x c e s s free e n e r g y (a) and the e x c e s s c h e m i c a l p oten-

•ft

tial (b) of the s y s t e m w i t h inv e r s e t w e l f t h p o w e r p o t e n t i a l vs . d at d i f f e r e n t v a l u e s for ^ • T h e solid c u r v e s w e r e o b t a i n e d from the e q u a t i o n s (lb), (la), (3) or (4) b y e l i m i n a t i n g o^0 , w h ile the r e s u l t s o b t a i n e d from the e q u a t i o n (5b) or (5a) for a. and

respectively, are i n d i c a t e d by d a s h e d lines. T h e Monte C a r l o r e s u l t s 10 are shown by d o t t e d lines. T h e a rrows s h o w the v a l u e s of d d e t e r m i n e d from the c o n d i t i o n g = J?0 ( e q u ation (6)) ( and o b t a i n e d from the AWC - c o n d i t i o n (+)•

F i g u r e 2.

The e q u a t i o n of state o f the s y s t e m w i t h i n v e r s e sixth p o w e r p o t ential. The d a s h e d c u r v e was o b t a i n e d fro m the AWC - - c o n d i t i o n b y n u m e r i c a l d i f f e r e n t i a t i o n of the free e n e r g y w i t h r e s p e c t to the density, w h i l e the d o t t e d c u rves s h o w the v i r i a l e q u a t i o n of state c o m i n g from b o t h the A W C - met h o d , and the c o n d i t i o n <j> = 5>0 . The p r e s s u r e o b t a i n e d from the c o n d i t i o n

£ =g0 b y n u m e r i c a l d i f f e r e n t i a t i o n is i n d i c a t e d b y s o lid line.

T h e M o n t e C a r l o r e s u l t s are d e n o t e d by full circles.

F i g u r e 3.

T h e e q u a t i o n of state for a L e n n a r d - J o n e s f l uid at high

•ft

t e m p e r a t u r e (T =5). The solid line shows the r e s u l t o b t a i n e d from the c o n d i t i o n g = / the MC - r e s u l t s 13 are i n d i cated b y b l a c k circles.

F igure 4.

The pa i r d i s t r i b u t i o n f u n c t i o n of the s y s t e m w i t h inverse twe l f t h p o w e r p o t e n t i a l at *4 = 0.4215. The z e r o t h o r d e r result o b t a i n e d from the c o n d i t i o n S - J ? * (solid line) is c o m p a r e d to that c a l c u l a t e d from the A W C t h e o r y (dashed line) and to the MC res u l t s (black circles).

F i g u r e 5.

The pai r d i s t r i b u t i o n f u n c t i o n of the s y s t e m w i t h inverse t w e l f t h p o w e r p o t e n t i a l at = 0.3746. The full line shows the r e s u l t of a first o r d e r c a l c u l a t i o n , w h e r e the t h r e e p a r t i c l e d i s t r i b u t i o n f u n c t i o n is g i v e n b y the K i r k w o o d s u p e r p o s i t i o n a p p r o x imation. T h e res u l t s of the z e r o t h o r der c a l c u l a t i o n a n d the MC - r e s u l t s are i n d i c a t e d b y d a s h e d line a n d b l a c k circles, res p e c t i v e l y .

F i g . 2

1 1.5

Fig. 5

1-5 r/cr 2.

Fig. 4

T a b l e C a p t i o n s

T a b l e 1.

T h e t h e r m o d y n a m i c p r o p e r t i e s of a s y s t e m w i t h inverse t w e l f t h p o w e r pot e n t i a l . MC = r e s u l t s of M o n t e - C a r l o e x p e r i ­ m e n t s 1 0 ; A W C = p r e d i c t i o n of the A W C - theory, § =£o ~ r esults o b t a i n e d from the e q u a t i o n (6). T h e e q u a t i o n of s t ate is

o b t a i n e d b y n u m e r i c a l d i f f e r e n t i a t i o n of the free energy;

A W C (V) and (v) are the p r e d i c t i o n s of the A W C - t h e o r y and the c o n d i t i o n jg> - £ o , r e s p e c t i v e l y , b a s e d o n the v i r i a l theorem. The v alues for the p r e s s u r e o b t a i n e d from the e q u a t i o n (la) u s ing the c o n d i t i o n £ =§ o are a l s o shown ( § ~ £ o (la)).

T h e AWC v a l u e s shown in the t a b l e d i f f e r s l i g h t l y fr o m those of Ref. 7 as a c o n s e q u e n c e of the d i f f e r e n t e x t r a p o l a t i o n p r o c e d u r e for у2° (r) inside the core.

T a b l e 2.

The e q u a t i o n of s t ate of a s y s t e m w i t h i nverse n i n t h p ower potential. T h e n o t a t i o n s are the same as in T a b l e 1. T h e MC v alues w e r e t a ken fr o m Ref. 12.

*

6

MC A W C MC A W C A W C (v) О = ^ 0(1/а)

0 . 1 0. 40 0. 404 0.404 1.45 1 . 4 5 0 1.455 1.450 1.449 1.454

0.2 0.9 1 0. 907 0 .908 2.12 2.123 2.155 2.124 2.123 2.143

го•о

1.53 1.537 1 . 5 4 0 3.12 3.116 3.232 3.123 3.119 3.166

0.4 2.33 2.331 2.337 4.58 4.556 4.873 4.580 4.571 4.631

in•О

3.34 3.332 3.347 6.66 6.616 7.334 6.665 6.669 6.640

0.6 4.61 4.599 4.623 9.56

!

9.539 10.945 9.591 9.688 9.256

T a b l e 1

MC AWC A W C (V) r s o

0 . 1 1.5 0 1.50 1.51 1.50 1.51

0. 2 5 2 . 7 0 2.69 2.84 2.70 2.77

0.5 6 . 6 0 6.48 7.74 6.61 6.65

T a b l e 2

< Kiadja a Központi Fizikai Kutató Intézet

Felelős kiadó: Krén Emil Szakmai lektor: Bergou János Nyelvi lektor: Tíittő István

Példányszám: 500 Törzsszám: 80-738 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1980. december hó