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Simple MSA solution and

thermodynamic theory in a hard-

sphere Yukawa system

Mitsuaki Ginozaa

aDepartment of Physics, College of Science, University of

the Ryukyus, Nishihara-Cho, Okinawa, 903-01, Japan

Available online: 11 Aug 2006

To cite this article: Mitsuaki Ginoza (1990): Simple MSA solution and thermodynamic theory in

a hard-sphere Yukawa system, Molecular Physics, 71:1, 145-156

To link to this article: http://dx.doi.org/10.1080/00268979000101701

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MOLECULAR PHYSICS, 1990, VO L. 71, N O. 1, 1 45 -15 6

S i m p l e M S A s o lu t io n a n d th e r m o d y n a m i c th e o r yin a hard-sphere Yukawa sys temB y M I T S U A K I G I N O Z A

D e p a r t m e n t o f P h y s i c s , C o l l e g e o f S c ie n c e,U n i v e r s i t y o f th e R y u k y u s , N i s h i h a r a - C h o ,

O k i n a w a 9 0 3 - 0 1 , J a p a n(Received 29 January 1990; accepted 11 April 1990)

A simple expression for the mean-spherical-approximation (M SA ) solutio nof the Orn stein-Ze rnike (OZ) equation in the Baxter formalism is presented forthe case with n comp onen ts and a s ingle Yu kaw a term with factorizable pre-factor: all coefficients of the MS A solution are given in term s of s imple rationa lfunctions of a parameter that is defined as the acceptable solution of a non-linear equation. The m anifold of solutions of the nonlinear equation an d th echoice o f the acceptable solution are diseussed. A therm ody nam ic theo ry is alsopresented in terms o f simple ration al functions of the above-men tioned par-ameter, and the effec t of the ' charge-non -neutra l i ty ' condi t ion on thermo dyna-mic functions is discussed.1. Introduction

S t u d ie s o f t h e t h e r m o d y n a m i c a n d s t r u c t u r a l p r o p e r t ie s o f a h a r d - s p h e r eY u k a w a (H S Y ) m i x t u r e , i n c l u d i n g a n e u t r a l h a r d - s p h e r e m i x t u r e a n d a c h a r g e dh a r d - s p h e r e m i x t u r e a s l im i t i n g ca se s, h a v e b e e n m a d e b y m a n y w o r k e r s [ 1, 2] . I nt h e s e s t u d i e s , t h e m e a n - s p h e r i c a l a p p r o x i m a t i o n ( M S A ) h a s o f t e n b e e n e m p l o y e d .T h e f o r m a l M S A s o l u t i o n o f t h e O r n s t e i n - Z e r n i k e ( O Z ) e q u a t i o n f o r t h e n -c o m p o n e n t m - Y u k a w a - t e r m e a s e h a s b e e n g i v e n b y B l u m a n d H o y e [ 3 ] i n t h eB a x t e r f o r m a l i s m a n d b y N i i z e k i [ 4 ] u s i n g t h e L a p l a c e - t r a n s f o r m t e c h n i q u e . I n t h ec a s e o f Y u k a w a t e r m s w i t h f a c t o r i z a b l e c o e ff ic i en t s, t h e p r e s e n t a u t h o r [ 5 ] s h o w e dt h a t t h e s y s t e m o f n o n l i n e a r a l g e b r a i c e q u a t i o n s d e f i n i n g t h e c o e f fi c i en t s o f t h eB l u m - H o y e s o l u t i o n c a n b e s im p l if ie d r e m a r k a b l y , a n d i n t h e s i n g l e - Y u k a w a - t e r mc a se , i n p a r t i c u l a r , t h e s y s t e m c a n b e r e d u c e d t o a s in g l e n o n l i n e a r e q u a t i o n .

I t is w e l l k n o w n t h a t t h e a b o v e s y s t e m o f n o n l i n e a r e q u a t i o n s h a s a m a n i f o l d o fs o l u t i o n s . R e c e n t l y , P a s t o r e [ 6 ] g a v e a n i n t e r e s t i n g d i s c u s s i o n r e g a r d i n g t h e m a n i -f o l d a n d s h o w e d t h a t t h e b r a n c h o f t h e a c c e p ta b l e M S A s o l u t i o n c a n b e s in g l ed o u tb y t h e c r i t e ri o n o f n o n - s i n g u l a r i t y o f th e B a x t e r m a t r i x i n t h e u p p e r h a l f o f th ec o m p l e x - w a v e n u m b e r p l a n e a s w e l l a s o n t h e r e a l a x i s . I n t h i s c o n t e x t , i t i s i n t e r -e s t in g t o i n v e s t i g a t e in d e t a i l t h e m a n i f o l d o f s o l u t i o n s o f th e s i n g l e n o n l i n e a re q u a t i o n f o r th e n - c o m p o n e n t s in g l e - Y u k a w a - t e r m c a se .

T h e H S Y m i x t u r e h a s o f t e n b e e n u s e d as a si m p l e m o d e l t o d e s cr i be c o m p o u n d -f o r m i n g a n d p h a s e - s e p a r a t i n g l i q u i d s d i re c t l y [1 , 2 ] , a n d i t m a y a l s o s e r v e a s ar e f e r e n c e s y s t e m i n p e r t u r b a t i o n a l o r v a r i a t i o n a l m e t h o d s f o r t h e d e t e r m i n a t i o n o ft h e t h e r m o d y n a m i c s a n d s t r u c t u r e o f a l i q ui d s y s te m w i t h m o r e c o m p l e x p o t e n ti a ls .F o r t h e s e p u r p o s e s, i f a s im p l e M S A s o l u t io n a n d t h e r m o d y n a m i c th e o r y c a n b eo b t a i n e d , i t m a y b e u s e fu l .

T h e a i m o f t h i s p a p e r i s (i) t o p r e s e n t s i m p l e e x p r e ss i o n s f o r th e M S A s o l u t i o na n d t h e t h e r m o d y n a m i c t h e o r y , ( ii ) t o d i sc u s s th e m a n i f o l d o f s o l u t io n s o f t h e

0026-8976/90 $3.00 9 1990 Taylor & Francis Ltd

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dedby[UniversityofGuanajuato]at12:1018October2011


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1 46 M . G i n o z an o n l i n e a r e q u a t i o n a n d t h e c h o i c e o f t h e a c c e p t a b l e s o l u t i o n a n d (iii) t o in v e s t i g a t et h e t h e r m o d y n a m i c e f f e c t o f t h e ' c h a r g e - n o n - n e u t r a l i ty ' c o n d i t io n a s a n a p p li c a ti o n .

I n s e c t i o n 2 , t h e M S A s o l u t i o n i n t h e B a x t e r f o r m a l i s m i s g i v e n f r o m o u rp r e v i o u s w o r k [ 5 ] , a n d i t i s s h o w n i n t h e A p p e n d i x t h a t , u n d e r t h e ' c h a r g e -n e u t r a l i t y ' c o n d i t i o n , t h i s M S A s o l u t i o n a g r e e s w i t h t h a t i n t h e o r i g i n a l f o r m a l i s mo f t h e O Z e q u a t i o n . I n s e c t i o n 3 , t h e m a n i f o l d o f s o l u t io n s o f t h e n o n l i n e a r e q u a t i o na n d t h e c h o i c e o f t h e a c c e p t a b l e s o l u t i o n a r e d i s c u ss e d . I n s e c t io n 4 , a t h e r m o d y n a -m i c t h e o r y is o b t a i n e d o n t h e b a s i s o f t h e a c c e p t a b l e s o l u t i o n a b o v e , a n d t h et h e r m o d y n a m i c e f fe c t o f t h e 'c h a r g e - n o n - n e u t r a l i t y ' c o n d i t io n w i ll b e in v e s t ig a t e d .C o n c l u d i n g r e m a r k s a r e g i v e n i n s e c t io n 5 .

2. Th e s imp le analyt ic M SA solutionL e t u s c o n s i d e r a n n - c o m p o n e n t H S Y m i x t u r e c o n s i s ti n g o f e q u a l -s i ze h a r d

s p h e r e s i n te r a c ti n g w i t h e a c h o t h e r v i a Y u k a w a p o t e n t i a ls o u t s i d e s p h e re s . T h et h e r m o d y n a m i c s a n d t h e s ta t ic s t r u c t u r e o f t h e m i x t u r e a r e i n v e s ti g a te d b y m e a n s o ft h e p a r t i a l c o r r e l a t i o n f u n c t i o n s h o & ) o r t h e p a r t i a l 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 s c o & .T h e l a tt e r a r e d e fi n e d b y t h e O Z e q u a t i o n

ho ( r ) = c o ( r ) + p ~ c , f d r 1 c , ~ [ r 1 - r I ) h o ( rl ), ( 1)w h e r e p is t h e t o ta l n u m b e r d e n s i t y a n d c , i s t h e c o n c e n t r a t i o n o f / - s p e c i e s s p h e re s .B a x t e r I - 7 ] g a v e a t r a n s f o r m a t i o n o f (1 ) i n a d i s o r d e r e d s y s t e m . I n t h e t r a n s f o r m -a t io n , h e in t r o d u c e d a m a t r ix Q ( k ) t h a t i s a s s u m e d t o b e n o n - s i n g u l a r i n t h e u p p e rh a l f o f t h e c o m p l e x - w a v e n u m b e r (k ) p l a n e a s w e ll a s o n t h e r e a l a x i s a n d w h o s e (i, j )e l e m e n t i s o f t h e f o r m 6 o - p ( c i c j ) l / 2 O o ( k ) , where 0 o ( k ) h a s t h e f o r m

a n d s a t i s f i e s~ 0~ 1 7 6Q . o (k ) = d r Q o ( r ) e ik " (2)

~ o ( k ) = - J d r c o ( r ) e i h " " (3 a )= Q o(k) + ~4 ,~- k ) - p ~ c t Q a ( k )Q j t ( - k ) . ( 3 b )l

T h e O Z e q u a t i o n i n th is B a x t e r f o r m a l i s m is t h u s [ 3 , 7 ]

fo ard | d rn r c o ( r ) = - Q o ( r ) + p ~ c~ Q ~ ( t + r ) Q j t( t) , (4 a )~ d ~ on r h o ( r ) = - Q o { r ) + 2 r i p ~ c t d t h,~ l t - r l ) ( r - t ) Q o { t ) . (4 b )

E q u a t i o n ( 4 a ) i s t h e r e a l s p a c e r e p r e s e n t a t i o n o f ( 3 b ), w h i l e ( 4 b ) is o b t a i n e d f r o m ( 1)a n d ( 3 ) u s i n g t h e n o n - s i n g u l a r i t y o f t h e B a x t e r m a t r ix , Q ( k ).T h e M S A f o r t h e n - c o m p o n e n t a n d s i n g i e - Y u k a w a - t e r m c a s e is , a s u s u a l, d e f in e db y t h e f o l l o w i n g c l o s u r e r e l a t i o n s f o r (1 ) o r ( 4 a , b ) :

h o ( r ) = - 1 ( r < a ) , (5 a )c o ( r ) = K _ ~ e - z , / , ( r > a ) , (5 b )r

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S i m p l e M S A s o lu t io n i n a H S Y m i x t u r e 147w h e r e a i s th e h a r d - s p h e r e d i a m e t e r . L e t u s d e f in e t h e f a c t o r i z a b l e ca s e b y

K i j = K e z Z i Z j , (5 c)a n d , w i t h o u t l o s s o f g e n e r a l i t y , w e a s s u m e ~ l cz Z ~ = 1.

B l u m a n d H o y e I - 3, 8 ] s o l v e d (4 a , b ) w i t h c l o s u r e r e l a t i o n s o f t h e m o s t g e n e r a lf o r m c o n s i st i n g o f a n a r b i t r a r y n u m b e r o f Y u k a w a t e r m s . F o r t h e c a s e o f (5 b), th e i rs o l u t i o n i s

( 8 9 - a ) + t i f f - a ) + C o ( e - z ' / " - e - z )Q o ( r ) = ~ + D o e - z r /" ( r < a ) , ( 6 a )

/ D e -z r/~ ( r > a ) . ( 6 b )k i jT h e c o e f f ic i e n ts i n ( 6 a , b ) a re d e f i n e d b y a n u m b e r o f e q u a t i o n s r e l a t e d t o t h ea c c e p ta b l e s o l u t i o n o f t h e s y s te m o f n o n l i n e a r e q u a t i o n s , a n d w e d o n o t q u o t e t h e mh e r e b e c a u s e o f th e i r l e n g t h .

T h e p r e s e n t a u t h o r [ 5 ] s h o w e d t h a t i n t h e f a c t o r i z a b l e c a s e s u c h a s (5 c ), t h i ss y s t e m o f n o n l i n e a r e q u a t i o n s c a n b e r e d u c e d t o t h e n o n l i n e a r e q u a t i o n f o r ap a r a m e t e r F , a n d t h e e x p r e s s io n s f o r t h e c o e ff i c ie n t s i n (6 a , b ) c a n b e c o m e e x t r em e l ys im p l e. A f t e r s o m e s t r a i g h t f o r w a r d c a l c u la t i o n a n d n o n - d i m e n s i o n a l i z a t i o n s u si n g o fa , w e o b t a i n t h e f o l l o w i n g f r o m t h e r e s u l t o f [ 5 ] :

Aj = 1 + + ~ PN a j , (7 a)

f l j = a - ~ + a A N a j , (7 b)D o = - a Z Z i a j e z , (7 c)

B i e - Z / 2 " ~ zC i j = ( 7 2 Z i -g J a i e , (7 d)

w h e r e

AN = - - ~ Z 2 l + 8 9 D x,

B i e # 2 = --- 8 9 ~ c j Z j ~ d t [ h o ( r ) + 1] e - Z ( r l r - 1 )) ,J r

= - - F X i + (O to + o q F ) D 1 ,r ~ F X i

a i - 3r/D 2 ,Z i - Z Z,X ~ - +C k o (z )F + 1 ~ , F + ~ o '

J

( 8 a )

( 8 b )

( 8 c )( 8 c ' )( 8 , / )

( 8 e )(8 f )

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1 48 M . G i n o z aw i t h

= E c j Z j ,Jrt = 6inp a 3,

A = 1 - ~ / ,1 - e - z

C o ( Z ) - - - ,Z

1 - - 8 9 + { z ) e - zl ( z) - z 3 ,

~Po = 1 + -~- qgo(Z - - 7 ~ l(z ) 1 + 89+ ,12r/

4 1 = 0 ( Z ) - - ~ I / / I( Z ) '

12rt tl= + 8 9

~ o = ~ l l + 8 9 - ~ .A s c a n b e s e e n f r o m ( 7 a - d ) a n d ( 8 a - f ) , al l coeff ic ien ts in (6 a ,b ) a r e g i v e n b y s i m p l er a t i o n a l f u n c t i o n s o f F , w h i c h s a ti sf ie s t h e f o l l o w i n g n o n l i n e a r e q u a t i o n :

F 2 + z F = - - 6K t /D2(F ) . (9 )O"

Since ~o(k) i s ob ta in ed f ro m (2 ) , (3 a ,b ) a n d (6 a,b) , i t is a ls o a r a t i o n a l f u n c t i o n o fF . M o r e o v e r , a s w il l b e d i s c u ss e d i n s e ct io n 4 , t h e r m o d y n a m i c f u n c t io n s c a n b ee x p r e s s e d i n t e r m s o f s i m p l e r a t i o n a l f u n c t i o n s o f F . T h e r e f o r e a l l w e h a v e t o d ob e f o r e in v e s t i g a ti o n o f th e s t a ti c s t r u c t u r e a n d t h e r m o d y n a m i c s o f th e m i x t u r e i s tos o l v e (9 ) . T h i s , h o w e v e r , h a s a m a n i f o l d o f s o l u t i o n s , a n d t h e c h o i c e o f t h e a c c e p t -a b l e s o l u t i o n w i l l b e d i s c u s s e d i n t h e n e x t s e c t io n .

I t m a y b e o f i n t er e s t t o p r o v e d i r ec t ly th e e q u a l i t y o f t h e M S A s o l u t i o n o b t a i n e da b o v e w i t h t h a t o f (1 ) a n d ( 5 ). A s f a r a s t h e a u t h o r i s a w a r e , h o w e v e r , t h e e x p l ic i te x p r e ss i o n o f th e l a t t e r h a s n o t b e e n g i v e n e x c e p t f o r t h e t w o - c o m p o n e n t c a s e u n d e rt h e ' c h a r g e - n e u t r a l i t y ' c o n d i t io n . W e d i sc u s s t h is p r o b l e m i n t h e A p p e n d i x .

3 . T h e m a n i f o l d o f s o l u t i o n s a n d t h e c h o i c e o f a c c e p t a b l e s o lu t i o nA s h a s b e e n s h o w n i n s e c t i o n 2 , o u r M S A s o l u t i o n i n t h e B a x t e r f o r m a l i s m

c o n t a i n s t h e q u a n t i t y F d e t e r m i n e d b y ( 9 ). T h i s e q u a t i o n c a n b e w r i t t e n e x p li c it lyw i t h t h e u s e o f (8 e , f ) a s f o l l o w s :

f ( x ; z , q , 6 ) = - 0 wi th x = F / z , ( 1 0 a )

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S im p le M S A s o lu tio n i n a H S Y m i x t u r e 149w h e r e

, I F ( 6 / : X I - ~ ) + (6/z2)6 7f(x; z, r/, 6)= (x 2 -[- x ) / L ( ~ o ZX - . ~ i~- (~l~iz -]- (~o)2J (10b)0 = kr/, ~b~ = ~bo(Z) (11 )(7t ~ ~ 2 .

I n o r d e r t o d e m o n s t r a t e t h e m a n i f o l d o f s o l u t io n s o f ( 10 a) , f ( x ; z , r l, 6 ) i s s h o w ni n f i gu re 1 a s a f un c t i o n o f x w i t h z = 1 , ~ / = 0 . 125 a nd 6 = 0 . 5 . I n t h i s f i gu re , t hes o l u t i o n s o f ( 1 0 a ) c o r r e s p o n d t o t h e i n t e r s e c ti o n s o f t h e c u r v e a n d t h e h o r i z o n t a ll in e g i v e n b y t h e o r d i n a t e v a l u e o f - 0 . I t c a n b e s e e n f r o m t h e f ig u r e t h a t t h e r e a r es ix d if f e r e n t b r a n c h e s o f s o l u t io n s f o r x : A B , B C , C D , D E , E F a n d F G . T h i s i s t h egen e ra l c ha rac t e r i s t i c o f ( 10 a ) w i t h 0 < 6 < 1 . I n t h e spec i a l ca ses o f 6 = 0 and 1 , a sw i ll b e d i s c u s s e d i n s e c t i o n 5 , t h e r e a r e o n l y f o u r b r a n c h e s .

T h e a s y m p t o t i c s o l u t i o n s o f ( 1 0 a ) o n t h e se s ix b r a n c h e s i n 101 ~ 1 c a n e a s i ly b eo b t a i n e d , a n d t h e y a r e a s fo l lo w s :

6 ( 6 ) 0X A B = - - ~ 1 - - 6 + ~ ,6 I 1 - 6 6 ]x ~,~ = - 1 + ~ ( , / ,o Z - - l Y + ( r - ~ o ) ~ o ,

l [ ( 6 / z ~ X l _ - 6 ) l o l ] 1/2,xco ~oZ -~

2X D E ~ ~ 0 Z X C D ~

r F" (6/Z2)~"~10] 71/2XEF = - - @IZ + L'~o(@o - ' ~ : ) J '

2@0X F G : r X E F "

0.03

CF b ~ /0 x-0,03B

Figu re 1. Plo t o ff (x ; z, ~/, 6) as a funct ion of x, where z = 1, ~/= 0.125 and 6 = 0.5 . Fo r agiven value o f 0, the solut ions of (10 a) corr esp ond to the intersections of the curve an dthe hor izo nta l line g iven by the ordinate va lue of - 0 . Th e point s B , C , D, E and F arethe po ints w here two different solut ions merge, and A a nd G are infinite points on thecurve.

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1 50 M . G i n o z aW e n o w d i s c u s s th e c h o i c e o f t h e a c c e p t a b l e b r a n c h f o r s o l u t i o n s o f (1 0 a ).

P a s t o r e [ 6 ] r e l a t e d t h i s p r o b l e m t o t h e n o n - s i n g u l a r i t y o f O ( k ) o n t h e r e a l a x i s a n di n t h e u p p e r h a l f o f th e c o m p l e x - k p l a n e . I n o r d e r t o i n v e s t i g a t e th e r e l a t i o n b e t w e e ns o l u t i o n s o f ( 10 a ) a n d t h e n o n - s i n g u l a r i t y o f O (k ), i t i s c o n v e n i e n t t o s t a r t w i t h t h eo r i g i n a l f o r m o f ( 10 a ), n a m e l y e q u a t i o n ( 1 6 ) o f [ 5 ] . I n t h e p r e s e n t c a s e t h e l a t t e r c a nbe w ri t ten , u s ing (5 c), (7 c) and (11) , as

B = Q( iz /~)A,w h e r e t h e / - c o m p o n e n t s o f t h e v e c to r s A a n d B a r e a z c ~ / 2 a n d - 2 7 r 0 Z t c ~ / e / z ~ lr e s p e ct i v el y . T h i s e q u a t i o n a n d t h e n o n - s i n g u l a r i t y o f ~ ( k ) l e a d t o t h e f o l l o w i n g :

2~ 0 /,~ .\ 1/2a , = - Z , I O l < < 1 . ( 1 2 )

z?] J k t q /w h e r e Q n s ( i z / a ) i s t h e B a x t e r m a t r i x c o r r e s p o n d i n g t o t h e h a r d - s p h e r e s y s t e m( 0 = 0 ). E q u a t i o n ( 12 ) i s a d i r e c t c o n s e q u e n c e o f t h e n o n - s i n g u l a r i t y o f Q (k ) i n t h eu p p e r h a l f o f th e c o m p l e x k p l a n e , a n d a ll t h e s o l u t i o n s o f ( 10 a ) t h a t d o n o t s a t is f y(12) m us t b e re jec ted .

N o w , b y c a l c u l a t i n g t h e a s y m p t o t i c b e h a v i o u r o f ai i n [ 0 [


8/13

S im p l e M S A s o lu t io n in a H S Y m i x t u r e 15 1w h i c h c a n b e w r i t t e n , u s i n g (8 c ', e , f ) , a s

B = (1 - 6 )F 6[(1 - - a l ) r - - ao ] (14)q~oF + 1 4~1F + 4~oS u b s t i t u t i o n o f (1 4) i n t o ( 1 3 ) g i v es th e d e s i r e d e x p r e s s i o n f o r E i n t e r m s o f F .

I n o r d e r t o o b t a i n t h e e x p r e s s i o n f o r F , l e t u s f ir s t r e g a r d K i n (5 c ) a n d ( 9) a s a' c o u p l i n g p a r a m e t e r ' . F o r t h e s ys te m u n d e r c o n s i d e ra t io n , F e y n m a n ' s th e o r e m [ 9 ]r e a d s

f lF = 2 -- SHS _ "':[-- d K ' B(F' ) , (15 a)d o O"w h e r e F ' i s t h e a c c e p t a b l e s o l u t i o n o f (9 ) w i t h K ' i n p l a c e o f K a n d

5 + ~ { ~_/~ : rr~ 3 / 2 1 1 o g - - l o g } r/(4 - - 3 r/) (1 5 b )Srls -~ __ Cj k2 ~f lh2 ] j Cj Cj ( 1 - - ? ] ) 2 'j = lm j a n d h b e i n g t h e m a s s o f t h e i th s p e ci e s o f s p h e r e a n d P l a n c k ' s c o n s t a n t d i v i d e db y 2 n r e s p e c t i v e l y [ 1 0, 1 1]. O n t h e o t h e r h a n d , w e c a n o b t a i n f r o m (9 ) a n d (1 4)

K I "2 + z f da &ID2(F ) d F B(F) = - - D2(Fr e sp e c ti v el y . T r a n s f o r m i n g t h e i n t e g r a t i o n v a r i ab l e f r o m K ' t o F ' in (15 a ) and us ingt h e e q u a t i o n s a b o v e , w e c a n e a s il y p e r f o r m t h e i n t e g r a t i o n i n ( 1 5 a ), w h i c h s h o u l d b ed o n e o n t h e a c c e p t a b l e b r a n c h A B d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n . T h e r e s u l t i sa s f o l l o w s :

1/ ~F = ~ - S HS --g - B ( r ) + ( ~ r 3 + 8 9 ( 1 6 )T h e s u b s t i t u t i o n o f ( 13 ) a n d ( 1 6) i n t o S = f l ( E - F ) y i e l d s

1s = s H s - ~ ( ~ r ~ + 89 (17)W e c a n s e e f r o m ( 13 ) , (1 6) a n d ( 1 7 ) t h a t t h e r m o d y n a m i c q u a n t i t i e s a r e g i v e n in

t e r m s o f s i m p l e r a t io n a l f u n c t i o n s o f F . W h e n t h e ' c h a r g e n e u t r a l i t y ' c o n d i t i o n i si m p o s e d ( Z = 0 o r 6 = 0 ), t h e s e e x p r e s s i o n s r e d u c e t o t h e c o r r e s p o n d i n g e x p r e s s i o n so f H a f n e r et a l . [12] ( see the A ppend ix ) .

A s a n a p p l i c a t i o n o f o u r s i m p l e t h e r m o d y n a m i c t h e o r y , le t u s i n v e st i g at e t h ee f fe c t o f t h e ' c h a r g e - n o n - n e u t r a l i t y ' c o n d i t i o n (6 ~ 0 ) o n t h e f o l lo w i n g q u a n ti f ie s :

0 (1 - 6 )F 6[(1 - cq )r - ~to]}A E = ~ ( E - e . s ) = ~ { +~o--T--- ~ ? T ~ o . '1 ( ~ r 3 + 89A S = S - S H S = - -

w h e r e E HS a n d SHS a r e E a n d S i n t h e h a r d - s p h e r e m i x t u r e ( K = 0 ) a n d w e h a v euse d (13), (14) an d (17) .T h i s e f f e ct m a y b e e m b o d i e d i n t h e d e p e n d e n c e s o f th e se q u a n t i t i e s o n 6 . T h i s

p a r a m e t e r i s i n t h e r a n g e o f 0 ~< 6 ~< 1 , a n d 6 = 0 m e a n s t h e ' c h a r g e n e u t r a l i t y ' .F i g u r e s 2 a n d 3 a r e t h e r e s u lt s fo r A E a n d A S r e sp e c t iv e l y . I n e a c h f i gu r e , w e s h o wt h e c a s e s 0 = - 0 - 3 , - 0 . 2 , - 0 . 1 , 0 . 1 , 0 .1 5 a n d 0 .2 , w h e r e t h e l a c k o f t h e c u r v e i n t h ec a s e o f 0 = 0 . 2 is d u e t o t h e f a c t t h a t t h e r e i s n o s o l u t i o n o f ( 10 a ) - - w h i c h h a s b e e n

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1 5 2 M . G i n o z a

-2 -0 .3~~ --1 -0.1

--2 i0;5 ~ 11

F i g u r e 2. P l o t o f A E a s a f u n c t i o n o f 6 , w h e r e z = 3 a n d ~ / = 0 .4 . T h e n u m b e r s b y t h e s ixcurves give the values of 0. Eq uat io n (10 a) has no solut ion at low ~ for 0 = 0.2.

011Figure 3. Plo t of AS as a funct ion of 6 , where z = 3 and ~/= 0 .4 . The num bers by the s ixcurves give the values o f 0. Eq uat io n (10 a) has no solut ion at low ~ for 0 = 0.2.i n t e r p r e t e d b y W a i s m a n [ 1 3 ] a s p h a s e s e p a r a t i o n o f t h e s y st e m . T h e s e r e s u lt s o b v i -o u s l y s h o w t h a t t h e e ff ec t o f t h e ' c h a r g e - n o n - n e u t r a l i t y ' c o n d i t i o n o n t h e t h e r m o d y -n a m i c f u n c t i o n s i s s i g n if ic a n t .

5. Con cluding emarksT h e r e s u l ts i n t h e p r e s e n t p a p e r c a n b e s u m m a r i z e d a s fo l lo w s .

( i ) A s i m p l e e x p r e s s i o n f o r t h e M S A s o l u t i o n i n t h e B a x t e r f o r m a l i s m h a s b e e np r e s e n t e d , a n d , u n d e r t h e ' c h a r g e - n e u t r a l i t y ' c o n d i t i o n , t h e e q u i v a l e n c e o ft h is t o t h e M S A s o l u t i o n i n th e o r i g i n a l f o r m a l i s m o f t h e O Z e q u a t i o n h a sb e e n p r o v e d . T h e a c c e p ta b l e b r a n c h f o r so l u ti o n s o f t h e n o n l i n e a r e q u a t i o nh a s b e e n d e t e r m i n e d .

(ii) S i m p l e e x p r e s s io n s f o r t h e r m o d y n a m i c f u n c t io n s h a v e b e e n p r e s e n t e d , a n d i th a s b e e n s h o w n t h a t t h e e ff ec t o f t h e ' c h a r g e - n o n - n e u t r a l i t y ' c o n d i ti o n o nt hese func t i ons i s s i gn i f i can t .

I n s e c t i o n 3 , w e h a v e i n v e s ti g a te d t h e m a n i f o l d o f s o l u t i o n s o f th e n o n l i n e a re q u a t i o n ( 1 0 a ) i n t h e r a n g e 0 < ~ < 1 w h e r e t h e t r e n d s h o w n i n f ig u r e 1 w a sg e n e ra l . A s c a n b e s e e n f r o m t h e d e n o m i n a t o r o f ( 10 b ), h o w e v e r , th i s t r e n d c a n n o tb e m a i n t a i n e d f o r 6 = 0 a n d 1 : f i g u r e 4 s h o w s f ( x ; z , rl, 3 ) a s a f u n c t i o n o f x w i t h

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10/13

Simple MSA so lu t i on in a HSY mix ture 1 5 3

\ f l ii ~ o . o a l ir\ - , . - - , t J

\ : / , , \xx~ . . . . . . . . ~ ~ : - 1~ " . . 0 x

~ " . . . " I\ . .. . , . . I' t / . , - 0 . 0 3

F i g u r e 4 . P l o t o f f ( x ; z , t/ , 6 ) a s a fu n c t i o n o f x , w h e r e z = 1 a n d ~ / = 0 . 1 2 5 : . . . . , 6 = 0 ;- - - , 6 = 1 . T h e p o i n t s A a n d B h a v e th e s a m e m e a n i n g s a s i n f ig u r e 1 .z = 1 a n d ~ / = 0 . 1 2 5 . I t c a n b e s e e n t h a t t h e r e a re o n l y f o u r b r a n c h e s o f s o l u t i o n s o f( 1 0 a ) f o r t h e s e v a l u e s o f 6 . A s i s e a s il y s h o w n f r o m a s i m i l a r d i s c u s s i o n t o t h a t i ns e c t i o n 2 , h o w e v e r , t h e a c c e p t a b l e b r a n c h i s a g a i n t h e b r a n ch A B e x c e p t i n th ev i c i n i ty o f t h e p o i n t B .

T h e r e l a t io n o f t h e b r a n c h e s o f s o l u t i o n s i n s e c t i o n 3 t o t h o s e in t h e w o r k o fP a s t o r e [ 6 ] c a n b e e s t a b l i s h e d b y i n t r o d u c i n g G * d ef i n ed a s fo l l o w s :

G * = ~ c ~ Z i c j Z J d r r [ h o ( r ) + 1 ] e - ~ ' / rZ, J

= + 1 + '

w h e r e w e h a v e u s e d ( 8 c ) a n d ( 1 4) . T h i s g i v e s a t r a n s f o r m a t i o n f r o m x ( = F/z ) t o G * .P a s t o r e [ 6 ] i n v e s t i g a te d s o l u t i o n s o f ( 1 0 a ) b y c o n s i d e r i n g f ( x ; z , ~7, 6 ) a s a f u n c t i o no f G * . I n fi g u r e 5 , w e s h o w f ( x ; z , t /, 6 ) a s a f u n c t i o n o f G * w i t h z = 1 a n d tl = 0 . 1 2 5 :

: II Ii f ~ II! /

: 1/:" /~ ].~ /.~176 /~ 1 7 6 1 7 6 J

= : . c ' 2 2 - " "

IIII

- 0 . 0 3 IIII

, ~ 1 7 6 . . . . . . . . . . . . . . . . . . . . . . . . 9 ' ~ . :9 " I i g G "" ' I I

~ : Z B I I\ i '- 0 . 0 a IqlF i g u r e 5 . P l o t o f f ( x ; z , ~1, 6 ) a s a f u n c t i o n o f G * , w h e r e z --- 1 a n d r / = 0 . 1 2 5 : . . . . . 6 = 0 ;, 6 = 1 . T h e p o i n t s A a n d B h a v e t h e s a m e m e a n i n g s a s in fi g ur e 1 .

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1 54 M. G i n o z athe b ranch AB in f igure 4 i s t r ansfo rmed to the b ranch AB in f igure 5 . F igure 2 o f[6] is for 6 = 1.

H a f n e r e t a l . [12 , 14-16] inves t iga t ed the rmodynamic and s t ruc tu ra l p rope r t i e sof l i qu id a l loys wi th s t rong chemica l shor t - range o rde r us ing a s imple mode l cor re -spo nd in g to the spec ia l ca se o f 6 = 0 in sec t ion 4. The re su l t on the e f fect o f the' c h a r g e - n o n - n e u t r a l i t y ' c o n d i t io n i n s e c t io n 4 p r o m p t s u s to u se t h e si m p le t h e o r yin sec t ion 4 a s a mode l .

A p p e n d i xT h e r e s u l t u n d e r th e " c h a r g e - n e u t r a l i t y ' c o n d i t io n ( 2 = O )

A.1 . T h e M S A s o lu t i o nW a i s m a n [ 1 3 ] a n d C o p e s t a k e e t a l . [17] ob ta ined an exp l ic i t so lu t ion o f (1) and

(5a ,b ) w i th (5c ) by us ing the ' cha rge -n eu t ra l i t y ' cond i t ion , 2 = 0 , fo r a two-c o m p o n e n t HS Y m i x tu r e. T h e e x t en s i o n o f th is so l u t i o n t o t h e c ase o f a n n -c o m p o n e n t m i x t u r e c a n b e o b t a i n e d u s i n g t h e s i m i l a r m e t h o d t o t h a t i n o u rprev ious work [18] . We g ive the re su l t he re wi thou t t he de r iva t ion :

c o ( r ) = c ~ ( r ) + Z i Z s c , ( r ) , (A 1 a)h o ( r ) = h ~ (r ) + Z , Z j h . ( r ) , (A 1 b)

w h e r e

h s ( r = c s ( r) + p J dt ,cs(I cx - t I)h~(r,),h ~ ( r ) = - I ( r < a ) ,c s ( r ) = 0 ( r > ~ ) ,h . ( r ) = c . ( r ) + p f d r l c . (I rx - r ] )h .( r l) ,h . ( r ) = 0 ( r < ~ ) ,C a(r = K e - ' ( ' / ' - I ) ( r > a ) .

r

( A 2 a )(A 2 b)( A 2 c )( A 3 a )(A 3 b)(A 3 c)

I t c a n b e sh o wn b y d i r e c t su b s t i t u t i o n t h a t ( A l a , b ) w i th ( A 2 a - c ) a n d (A 3 a - c )sat isfies (1) and (5 a - c ) .

Eq ua t io n (A 2 a ) was so lved by W er the im [19] and Th ie le [20] , whi l e (A 3 a ) wasso l v e d b y W a i sm a n [ 1 3 ] : f o r r < a ,

Cs(r A o + A , r ( r ) a-+A 3 ,O"o o o { o - ; _ 1 ( z r ) ]c . ( r ) = - f f - + 2 - - 7 1 1 - c o s h - ; , (A 4)( A 5 )

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S i m p le M S A s o lu t io n in a H S Y m i x t u r e 155w h e r e

(1 + 2 q ) 2 _ 2A 3 6 t /(1 + 89 2A o = - - , A 1 -(1 - ~/)4 q (1 - ~/)4a n d co i s t h e a c c e p t a b l e s o l u t io n o f t h e n o n l i n e a r e q u a t i o n a s

t20[89 + 13" = co(z - 89 (A 6)O n t h e o t h e r h a n d , t h e M S A s o l u t i o n i n se c t io n 2 b e c o m e s v e ry s i m p l e w h e n

2 = 0 . S i nc e (8 e , f ) a n d 2 = 0 g i v e D a = 0 , w e g e t AN = 0 a n d P N = 0 f r o m ( 8 a,b).T h e r e f o r e ( 7 a - d ) , ( 8 a - f ) a n d 2 = 0 g i v e t h e f o l l o w i n g c o e f f ic i e n t s o f ( 6 a,b):

A j = A , f l j = a f t , D i j = Z i Z j t 7 2 D o , C i j = Z i Z j a 2 C o ,w h e r e (= --~ 1 + f l = - ~ , D o = - 2 r [ c k o ( z ) r + 13 e'---~n C O = 2 r 1 +' 6 q ' ~ -~ "W i t h 2 = 0 , (9 ) b e c o m e s

6 0F 2 + z F + = O .[ ~ o ( ~ ) r + 13 5S u b s t i t u t i o n o f th e s e c o e f f i c ie n t s i n t o ( 6 a ,b ) y i e l d s

Q d r ) = Q o (r ) + Z ~ Z j Q , ( r ) ,w h e r e

( A 7 )

(A8a)

c , j r ) = _ O a 2 F { e - Z ' / ~ - 1 e - " 2 F [ 1 - c ~ ( rz /a )] ~ ( A 9 b ). r O o ( z ) r + 1 2 z + T j '

a n d es(r a n d ca(r) a r e g i v e n b y ( A 4 ) a n d ( A 3 e ) re s p e c t iv e l y .N o w c o m p a r i s o n o f ( A 9 b ) w i t h ( A 5 ) s u g g e s ts t h a t co a n d F a r e r e l a t e d a sf o l l o w s :

2 Fco = dpo(z)F + 1" (A 10)U s i n g t h i s r e la t i o n , i t i s e a s y t o s h o w t h e e q u a l i t y o f t h e s o l u t i o n g i v e n b y ( A 9 a ,b )a n d ( A 7 ) w i t h t h a t g i v e n b y ( A 1 a ) , ( A 4 ) , ( A 5 ) a n d ( A 6 ) .

w h e r e

Q o ( r )= { ~ A r ( r - a ) + p a ( r - a ) ( r < ~ ) ,( r > ~) , (A 8 b)) ' a 2 C o ( e - z r / r _ e - Z ) 4- a 2 D o e - z ' /a ( r < a ) , (A 8 c )

Q l ( r ) = [ a 2 D ~ e - Z ' / a ( r > a ).I t i s s t r a i g h t f o r w a r d t o o b t a i n t h e f o ll o w i n g w i t h th e s u b s t i t u t i o n o f ( A 8 a-c ) i n t o( 4 a ) a n d t h e u s e o f ( A 7 ) :

~c~(r) + Z~ Z j c '~(r) (r < a) ,c i j( r ) = ~ Z i Z j Ca(r ( r > a ) , (A 9 a )Do

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1 56 M . G i n o z aA . 2 . T h e t h e r m o d y n a m i c t h e o ry

I t i s e a s y t o s h o w t h a t t h e u s e o f c o d e f i n e d b y ( A 1 0 ) i n p l a c e o f F i n ( 1 3) , ( 14 ),( 16 ) a n d ( 17 ) y i e ld s th e c o r r e s p o n d i n g e x p r e s s i o n s f o r th e r m o d y n a m i c f u n c t io n sg i v e n b y H a f n e r e t a l . [ 1 2 ] .

References[ 1 ] M A R C H ,N. H . , and TosI , M . P . , 1984, C o u l o m b L i q u i d s (Academic Press) .[2 ] Y O U N G ,W . H. , 1987, Can. J . Phys . , 65, 241.[3 ] B L U M ,L. , and H o w , J . S., 1978, J . s t a t i s t. P h ys . , 19, 317.[4] NnZEK I, K . , 1984, M o l e c . P h y s ., 43, 251.I-5] GINO ZA, M . , 1985, J . p h ys . S o c . Ja p a n , 5 4 , 2783; 1986a , Ib id . , 55, 95 ; 1986b, Ib id . , 55,1782.[6] PASTOP.E,G. , 1988, M o l e c . P h y s . , 63, 371.I-7] .BAXT ER, R. J., 197 0, J . ch em. P h ys . , 52, 4559.[8 ]. BLUM, L. , 1 980, J . s ta t i s t . Phys . , 22, 661.[9] FEVNMAN,R. P., 197 2, S t a t is t i c a l M e c h a n i c s (Benjamin).[10] CARNArlAN,N. F. , and STARLING,K. E. , 1969, J . ch em. Ph ys . , 51, 635.[11] CARNAnAN,N. F. , and STARLING,K. E. , 1970, J . ch em. Ph ys . , 53, 600.[12] HAFNER, . , PASTUREL,A., an d HICTER, P. , 1984 , J . P h y s . F, 14, 1137.1-13] W~dSMAN,E., 1973, J . ch em. Ph ys . , 59, 495.[14 ] HAFN~R, . , PAS ~RE L, A. , an d HICTER,P. , 1984, J . P h y s . F, 14, 2279.[15] PASa'UPa.ZL,A. , HAF r~R, J . , and HICT~R, P. , 1985, P h y s . R e v . B, 32, 5009.[16 ] HA t,mR , J . , and JANK, W . , 1988, J . P h y s . F, 18, 333.[17] COPESTAK~,A. P., EVANS, R., RUPPERSBERG,H . , an d SCrlmMACrmR, W . , 1983, J . Ph ys . ,

13, 1993.[18 ] G IN O Z A ,M., 1987, J . p h ys . S o c . Ja p a n , 5 6 , 5 .[19] WERTHEIM,M . S., 1963, Ph ys . Rev . Le t t . , 10, 321.[20] TmELE, E . , 1963, J . ch em. Ph ys . , 39, 474.

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