PHASE TRANSITIONS IN COULOMB SYSTEMS

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J. Hansen

To cite this version:

J. Hansen. PHASE TRANSITIONS IN COULOMB SYSTEMS. Journal de Physique Colloques, 1985, 46 (C3), pp.C3-9-C3-20. �10.1051/jphyscol:1985302�. �jpa-00224619�

DOURNAL DE PHYSIQUE

Colloque C3, supplément au n°3, Tome 46, mars 1985 page C3-9

PHASE TRANSITIONS IN COULOMB SYSTEMS J. P. Hans en

Laboratoire de Physique Theorique des Liquides*, Universite P. et M. Curie, 75230 Paris Cedex 05, France

Résumé - On donne une brève revue de certains résultats théoriques concernant les transitions de phase dans des mo- dèles simples de systèmes coulombiens simples. On met l'ac- cent sur l'application éventuelle de ces résultats à des suspensions de colloïdes chargés. L'application de la mé- canique statistique à une telle suspension révèle l'exis- tence possible d'une transition réversible "liquide-gaz"

distincte de la coagulation irréversible.

Abstract - A brief review is given of some theoretical re- sults on phase transitions in simple models of classical Coulomb systems. The possible relevance of these results for charged colloidal suspensions is pointed out. A Statis- tical Mechanics treatment of a suspension of charged colloi- dal particles indicates the possible existence of a rever- sible "liquid-gas" transition distinct from irreversible coagulation, under favourable physical conditions.

1. INTRODUCTION

Solid colloidal particles suspended in water are frequently highly char- ged, due to the dissociation of surface radicals in the highly polar

solvent. Consequently Coulomb forces play an important role in deter- mining the stability, the structure and the thermodynamics of such sus- pensions. The Statistical Mechanics of Coulomb systems is presently a very active area of research and discovery, and has recently been the object of another Les Houches workshop(l). In this review an attempt will be made to show how some of the results obtained for simple models of classical Coulomb fluids might be relevant for the study of charged colloids. The emphasis will be on phase transitions in such models, including melting, condensation and ion pairing ("conductor-insulator"

transition), both in three and in two dimensions. In the second part of this paper, the possible existence of a reversible "liquid-gas"

transition in charged colloidal suspensions, distinct from irreversible coagulation, will be examined. It will be shown that such a transi- tion should be expected in a relatively narrow range of stray salt con- centrations and for particles typically one micron in diameter.

2. SINGLE COMPONENT MODELS

The familiar DLVO potention between the double-layers surrounding spher- ical particles of diameter c is of the form^ ' ':

(D

*Equipe associee au C.N.R.S.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985302

where r is the distance between the centers of the spheres,#-'is the Debye screening length and

where the second line holds if the surface potential JI, of the colloi- dal particles is less than kgT/e (the counter-ions and salt ions are assumed to be monovalent). In the weak screening (linear) regime, qo is proportional to the total surface charge Ze carried by the macrol- ons, so that v(r) takes the form of a screened Coulomb potential:

-6' L .

where e is expressed In electrostatic units. For very weak screening

( K + O ) , a regime corresponding typically to the absence of added salt,

the colloidal particles interact via the bare Coulomb potential and move in a pr,actically uniform neutralizing background of counterions.

Neglecting steric repulsion for the moment, which amounts to setting a = o (this point will be reexamined later), the colloidal system is then reasonably well described by the "one component plasma" (OCP) model of N point charges immersed in a uniform background of opposite

charge. The total potential energy V for this model includes part- icle-particle, particle-background an! background-background Coulomb terms. For a periodic system in a volume R , ^V is most compactly expressed in terms of the Fourier compznents of phe microscopic den- sity:

- L e

where the primed sum runs over all wavz-nugbers k f + o compatible with the periodic boundary conditions, the k = o term being left out to account for the background interactions (electroneutrality) ; Cf[k) is the Fourier transform of the Coulomb potential

The OCP is the simplestmodel of a Coulomb system, and it is the clas- sical equivalent of Wigner's "jellium" model for the electron gas

(where Z = -1 and E = 1). In view of its simplicity it is one of the most yjyely studied and best understood models of Statistical Mech- anics

.

(non-ideal) properties are functions of the single dimensionless para- meter r , the ratio of the Bjerrum (or Landau) length b over the

ion-sphere radius a:

-

&

v = -

= m

In the strong coupling limit I' >>l, the thermodynamic properties of the model are given quite accurately (within a few percent) by the

s i m p l e i o n - s p h e r e model, i n which e a c h i o n i s c o n f i n e d t o a s p h e r e o f r a d i u s a and i n t e r a c t s o n l y w i t h t h e u n i f o r m background c o n t a i n e d i n t h a t s p h e r e , which e x a c t l y c a n c e l s t h e i o n i c c h a r g e . An e l e m e n t a r y e l e c t r o s t a t i c c a l c u l a t i o n shows t h a t t h e i o n i s s u b j e c t t o a h a r m o n i c p o t e n t i a l ; t h e r e d u c e d e x c e s s i n t e r n a l e n e r g y p e r i o n i s t h e n g i v e n by:

-

. " .

where t h e f i r s t t e r m i s p u r e l y s t a t i c ( i o n a t t h e c e n t e r o f i t s s p h e r e ) and t h e s e c o n d t e r m i s t h e t h e r m a l e n e r g y ( e q u i p a r t i t i o n t h e o r e m ) . I n t h e s t r o n g c o u p l i n g l i m i t , t h e p a i r d i s t r i b u t i o n f u n c t i o n g ( r ) a s c a l - c u l a t e d from " e x a c t " s i m u l a t i o n s o r from a p p r o x i m a t e i n t e g r a l e q u a t i o n

7.4) ( o f which t h e HNC e q u a t i o n i s by f a r t h e m o s t a c c u r a t e f o r t h i s m o d e l , e x h i b i t s b o t h a w e l l - d e f i n e d " c o r r e l a t i o n h o l e " ( g ( r ) s o f o r r < a ) and pronounced o s c i l l a t i o n s ( s h o r t r a n g e o r d e r ) . The r e s u l t i n g s t a t i c s t r u c t u r e f a c t o r :

00

i s v e r l i q u i d - l i k e , e x c e p t a t s m wavenumbers, where i t v a n i s h e s l i k e k Y , due t o p e r f e c t s c r e e n i n g a t f , i n s t e a d of t e n d i n g t o t h e 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 . A v e r y c h a r a c t e r i s t i c f e a t u r e of t h e s t r o n g - l y c o u p l e d OCP ( r t h a t t h e p o s i t i o n o f t h e main peak i n S ( k ) , kmax s c a l e s l i k e :l7') ( l b S , i n o t h e r words:

s u c h a s c a l i n g b e h a v i o u r i s p r e c i s e l y o b s e r v e d i n S ( k ) o f d i s p e r s i o n s of c h ~ ~ f g y p o l y s t y r e n e s p h e r e s a s measured b y l i g h t s c a t t e r i n g e x p e r i - m e n t s

.

d e l have b e e n u s ~ ~ ) t o a n a l y s e t h e l i g h t s c a t t e r i n g d a t a , i n t h e weak s c r e e n i n g regime

.

I t i s i n t e r e s t i n g t o n o t e i n p a s s i n g t h a t t h e OCP model i s a l s o an ex- c e l l e n t s t a r t i n g p o i n t a n a l y s i s o f n e u t r o n o r X-ray s c a t t e r i n g d a t a from l i q u i d a l k a l i f P 6 ' f P f , where t h e s c a l i n g r e l a t i o n (9) i s a l s o w e l l obeyed.

F o r s u f f i c i e n t l y s t r o n g c o u p l i n g , t h e OCP u n d e e s a f i r s t o r d e r t r a n - s i t i o n f r o m a f l u i d t o a c r y s t a l l i n e BCC p h a s e f r 9 . The p r e c i s e l o - c a t i o n o f t h e t r a n s i t i o n i s a v e r y d e l i c a t e t a s k , s i n c e t h e f r e e e n e r - g i e s o f b o t h p h a s e s , a s d e t e r m i n e d by computer s i m u l a t i o n s , l i e v e r y c l o s e o v e r a n e x t e n s i v e r a n g e of r v a l u e s . T h i s p o i n t c a n b e u n d e r - s t o o d from e q . ( 7 ) f o r t h e i n t e r n a l e n e r g y : t h e t h e r m a l p a r t (which i s j u s t 3 / 2 i n t h e s i m p l e i o n s p h e r e model) w h i c h e s s e n t i a l l y d i s t i n g u i s h - e s b o t h p h a s e s , i s o n l y a s m a l l f r a c t i o n o f t h e t o t a l when > 100.The l a t e s t r e s u l t s o b t a i n e d from v e r y l o n g r u n s on t h e Los Alamos c o m p u t e r s and a c a r e f u l a n a l y s i s o f t h e N-dependenc t h e s i m u l a t e d s a m p l e s , l e a d t o t h e e s t i m a t e r m % 1 7 8 a t m e l t i n g e l q f . Remembering e q s . ( 6 ) t h i s l e a d s t o t h e f o l l o w i n g form o f t h e m e l t i n g c u r v e :

- 5 p - A.I 73

T,(K)

--Lp,[--?]

The e n t r o p y c h a n g e p e r p a r t i c l e a t m e l t i n g , AS/Nk

=

The OCP mo'ael c a n b e accommodated t o a c c o u n t f o r t h e f i n i t e s i z e ( o f o ) o f t h e c o l l o i d a l p a r t i c l e s . The p r e s e n c e o f a n e x t e n d e d c o r r e l a t i o n h o l e i n t h e s t r o n g c o u p l i n g r e g i m e means t h a t c o n t a c t c o n f i g u r a t i o n s

between t h e p a r t i c l e s w i l l be e f f e c t i v e l y p r e v e n t e d by t h e Coulomb r e - p u l s i o n a s long a s U < a , i f r ^>> 1. I n o t h e r words i t i s e x p e c t e d t h a t t h e s t r u c t u r a l and thermodynamjc p r o p e r t i e s of a model of

e q u a l l y c h a r g e d h a r d s p h e r e s i n a n e u t r a l i z i n g background w i l l be i d e n - t i c a l t o t h o s e o f t h e OCP i f t h e p r e c e d i n g c o n d i t i o n i s f u l f i l l e d . This o b s e r v a t i o n i s i n t a c t t h e b a s i s o f a v e r y s u c c e s s f u l t h e o r y of t h e OCF s t r u c t u r e based on an a n a l y t i c s o l u t i o n of t h e ' ( y g ~ n s p h e r i c a l a p p r o x i - mation" (MSA) f o r t h e charged h a r d s p h e r e model

.

l y s i s . F o r t u n a t e l y b o t h c a s e s can be shown t o be e q u i v a l e n t , e x c e p t f o r a r a t h e r t r i v i a l r e n o r m a l i z a t i o n of t h e c h a r g e : t h e f r a c t i o n of t h e background i n s i d e t h e s p h e r e s r e d u c e s t h e a p p a r e n t c h a r g e s o f t h e s e s p h e r e s , s o t h a t a c c o r d i n g t o Newt n ' s theorem a system o f h a r d s p h e r e s 9

of c h a r g e Ze/ 1-q ) (where n = I I p a /6 i s t h e packing f r a c t i o n ) i n a pene- t r a t i n g background i s e x p e c t e d t o be e q u i v a l e n t t o a system of

s p h e r e s of r e d u c e d c h a r g e Ze immersed i n an e x c l u d e d background

.

The m e l t i n g t r a n s i t i o n w i l l be s h i f t e d a c c o r d i n g l y .

With i n c r e a s i n g s a l t c o n c e n t r a t i o n , s c r e e n i n g e f f e c t s become more and more i m p o r t a n t , and t h e b a r e Coulomb p o t e n t i a l between c o l l o i d a l p a r t -

i c l e s must be r e p l a c e d by a s c r e e n e d p o t e n t i a l of t h e form ( 3 ) . As long a s K O < < l , t h e d i f f e r e n c e between s c r e e n e d and b a r e Coulomb p o t e n t i a l s can be looked upon a s a p e r t u r b a t i o n , t h e OCP b e i n g c o n s i d e r e d a s a z e r o t h o r d e r " r e f e r e n c e system". N e g l e c t i n g a g a i n c o r e e f f e c t s which a r e n e g l i g i b l e f o r s u f f i c i e n t l y s t r o n g Coulomb c o u p l i n g s , t h e s t r u c - t u r e f a c t o r S ( k ) o f t h e s c r e e n e d system can be r e l a t e d t o t h a t of t h e OCP r e f e r e n c e s y s t e m , S o ( k ) , v i a t h e random p h a s e a p p r o x i m a t i o n

(RPA)(15,

where Q ( k ) i s t h e " p e r t u r b a t i o n " , i . e . t h e d i f f e r e n c e between s c r e e n e d and b a r e Coulomb p o t e n t i a l s :

The e x a c t s m a l l k e x p a n s i o n of t h e OCP s t r u c t u r e f a c t o r r e a d s ( 4 )

where KO i s t h e i n v e r s e Debye s c r e e n i n g l e n g t h f o r t h e b a r e OCP(i.e.

w i t h o u t . c o u n t e r - i o n o r s a l t c o n t r i b u t i o n s ) and $. ₌

p

i s t h e reduced 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 of t h e OCP (which becomes i n f a c t n e g a t i v e f o r r > 3 ) . Combination o f eqs.11-13 l e a d s immediately t o t h e k 4 o l i m i t and hence t o t h e reduced o s m o t i c c o m p r e s s i b i l i t y )

S w i t c h i n g from s u r f a c e c h a r g e t o s u r f a c e p o t e n t i a l ( c f . e q s . 1 - 3 ) , t h i s r e l a t i o n can i n p r i n c i p l e be u s e d e s t i m a t e t h e s u r f a c e p o t e n t i a l

yo from a measurement o f S(k = o )

.

An a l t e r n a t i v e , v a r i a t i o n a l approach f o r t h e c a l c u l a t i o n o f t h e thermo- dynamic p r o p e r t i e s i s based on t h e Gibb-Bogoliubov i n e q u a l i t y f o r t h e f r e e e n e r g i e s of t h e s c r e e n e d system (F)and of t h e r e f e r e n c e OCP s y s - tem (F-) :

where t h e second term i s t h e a v e r a g e o f t h e t o t a l p e r t u r b a t i o n :

c a l c u l a t e d o v e r t h e r e f e r e n c e system ensemble. The r . h . s . o f e q . ( 1 5 ) c a n be minimized w i t h r e s n e c t t o a v a r i a t i o p $ l ) p a r a m e t e r , e . g . an e f - f e c t i v e v a l e n c e Zeffof t h e r e f e r e n c e system

.

(1 71 a l k a l i m e t a l s (where t h e s c r e e n i n g i s due t o t h e c o n d u c t i o n e l e c t r o n s ) , b u t h a s a p p a r e n t l y n o t y e t been t r i e d f o r c h a r g e d c o l l o i d a l s y s t e m s . A " r u l e of t h e thumb" would s t a t e t h a t m e l t i n g of a s c r e e n e d c o l l o i d a l c r y s t a l w i l l occur whenever r a s d e t e r m i n e d by minimizing e q . ( 1 3 ) , r e a c h e s i t s OCP v a l u e rm = 1 7 8 e f f 2

When t h e s c r e e n i n g becomes v e r y s t r o n g ( K o > > l ) , i t i s e x p e c t e d t h a t t h e OCP model c e a s e s t o be a u s e f u l r e f e r e n c e s y s t e m , because t h e p e r - t u r b a t i o n (12) becomes t o o l a r g e . The h a r d s p h e r e model, w i t h a n e f - f e c t i v e d i a m e t e r a f f > a , w i l l t h e n be t h e more u s e f u l r e f e r e n c e system.

T h i s optimum d i a m e f e r a e f f c a n a l s o be d e t e r m i n e d by a v a r i a t i o n a l c a l - c u l a t i o n based on t h e i n e q u a l i t y ( 1 5 ) , o r b o t h e r s t a n d a r d p e r t u r - b a t i o n t h e o r i e s of t h e t h e o r y of l i q u i d s ( l 8 T ; an i l l u s t r a t i o n o f s u c h a p r o c e d u r e w i l l be g i v e n i n t h e l a s t s e c t i o n of t h i s p a p e r . I n t h e s t r o n g s c r e e n i n g r e g i m e , which c o r r e s p o n d s t o h i g h c o n c e n t r a t i o n s of ad- ded s a l t ( b u t below t h e c o a g u l a t i o n t h r e s h o l d ) , c r y s t a l l i z a t i o n i s ex- p e c t e d t o o c c u r whenever t h e e f f e c t i v e p s a k i n g f r a c t i o n 9 ^{f f} =

apo3 / 6 r e a c h e s t h e c r i t i c a l v a l u e f o r t h e h a r d s p h e r e ? r a n s i t i o n , i . e . e f f f s ~ . ~ .

I n t h e d i f f i c u l t i n t e r m e d i a t e s c r e e n i n g regime ( K a ~ l ) , n e i t h e r r e f e r - e n c e system i s t r u l y s a t i s f a c t o r y , and one must r e l y on computer simu- l a t i o n ~ ( l 9 ) o r on s p e c i a l l y a d a t e d t h e o r i e s , l i k e t h e " r e s c a l e d " mean s p h e r i c a l a p p r o x i m a t i o n (MSA) ($01 t o p r e d i c t a c c u r a t e s t r u c t u r e f a c t o r s o r thermodynamic p r o p e r t i e s . No s i m p l e p r e d i c t i o n s f o r t h e m e l t i n g c u r v e a r e a v a i l a b l e i n t h a t c a s e .

3 . ONE COMPONENT COULOMB SYSTEMS IN TWO DIMENSIONS

C o l l o i d a l s u s p e n s i o n i n r e s t r i c t e d ( s l a b ) g e o m e t r i e s , o r a t t h e l i q u i d - a i r i n t e r f a c e , form f a s c i n a t i n g two-dimensional s y s t e m s , which have a t - t r a c t e d a keen i n t e r e s t o f t h e e x p e r i m e n t a l i s t s ( 2 2 ) . For t h a t r e a s o n i t i s of i n t e r e s t t o r e v i e w b r i e f l y o u r p r e s e n t knowledge on phase t r a n s i t i o n s i n two-dimensional (2d) Coulomb s y s t e m s . Like t h e p r e c e - d i n g s e c t i o n , t h e p r e s e n t one i s r e s t r i c t e d t o one-component s y s t e m s .

One must c a r e f u l l y d i s t i n g u i s h between 2d systems of p a r t i c l e s i n t e r - a c t i n g v i a t h e 3d Coulomb p o t e n t i a l ( l / r ) and t r u l y 2d s y s t e m s , where t h e Coulomb p o t e n t i a l , i . e . t h e s o l u t i o n of t h e 2d P o i s s o n e q u a t i o n , i s l o g a r i t h m i c .

a) l / r p o t e n t i a l

T h i s i s t h e p h y s i c a l l y r e l e v a n t p o t e n t i a l f o r l a y e r s of s p h e r i c a l c o l l o i d a l ~ a r t i c l e s . There i s an i n t e r e s t i n e a n a l o e v w i t h two-dimen- _{" ",} s i o n a l e l e c t r o n l a y e r s t r a p p e d a t t h e s u r f a c e of l i q u i d Helium, which have been e x t e n s i v e l y s t u d i e d a t B e l l ~ a b o r a t o r i e s ( 2 3 ) a n d a t s a c l a y ( 2 4 ) .

I n t h e s e l a y e r s , t h e e l e c t r o n s c r y s t a l l i z e i n t o a 2 d hexagonal l a t - t i c e a l t h o u g h t h e y s t a y f a r a p a r t ( t h e l a t t i c e s p a c i n g i s t y p i c a l l y 1 micron) ; b u t due t o t h e low t e m p e r a t u r e (T S 1K) , t h e c o u p l i n g cons- t a n t r d e f i n e d i n e q . (6) ( w i t h a = ( 1 / 1 ~ p ) l / 2 i n 2d. p b e i n g t h e

a r e a 1 d e n s i t y ) i s t y p i c a l l y l a r g e r t h a n 100 ! I n f a c t e x p e r i m e n t s (23) and computer s i m u l a t i o n s ( 2 5 ) a g r e e on t h e v a l u e Tm 130 a t m e l t i n g . The t r a n s i t i o n a p p e a r s t o be f i r s t o r d e r a s i n t h e c a s e of 2d systems of h a r d c o r e p a r t i c l e s ( 2 6 ) , b u t a t t e m p t s have been made t o u s k t h e d i s - l o c a t i o n - u n b i n d i n g m e ~ h a n i s r n ( ~ ~ ) t o s t u d y t h e m e l t i n g of t h e 2d e l e c - t r o n l a t t i c e ( 2 8 ) . I t i s i m p o r t a n t t o s t r e s s t h a t t h e e l e c t r o n l a y e r s obey e s s e n t i a l l y c l a s s i c a l S t a t i s t i c a l Mechanics, d e s p i t e t h e low tem- p e r a t u r e , b e c a u s e t h e i r s p a c i n g i s much l a r g e r t h a n t h e i r t h e r m a l d e B r o g l i e w a v e l e n g t h . The a n a l o g y w i t h charged c o l l o i d a l s p h e r e s con- s t r a i n e d t o 2d i s hence c o m p l e t e , p r o v i d e d t h e s c r e e n i n g i s s u f f i c i e n t - l y weak s o t h a t Coulomb c o r r e l a t i o n s mask any s t e r i c e f f e c t s . The t h e r - mal f a c t o r E kBT f o r c o l l o i d a l s u s p e n s i o n s i s t y p i c a l l y 20000 t i m e s

l a r g e r t h a n i n t h e c a s e o f e l e c t r o n l a y e r s , b u t t h i s i s w e l l compen- s a t e d by t h e v a l e n c e f a c t o r 22, s o t h a t comparable v a l u e s of r a r e r e - c o v e r e d . I n t h e s t r o n g s c r e e n i n g r e g i m e , t h e h a r d d i s k system becomes t h e r e l e v a n t model f o r t h e s t u d y of t h e 2d m e l t i n g t r a n s i t i o n . A s i z e - a b l e volume change o n - m e l t i n g (of t h e o r d e r of 1 0 % ) s h o u l d be o b s e r v e d i n t h e l a t t e r c a s e ( 2 6 ) , w h i l e t h i s volume change i s e x p e c t e d t o d r o p t o z e r o when t h e s a l t c o n c e n t r a t i o n , and hence t h e s c r e e n i n g , d e c r e a s - e s . But t h e e n t r o p y of m e l t i n g remains f i n i t e i n t h e weak s c r e e n i n g l i m i t , b e i n g of t h e o r d e r of AS/Nkg = 0.3 ( 2 5 ) .

P o l y s t y r e n e s p h e r e s t r a p p e d a t t h e w a t e r - a i r i n t e r f a c e c a r r y l a r g e d i - p o l e moments 3 p e r p e n d i c u l a r t o . t h e i n t e r f a c i a l p l a n e , due t o t h e asym- m e t r i c of t h e c h a r g e d i s t r i b u t i o n around t h e p a r t l y immersed p a r t i c l e s . The p a r a l l e l d i p o l e s r e p e l e a c h o t h e r - v i a t h e d i p o l a r p o t e n t i a l :

/ L

The c o r r e s p o n d i n g d i m e n s i o n l e s s c o u ~ l i n g c o n s t a n t i s now

1 7 = 2%-

For s u f f i c i e n t l y l a r g e v a l u e s o f r , t h e i n t e r f a c i a l c o l l o i d i s e x p e c t - e d t o c r y s t a l l i z e . An i n t e r f a c i a l hexagonal c r y s t a l h a s i n d e e d been ob- s e r v e d e x p e r i m e n t a l l y (29) and s i m u l a t i o n s l e a k t o t h e e s t i m a t e r , 7 6 0 a t m e l t i n g ; t h e c o r r e s p o n d i n g e n t r o p y change p e r p a r t i c l e i s v e r y s i - m i l a r t o t h a t imated f o r t h e i n t e r f a c i a l Coulomb c r y s t a l

( bS/NkgL.0.3(08f. I t s h o u l d be s t r e s s e d however t h a t a d i r e c t compari- son between t h e o r y and e x p e r i m e n t i s n o t p o s s i b l e , s i n c e t h e s i m u l a - tions.assume t h e p o i n t d i p o l e i n t e r a c t i o n (17) w h i l e t h e t r u e i n t e r -

a c t i o n between e x t e n d e d c h a r g e d i s t r i b u t i o n s i s c e r t a i n l y more compli- c a t e d .

b ) Log r i n t e r a c t i o n

The s o l u t i o n o f t h e 2d P o i s s o n e q u a t i o n i s p o t e n t i a l :

where L i s a n a r b i t r a r y l e n g t h s c a l e . ( l 9 ) i s p r e c i s e l y t h e i n t e r a c t i o n between i n f i n i t e l y l o n g p a r a l l e l r o d s c a r r y i n g a l i n e a r c h a r g e d e n s i t y Ze. The t w o - d i m e n s i o n a l OCP model o f p o i n t c h a r g e d i n t e r a c t i n g t h r o u g h t h e p o t e n t i a l (19) may h e n c e b e a r e a s o n a b l e a p p r o x i m a t i o n f o r t h e m o t i o n o f p a r a l l e l c h a r g e d r o d s ( l i k e t h e Tobacco Mosaic V i r u s ) i n t h e p l a n e p e r p e n d i c u l a r t o t h e i r a x i s , p r o v i d e d t h a t t h e s c r e e n i n g l e n g t h of t h e c o u n t e r - i o n s and t h e s t r a y i o n s i s s i g n i f i c a n t l y l a r g e r t h a n t h e s p a c i n g between t h e r o d s .

The d i m e n s i o n l e s s c o u p l i n g c o n s t a n t f o r t h e 2d O C P , r=zk'/gs~

i s i n d e p e n d e n t o f d e n s i t y , a n d a s i m p l e s c a l i n g a r g u m e n t shows t h a t t h e e q u a t i o n - o f - s t a t e i s r i g o r o u s l y g i v e n a t a l l t e m p e r a t u r e s

(2 0 1

u n d e r g o e s a f i r s t o r d - c r y s t a l w h e n P = 1 4 0 ( ~ ~ , o f Choquard and ~ l b r o u i n ( ~ ~ ) h a v e r e v e a l e d v e r y i n t e r e s t i n g c o o p e r a t i v e p r e m e l t i n g m o t i o n s o f g r o u p s o f p a r t i c l e s , j u s t below t h e m e l t i n g t e m p e r a t u r e ; t h e s e m o t i o n s i n c l u d e c y c l i c p e r m u t a t i o n s o f 6 , s o r more p a r t i c l e s , s i m i l a r t o p r e m e l t i n g m o t i o n s o b s e r v e d f o r t h e h a r d d i s k f l u i d , p o i n t i n g t o some k i n d o f u n i - v e r s a l m e l t i n g b e h a v i o u r i n 2d. T h i s " U n i v e r s a l i t y " i s c o n f i r m e d on

t h e f l u i d s i d e o f t h e t r a n s i t i o n by t h e s t r i k i n g s i m i l a r i t y o f t h e main p e a k o f t h e s t r u c t u r e f a c t o r s S ( k ) i n t h e s e r a d i c a l l y d i f f e r e n t s y s t e m g 3 ? )

4. TWO COMPONENT COULOMB SYSTEMS

Up t o now t h e d i s c u s s i o n h a s b e e n r e s t r i c t e d t o o n e component b a r e o r s c r e e n e d Coulomb s y s t e m . I n s u c h m o d e l s t h e c o u n t e r i o n s and s t r a y i o n s a r e r e p r e s e n t e f by a c o n t i n u u m which i s c h a r a c t e r i z e d o n l y by i t s s c r e e - n i n g l e n g t h K

.

v e r s i o n (RPM) where t h e d i a m e t e r s a n d a b s o l u t e c h a r g e s o f a l l i o n s a r e e q u a l , s o t h a t :

A thermodynamic s t a t e of T h i s system i s c h a r a c t e r i z e d by two v a r i - a b l e s chosen t o be t h e c o u p l i n g r i n e q . ( 6 ) , o r e q u i v a l e n t l y

and t h e p a c k i n g f r a c t i o n = IIpa3/6, where p i s t h e t o t a l number of i o n s p e r u n i t volume. The RPM i s a r e a s o n a b l e model f o r t h e s t u d y o f t h e s t r u c t u r e of m o l t e n a l k a l i h a l i d e s , b u t i t i s much l e s s s a t i s f a c t - o r y f o r i o n i c s o l u t i o n s b e c a u s e i o n - s o l v e n t e f f e c t s a r e c o m p l e t e l y i g n o r e d ; t h i s i s n o t j u s t i f i e d when i o n s and s o l v e n t m o l e c u l e s a r e of comparable s i z e .

Molten s a l t s undergo a l i q u i d - s o l i d t r a n s i t i o n f o r c o u p l i n g s r cn 60, which i s accompanied by an u n u s u a l l y l a r g e r e l a t i v e volume change, AV/VR whfch can be a s l a r g e a s 20%. I n t h e l i q u i d t h e r e i s a s t r o n g c o m p e t i t i o n between s c r e e n i n g and s t e r i c (excluded volume) e f f e c t s which l e a d s t o t h e v e r y c h a r a c t e r i s t i c f e a t u r e of c h a r g e o r d e r i n g , a r e g u l a r a l t e r n a t i o n of o p p o s i t e l y charged n e i g h b o r s h e l l s around a c e n t - r a l i o n ( 3 5 ) . T h i s l e a d s t o an o s c i l l a t o r y b e h a v i o u r of t h e p a i r d i s t - t r i b u t i o n f u n c t i o n s (A)

(,,p

Molten s a l t s undergo a l i q u i d - g a s t r a n s i t i o n which t e r m i n a t e s a t a

P

c r i t i c a l p o i n t c o r r e s p o n d i n g r o u g h l y t o l0 and 0 . 1 .

T h i s p h a s e t r a n s i t i o n i s p a r t i c u l a r l y i n e r e s t i n g b e c a u s e i t i s a c - companied by a k i n d of " c o n d u c t o r - i n s u l a t o r " ( o r Mott) t r a n s i t i o n due t o i o n p a i r f o r m a t i o n i n t h e low d e n s i t y v a p o u r . While i o n s a r e l a r g e - l y d i s s o c i a t e d i n t h e d e n s e r l i q u i d p h a s e , which i s e s s e n t i a l l y Coulom- b i c i n n a t u r e , a c o r r e c t S t a t i s t i c a l Mechanics d e s c r i p t i o n of t h e d i - l u t e vapour p h a s e must t a k e i o n p a i r s (forming d i p o l a r m o l e c u l e s ) , a n d p o s s i b l y l a r g e r c l u s t e r s i n t o a c c o u n t . F a i l u r e t o do s o h a s l e d t o h i g h l y e r r o n e o u s e s t i m a t e s of t h e c r i t i c a l p a r a m e t e r s i n t h e framework of t h e RPM of eq. (21) ( 3 7 ) . I n p a r t i c u l a r , t h e f a m i l i a r "mean s p h e r i c -

a l approximation1' (MSA) l e a d s t o a c r i t i c a l d e n s i t y ?, ^{^.} O.Ol,which i s t o o low by an o r d e r o f magnitude.

To o b t a i n a c o r r e c t d e s c r i t i o n of t h e low d e n s i t y vapour and o f l i q u i d g a s c o e x i s t e n c e , B . e r r u m 1 s P 3 ~ ) c l a s s i c " i o n p a i r i n g " i d e a h a s r e c e n - t l y been r e v i v e d ( 3 d ) . I n i t s c r u d e s t form t h i s t h e o r y i s b a s e d on t h e o b s e r v a t i o n t h a t t h e i n t e g r a n d of t h e p a r t i t i o n f u n c t i o n f o r a p a i r of o p p o s i t e l y c h a r g e d i o n s , i . e . :

-

has a minimum a t r ⁼ b/Z, i . e . a t one h a l f t h e Bjerrum l e n g t h ; t h i s d i s t a n c e d e f i n e s an i o n p a i r a s one whose s e p a r a t i o n i s l e s s t h a n b / 2 , a s l o n g a s b/2 > a

.

I n t h e h i g h l y d i s s y m m e t r i c c a s e of macroions i n t h e p r e s e n c e of c o u n t e r - i o n s and s t r a y i o n s , t h e mechanism s i m i l a r t o i o n p a i r i n g i n symmetric- a l e l e c t r o l y t e s i s t h e f o r m a t i o n of S t e r n l a y e r s and t h e phenomenon o f - " i o n c o n d e n s a t i o n " which w i l l be b r i e f l y examined i n t h e f o l l o w i n g

s e c t i o n .

5. THE TWO-DIMENSIONAL COULOMB GAS

The s y m m e t r i c a l two-dimensional Coulomb g a s i s a system of o p p o s i t e l y charged h a r d d i s k s i n t e r a c t i n g v i a t h e 2d Coulomb p o t e n t i a l , and i s hence t h e e x a c t c o u n t e r p a r t of t h e 3d RPM of e q . ( 2 1 ) ; s e t t i n g E = 1, t h e 2d p o t e n t i a l r e a d s :

"r,p (11) =

*

1 (241

= - t , ~ ~ e ~ & ( n / ~ )

;

A v e r y i n t e r e s t i n g f e a t u r e i s t h a t , c o n t r a r i l y t o t h e 3d c a s e , a s y s - tem of p o i n t c h a r g e s ( a = o) w i l l be s t a b l e a s long a s

T > ^e 2 ^/2kg, or <2.

The p a r t i t i o n f u n c t i o n w i l l be f i n i t e under t h i s c o n d i t i o n , s i n c e t h e c o n t r i b u t i o n o f a p a i r of o p p o s i t e l y c h a r g e d p a r t i c l e s w i l l t h e n r e - main i n t e g r a b l e a t v a n i s h i n g d i s t a n c e s .

A t r ^{= 2 ,} t h e i n t e r n a l e n e r g y and t h e s p e c i f i c h e a t d i v e r g e and t h e plasma of N p o s i t i v e and N n e g a t i v e i o n s recombines i n t o an i d e a l gas of N " c o l l a p s e d " p a i r s . I f p = 2N/S i s t h e t o t a l a r e a 1 d e n s i t y , t h e e q u a t i o n - o f - s t a t e f o r r < 2 i s t h e same a s t h a t o f t h e 2d OCP ( e q . (ZO)), w h i l e f o r r ^{> 2} i t r e d u c e s t o t h a t of a n i d e a l gas(31) :

I n t h e d i s s y m m e t r i c c a s e of N i o n s of c h a r g e + Ze and N Z i o n s of c h a r - g e -.e t h e plasma w i l l be s t a b l e a s l o n g a s T > e 2 / ( k B 2 Z ) , and t h e r e w i l l be Z s u c c e s s i v e " r e c o m b i n a t i o n s " , w i t h a s many b r e a k s i n t h e equa- t i o n - o f - s t a t e , which now r e a d s (40) :

The a n a l o g y betwe'kn t h e s e " c o l l a p s e t r a n s i t i o n s " i n t h e 2d Coulomb g a s and Manningrs(41) " i o n c o n d e n s a t i o n " on l i n e a r p o l y e l e c t r o l y t e s i s s t r i k i n g . According t o Manning's p i c t u r e monovalent c o u n t e r i o n s w i l l

"condense" on 'a l i n e a r p o l y e l e c t r o l y t e c h a i n when t h e r a t i o of t h e Bjerrum l e n g t h b o v e r t h e s p a c i n g a between monovalent monomers on t h e c h a i n w i l l exceed t h e c r i t i c a l r a t i o 5 = 1.

For i o n s of f i n i t e s i z e ( a f o ) , t h e c o l l a p s e t r a n s i t i o n goes o v e r i n t o a t r a n s i t i o n between a h i g h t e m p e r a t u r e Coulomb g a s , and a low temper- a t u r e d i p o l a r g a s i n which o p p o s i t e l y charged i o n s a r e p a i r e d i n t o n e u t r a l molecule^", v e r y much a s i n t h e 3d c a s e 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 ( 2 7 ) . Although t h e e x i s t e n c e of such a " c o n d u c t o r - i n - s u l a t o r " t r a n s i t i o n i n 2d h a s been r i g o r o u s l y e s t a b l i s h e d ( 4 2 ) , t h e

p r e c i s e phase diagram i s n o t y e t known q u a n t i t a t i v e l y . C a l c u l a t i o n s a l o n g Bjerrum o l d i d e a s ( 3 8 ) a r e p r e s e n t l y under way. Ion p a i r i n g p l a y s a l s o a c e n t r a l r o l e i n t h e c a s e of p o i n t i o n s ( U = o ) , i n t h e range 1 < r < 2 , where i t s t r o n g l y i n f l u e n c e s p a i r c o r r e l a t i o n s ( 4 2 ,4 0 ) . 6 . "LIOUID-GAS" TliANSITION IN CHARGED COLLOIDAL SUSPENSIONS

I t i s w e l l known t h a t f o r s u f f i c i e n t l y s t r o n g s c r e e n i n g ( i . e . i n t h e p,resence o f a s u f f i c i e n t amount of added s a l t ) , t h e e l e c t r o s t a t i c r e - p u l s i o n (1) between d o u b l e - l a y e r s becomes s h o r t - r a n g e and "uncover"

t h e v a n d e r Waals a t t r a c t i o n between c o l l o i d a l p a r t i c l e s . T h i s a t t r a c - t i o n i s u l t i m a t e l y r e s p o n s i b l e f o r t h e c o a g u l a t i o n of t h e c o l l o i d ( 2 ) . The t o t a l p o t e n t i a l between two c o l l o i d a l p a r t i c l e s i s t h e n of t h e

form:

-

4

X

-

where X = r / u i s t h e reduced d i s t a n c e between t h e c e n t e r s of two p a r t - i c l e s K = K d i s t h e reduced i n v e r s e Debye l e n g t h , J = $ / d i s t h e e l e c t r o s t a t i c c o u p l i n g c o n s t a n t , A i s Hamaker's c o n s t a n t and t h e f u n c - t i o n h ( X ) d e s c r i b e s t h e van d e r Waals a t t r a c t i o n between c o l l o i d a l _{. .} s p h e r e s :

.R

x L - 1 + -

+2&[1-+)

Whenever t h e r a t i o

3

m ' C o a g u l a t i o n w i l l be p r e v e n t e d a s soon a s t h e Coulomb b a r r i e r v(xM) i s s u b s t a n t i a l l y l a r g e r t h a n t h e t h e r m a l e n e r g y k T , and t h e p h y s i c a l d i a - m e t e r a o f t h e p a r t i c l e s can be s a f e l y r e p l a c e 8 by t h e e f f e c t i v e d i a - m e t e r ueff = 0 xN. I f t h e s e c o n d a r y minimum i s s u f f i c i e n t l y deep

( ) t h e r e e x i s t s t h e n t h e p o s s i b i l i t y t h a t c h a r g e - s t a b i l i z e d d i s p e r s i o n s of c o l l o i d a l p a r t i c l e s w i l l undergo a r e v e r s i b l e phase s e p a r a t i o n i n t o a d i l u t e "gas" p h a s e c o e x i s t i n g w i t h a c o n c e n t r a t - ed " l i q u i d " p h a s e , under t h e combined a c t i o n of t h e s c r e e n e d e l e c t r o - s t a t i c r e p u l s i o n and t h e van d e r Waals a t t r a c t i o n . The a n a l o g y w i t h t h e diagram of s i m p l e m o l e c u l a r s u b s t a n c e would t h e n be c o m p l e t e , w i t h t h e c o l l o i d a l d i s p e r s i o n e x h i b i t i n g t h e u s u a l g a s , l i q u i d and s o l i d p h a s e s . T h i s p o s s i b i l i t y has r e c e n t l y been e x p l o r e d w i t h i n t h ame- work of

($97

( 4 i f

Grimson and V i c t o r and Hansen . The f i r s t of t h e s e p a p e r s a p p l i e s o n l y t o weakly charged m i c r o e m u l s i o n s , w h i l e t h e second c o n s i d e r s t h e c a s e of h i g h l y charged s p h e r i c a l c o l l o i d s of a r b i t r a r y s i z e . The l a t - t e r work d e t e r m i n e d t h e r a n g e of p h y s i c a l p a r a m e t e r s (J,A,K) o v e r which t h e p r e d i c t e d h a s e s e p a r a t i o n may be o b s e r v a b l e e x p e r i m e n t a l l y

i n aqueous s o l u t i o n s ? 4 4 ) . The c o n c l u s i o n s of t h i s work a r e t h e f o l - lowing. A " l i q u i d - g a s " t r a n s i t i o n s h o u l d i n d e e d be o b s e r v a b l e a t p h y s i c - a l l y a c c e p t a b l e t e m p e r a t u r e s (T > 273K), p r o v i d e d t h e v a l u e s of t h e e l e c t r o s t a t i c and Hamaker c o u p l i n g c o n s t a n t s l i e above a c r i t i c a l c u r v e shown i n f i g . 1 . I n t h a t r a n g e t h e t e m p e r a t u r e T and p a c k i n g f r a c t i o n

of t h e l i q u i d - g a s c r i t i c a l p o i n t i n c r e a s e s w i t h i n c r e a s i n g s a l t Z ~ n c e n t r a t i o n (i . e . K ) , b u t t h e two-phase system w i l l be s t a b l e a g a i n s t c o a g u l a t i o n o n l y o v e r a v e r y l i m i t e d r a n g e of s a l t c o n c e n t r a t i o n s .

F i g u r e 1

T y p i c a l l y t h e s e c o n d i t i o n s c a n b e m e t , f o r r e a s o n a b l e v a l u e s of Hamak- e r ' s c o n s t a n t (A52.5 X 1 0 - ~ 0 J o u l e ) and of t h e s u r f a c e p o t e n t i a l

( + o s 2 5 m ~ ) , i f t h e d i a m e t e r of t h e c o l l o i d a l p a r t i c l e s i s 0 0 , s I-r o r l a r g e r , a n d f o r s a l t c o n c e n t r a t i o n s o f t h e o r d e r o f 0.01M ( t h e n

X = K t > 100). The r e s u l t i n g l i q u i d g a s c o e x i s t e n c e and s p i n o d a l c u r v e s l o o k v e r y much l i k e t h o s e t y p i c a l o f s i m p l e m o l e c u l a r f l u i d s . Un-, d e r v e r y f a v o u r a b l e c o n d i t i o n s a g a s - l i q u i d - s o l i d t r i p l e p o i n t s h o u l d b e o b s e r v a b l e , b u t u n d e r t h e c o n d i t i o n s o f s t r o n g s c r e e n i n g n e c e s s a r y f o r s u c h a c o e x i s t e n c e , t h e r e p u l s i o n i s s u f f i c i e n t l y s t e e p t h a t a n FCC, r a t h e r t h a n a BCC c o l l o i d a l c r y s t a l would b e t h e more s t a b l e s t r u c - t u r e . I t i s s p e c u l a t e d t h a t w i t h i n c r e a s i n g s a l t c o n c e n t r a t i o n a t r a n s - i t i o n from BCC t o FCC s h o u l d b e o b s e r v a b l e i n t h e s o l i d p h a s e .

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