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HAL Id: jpa-00224633

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Submitted on 1 Jan 1985

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STRUCTURE OF LIQUIDS AND SOLIDS IN NONEQUILIBRIUM

S. Hess

To cite this version:

S. Hess. STRUCTURE OF LIQUIDS AND SOLIDS IN NONEQUILIBRIUM. Journal de Physique

Colloques, 1985, 46 (C3), pp.C3-191-C3-209. �10.1051/jphyscol:1985316�. �jpa-00224633�

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JOURNAL DE PHYSIQUE

Colloque C3, supplément au

n03,

Tome

46,

mars

1985

page C3-191

STRUCTURE OF LIQUIDS AND SOLIDS IN NONEQUI LIBRIUM

S . Hess

I n s t i t u t Laue Langevin, 156X, 38042 Grenob l e Cedex, France

and I n s t i t u t f ü r Theoretische Physik, Universitut Erlangen-Nümberg, 0-8520 Erlangen, F.R.G.

Résumé - Ce travail concerne l'anisotropie de la fonction de paire hors d'équilibre. On considère d'abord la distorsion de la structure d'un liquide soumis

à

un écoulement de Couette.

Ensuite, une analyse de la décroissance d'un état cristallisé instable et de la relaxation de l'ordre de liaison est effec- tuée.

Abstract - The theoretical description and some recent com- puter simulation results are presented for the anisotropy of the pair-correlation function in non-equilibrium situations.

Firstly, the shear-flow-induced distortion of the structure of a liquid subjected to a plane Couette flow is treated.

Secondly , the decay of an unstable crystalline state and the relaxation of the pertinent bond-orientational order are analyzed in the prefreezing regime.

INTRODUCTION

The structure of a simple liquid as expressed in terms of the pair-cor- relation function is isotropic in thermal equilibrium. A structural anisotropy can be induced by a transport process, e.g. by a viscous flow. An increasing local anisotropy as a precursor to a crystalline ordering is expected if one approaches the freezing transition.

In this talk, some theoretical ideas and recent computer simulation re- sults are presented firstly, for the structure of a fluid distorted by a plan Couette flow and, secondly, for the decay of an unstable crystal- line structure and the relaxation of the pertinent bond orientational order specifying the anisotropy of the first coordination shell.It should be stressed that the following considerations for simple fluids and solids, in many respects, also apply to model liquids and crystals composed of spherical colloidal particles.

1. SHEAR-FLOW-INDUCED DISTORTION OF THE STRUCTURE OF

A

FLUID

The pair-correlation function g(x), which is a measure for the proba- bility of finding a particle at position x if a reference particle is located at

= 0,

becomes anisotropic when a fluid is subjected to a shear flow. The mere existence of the (dominating) potential contribu- tion of the shear viscosity is an indirect evidence for this distortion 111. Direct experimental observations in a physical system have been made by light scattering techniques from colloidal model fluids 1 2 1 ,

for a theoretical treatment where the importance of non-linear effects has been pointed out, see ref

. ~ 3 ] .

The above-mentioned structural dis- tortion can also be studied in non-equilibrium molecular dynamics simu- lations

[4-61.

Recently, also terms nonlinear in the shear rate have been analysed c7-107 which are closely associated with non-Newtonian viscous behavior 110-14. A comparison between the simulation results and the light scattering data has been given in ref. L13J.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985316

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C3-192 JOURNAL DE PHYSIQUE

1.1 EXPANSION OF THE PAIR-CORRELATION FUNCTION, PLANE COUETTE SYM-

MET RY

The a n g u l a r dependence o f g ( f ) can be t a k e n i n t o a c c o u n t _ e x p l i c i t e l y by an e x p a n s i o n w i t h r e s p e c t t o s p h e r i c a l harmonics

g L r n ( r ) = p p m ( z ) g ( L ) d 2 i; (1.3)

n o t i c e t h a t

r

= r?,

r=lrl .

The " s c a l a r " p a r t gS ( o r i e n t a t i o n a l a v e r - age) and t h e c o e f f i c i e n t f u n c t i o n s gLm depend on t h e d i s t a n c e r between two p a r t i c l e s , due t o g ( r ) = g ( - r ) c o r r e s p o n d i n g t o an i n t e r c h a n g e of t h e two p a r t i c l e s , o n l y even r a n k t e n s o r s ( R=2,4,

....)

o c c u r i n ( 1 . 1 ) . I n s t e a d o f ( 1 . 1 ) , t h e e q u i v a l e n t e x p a n s i o n w i t h r e s p e c t t o i r r e d u c i b l e C a r t e s i a n t e n s o r s [13,14,1521 c o n s t r u c t e d from t h e components of

2

c a n

be u s e d .

The t e n s o r i a l b e h a v i o r o f g ( 2 ) i s somewhat s i m p l e r f o r s p e c i f i c geo- m e t r i e s . I n t h e f o l l o w i n g , a p l a n e C o u e t t e f l o w w i t h t h e f l o w v e l o - c i t y 1 i n t h e x - d i r e c t i o n and i t s g r a d i e n t i n t h e y - d i r e c t i o n i s con- s i d e r e d . The n o t a t i o n

~ V X

Y = a y

(1.4)

i s u s e d f o r t h e s h e a r r a t e .

I n t h i s c a s e , o n l y 3 of t h e 5 components of t h e 2nd r a n k c o e f f i c i e n t s of g ( r ) a r e n o n z e r o . Up t o R=2, t h e e x p a n s i o n f o r g ( r ) c a n be w r i t - t e n a s C3,7-91

1 2 2 2

g ( 5 ) = g, + g-w + g-?(ii -7 ) +gO(Z - 1 / 3 ) + .

. . .

( 1 .5) where X ,

y ,

? a r e t h e components of

F .

The g +

-

a r e l i n e a r combi- n a t i , o n s of g 2 + 2 m g o i s e s s e n t i a l l y g Z 0 of (1.1'). More s p e c i f i c a l l y ,

t h e f u n c t i o n s - g , , ( r ) , k:+,-,O of ( 1 . 5 ) a r e g i v e n by w i t h

f o r gs s e e ( 1 . 2 ) . The d o t s i n (1.5) s t a n d f o r t e n s o r s of r a n k s L = 4 , 6 , .

.

I n t h e r m a l e q u i l i b r i u m , g r e d u c e s t o t h e e q u i l i b r i u m 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 g and tRe gk ( a s w e l l a s a i l o t h e r h i g h e r r a n k t e n - s o r s ) v a n i s h . 1neq t h e l i n e a r f l o w regime where terms n o n l i n e a r i n t h e s h e a r r a t e a r e d i s r e g a r d e d , one s t i l l h a s g s w ge g- = g o b u t g+ = O . I t h a s been c o n v e n t i o n a l t o w r i t e [ l , 4-61 8:(r) =y" ( r ) i n t h a t c a s e .

1.2 KINETIC EQUATION, STOKES-MAXWELL RELATIONS

A r e l a t i o n between g+ and r g

,

( t h e prime d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o r ) h a s been t e s t e s q some time ago[4]. I t i s based on t h e i d e a proposed by Maxwell and S t o k e s t h a t t h e d i s t o r t i o n of t h e s t r u c - t u r e of a f l u i d s u b j e c t e d t o a s h e a r r a t e y e q u a l s t h a t of a d i s t o r - t i o n c a u s e d by a s h e a r d e f o r m a t i o n w i t h magnitude y ~ where T i s a s t r u c t u r a l r e l a x a t i o n t i m e . A g e n e r a l i z a t i o n of Maxwell-Stokes r e l a - t i o n s between t h e f u n c t i o n s g s ,

g+,

g - ? go and a f i r s t t e s t - ' a non- e q u i l i b r i u m m o l e c u l a r dynamics s i m u l a t i o n (108 p a r t i c l e s , r '%oten- t i a l ) have been p r e s e n t e d r e c e n t l y [9]. The b a s i c i d e a s u n d e r l y i n g t h i s d e r i v a t i o n and a f u r t h e r g e n e r a l i z a t i o n a r e g i v e n n e x t .

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P o i n t o f d e p a r t u r e i s k i n e t i c e q u a t i o n (Kirkwood-Smoluchowçki e q u a t i o n ) f o r t h e p a i r - c o r r e l a t i o n f u n c t i o n :

% +

YY& + R(g) = O (1.8)

where t h e 2nd term r e s u l t s from t h e p l a n e C o u e t t e f l o w and

R

i s a ( l i n e a r i z e d ) damping term w i t h t h e p r o p e r t y R (g ) = O i n o r d e r t o g u a r a n t e e t h a t g ( r ) r e l a x e s t o g i n t h e absenceeq o f a f l o w . F u n c t i o n - a l forms f o r

R

(g) a r e t h e ~ i r c h W 8 o d - ~ m o l u c h o w s k i C l ] e x p r e s s i o n o r i t s g e n e r a l i z a t i o n s n1,16] which a r e r a t h e r c o m p l i c a t e d . To p r o c e e d f u r - t h e r , t h e s i m p l e a p p r o x i m a t i o n

-

1

Q(gs) = T~ (gS-geq), ( 1 . 9 )

-

1

Q(gR. .) = T L g Q - - . (1.10)

i s made where g R . . . s t a n d s f o r components of a R-th r a n k t e n s o r , R = 2,4

...

The T

, are

r e l a x a t i o n t i m e s which, i n p r i n c i p l e , can be c a l c u l a t e d from R Oby a k i n d of a v a r i a t i o n a l a n s a t z r 3 3 .

Next, a remark on t h e flow term i s i n o r d e r . I t stems from a term of

t h e form v(r 1 a Z r . ( & ) . -

a

+ ~ ( L ~ I

."r; - -

-1 ar

-1 a 5 (1.11)

where i s r e c a l l e d a s t h e f l o w f i e l d assiimed t o be c h a r a c t e r i z e d by t h e c o n s t a n t v e l o c i t y g r a d i e n t t e n s o r

1

; one h a s

r

= El

-

r and t h e p a i r - c o r r e l a t i o n f u n c t i o n g i s assumed t o depend on

r

only-and n o t on

rl

+ r

.

For p l a n e C o u e t t e symmetry, (1.11) r e d u c e s t o t h e f l o w term o f - f 1 . 8 ) . I n g e n e r a l , c a n be decomposed i n t o i t s i r r e d u c i b l e p a r t s , i n v o l v i n g t h e d i v e r g e n c e o.^, t h e v o r t i c i t y 1

w

= 2 0 x x

(1.12)

and t h e symmetric t r a c e l e s s s h e a r r a t e t e n s o r

-

v = V v u (1.13)

The symbol C-+

. . . .

r e f e r s

- - -

a b ;O = - ( a b

tihë

2 - - 1 symmetric t r a c e l e s s p a r t of a t e n s o r , e . g .

+ k a )

- - a . b 3 1

6

(1.14)

f o r t h e d y a d i c c o n s t r u c t e d from t h e components of two v e c t o r s 2 and

b.

For a d i v e r g e n c e f r e e f l o w . 11.11) becomes

- .

.

r.

(1

y)-" ar =

- r: f.

~ - - - - f (1.15)

The d i f f e r e n t i a l o p e r a t o r s =

a

-

f = &

=

- a r

(1.16)

ar

-

a r e t h e g e n e r a t o r s of SU(3). N o t i c e t h a t

1

i n (1.11,15) r e f e r s t o a d i f f e r e n t i a t i o n i n r e a l s p a c e , whereas

tL

f s t h e d e r i v a t i v e w i t h r e s - p e c t t o t h e r e l a t i v e p o s i t i o n v e c t o r o c c u r r i n g i n g ( r ) . For t h e p l a n e C o u e t t e symmetry, t h e d e c o m p o s i t i o n of t h e v e l o c i t y g r a d i e n t f i e l d V

x

i n t o and

x

c o r r e s p o n d s t o w r i t i n g

1 x

a s

O10

y

( )

=

-

y;

[?;O)

+

(

1.0 (1.17)

000 O 0 0 O00

.

The f i r s t term on t h e r . h . s . of (1.17) i s t h e a n t i s y m m e t r i c t e n s o r a s - s o c i a t e d w i t h t h e a x i a l v e c t o r w

,

t h e second term r e p r e s e n t s

x .

The decomposition r e v e a l s two d i s t i n c t e f f e c t s c a u s e d by t h e flow: a p l a n a r b i a x i a l d e f o r m a t i o n due t o and a r o t a t i o n induced by t h e v o r t i c i t y g

.

There a r e a l s o m a t h e m a t i c a l d i f f e r e n c e s a s s o c i a t e d w i t h t h e s e t e r m s which have i m p o r t a n t p h y s i c a l consequences. The v e c t o r o p e r a t o r v a n i s h e s when a p p l i e d on a s p h e r i c a l symmetric f u n c t i o n ('%=O) and i t c o u p l e s components of t e n s o r s o f r a n k R w i t h t h o s e of t h e same r a n k ; t h e t e n s o r o p e r a t o r

4

c o u p l e s t e n s o r s of r a n k R w i t h t h o s e of r a n k R and R I Z .

I n s e r t i o n o f t h e t e n s o r i a l e x p a n s i o n o f g ( r ) i n t o t h e k i n e t i c e q u a t i o n (1.8) l e a d s t o coupled e q u a t i o n s f o r t h e e x p a n s i o n t e n s o r s . For t h e s p e c i a l c a s e ( 1 . 5 ) , one o b t a i n s , i n a s t a t i o n a r y s i t u a t i o n where t h e t i m e d e r i v a t i v e &i can be d i s r e g a r d e d ,

a t

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JOURNAL DE PHYSIQUE

with coefficients Bk which only depend on the shear rateybut not on r.

In lowest order in

y

they are given by B

=

YTO, B

= B =

B

=

yT

O 1 2 3 (1.22)

Relation (1.20) stems from the

g.

4 term, i:e. B2 vanishes for a vort- icity free flow field (e.g.in four roller geometry) . The other three relations (1.18, 19, 21) stem from

3 :

5 term.involving the deformation rate tensor y .

The dots in (1.19) and (1.21) stand for terms of higher order in

y

due to the

y :

4 inducéd coupling of the 2nd rank with the 4-th rank ex- pansion tenzors.

On the 4-th rank tensorial level, one finds a term with 2 ' ~ ~ symmetry coupled to g+. It leads to the additional relation

2 1

gq =

JTI~ B4(rg+) - 2g+) (1.23)

with

Here g4 is an expansion function of rank 4 with full cubic symmetry to be discussed later.

In the linear flow regime, (1.18-21, 1.23) reduce to gs=geq,g-=go=g4=0, and the Maxwell-Stokes relation

g+ = -YT rg

'

2 eq

(1.25)

1.3 SIMULATION OF SHEAR FLOW

In a nonequilibrium molecular dynamics simulation, the equations of motion of N particles are solved subject to certain constraints.

"Macroscopic" quantities of interest, e.g. the average energy, the kinetic and potential contributions of the pressure tensor, as well as the pair-correlation functions are extracted as N-particle averages involving the dynamic variables (positions, velocities, ....) of the

particles. Results are presented for ~ = 8 ~ = 5 1 2 particles interacting with the Lennard-Jones potential

@ = 4cLJ [(0/r)12

- (1.26)

or the repulsive r-lZpotential ("soft spheres")

@ =

~ ~ ~ ( s / r )

1 2

(1.27)

where

<s

and s,

E~~

and

E

are characteristic scales for the length and the energy. ~cale8~variables are used, e.g. for the Lennard- Jones system, the densities, temperatures, pressures and times are exaressed in units of

where m is the mass of a particle; kB is the Boltzmann constant. Shear

rares are given in units of (toa -7.

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F o r t h e s ~ f t s p h e r e s y s t e m , one h a s t h e a n a l o g o u s q u a n t i t i e s b u t E

and s r a t h e r t h a n E and a

.

To c o n v e r t from t h e s t a n d a r d s o f t S S s p h e r e s c a l e d v a r i a k g e s t o L e n n a r d - J o n e s s c a l e d v a r i a b l e s s u c h t h a t

(1.27) i s j u s t t h e r e p u l s i v e p a r t of (1.26)

,

one may c h o o s e E~~ -

- 4 E ~ ~

a n d a = S . T h i s means t h a t t h e r e d u c e d d e n s i t i e s a r e e q u a l , however, one h a s t o i n s e r t s i m p l e f a c t o r s i n t h e c o m p a r i s o n o f t e m p e r a t u r e s , p r e s s u r e s , t i m e s , and s h e a r r a t e s ; e . g . a s o f t s p h e r e r e d u c e d temper- a t u r e of 0.25 and a r e d u c e d s h e a r r a t e of 0.5 c o r r e s p o n d t o t h e r e - duced L e n n a r d - J o n e s t e m p e r a t u r e 1 . 0 and t h e s h e a r r a t e 1 . 0 . F o r con- v e n i e n c e , r e d u c e d v a r i a b l e s a r e d e n o t e d by t h e o r i g i n a l s y m b o l s , e . g .

i n s t e a d $ f 2 ( l L 8 5 , 2 6 ) t h e r ~ q y c e d i n t e r a c t i o n p o t e n t i a l s a r e w r i t t e n a s Q = 4 ( r -r ) and Q = r

,

r e s p e c t i v e l y . I n b o t h c a s e s , t h e i n t e r - a c t i o n s i s c u t o f f a t r = 2.5.

The e q u a t i o n s of m o t i o n a r e i n t e g r a t e d u s i n g a p r e d i c t o r - c o r r e c t o r meth- od ( 5 - t h o r d e r Gear method [17] which h a s b e e n t e s t e d b e f o r e 15-91 t o work e f f i c i e n t l y f o r n o n e q u i l i b r i u m p r o b l e m s . P e r i o d i c b o u n d a r y con- d i t i o n s a r e u s e d which t a k e i n t o a c c o u n t t h a t t h e p a r t i c l e s a r e s u b - j e c t e d t o a p l a n e C o u e t t e f l o w a s m e n t i o n e d b e f o r e w i t h a p r e s c r i b e d s h e a r r a t e y .The s h e a r f l o w , of c o u r s e , h a s a l s o t o b e t a k e n i n t o c o n s i d e r a t i o n when one c a l c u l a t e s t h e "images" of p a r t i c l e s i n t h e n e i g h b o r i n g p e r i o d i c i t y b o x e s . The s i z e o f t h e c u b i c p e r i o d i c i t y box i s d e t e r m i n e d by t h e number o f p a r t i c l e s N and t h e d e n s i t i y n . A l i - n e a r v e l o c i t y p r o f i l e i s m a i n t a i n e d by imposing t h e c o n s t r a i n t s [18]

( P , v = 1 , 2 , 3 ) N

C ri c i = O,

i-1 lJ ( 1 . 2 9 )

where

i . i

c = r,,

-

r y ( V v )

!J v P

i s t h e p e c u l i a r v e l o c i t y of p a r t i c l e "i" i n t h e p r e s e n c e of a l i n e a r f l o w v e l o c i t y f i e l d

x.

C a r t e s i a n components a r e d e n o t e d by Greek sub- s c r i p t s , t h e summation c o n v e n t i o n i s u s e d f o r them. The t e m p e r a t u r e T i s d e f i n e d by

T = ( 3 ~ ) - ' 1 c i c i (1 . 3 1 )

i

,,

P.

I t i s k e p t c o n s t a n t by r e s c a l i n g t h e m a g n i t u d e o f t h e

ci.

From t h e h e a t e x c h a n g e w i t h t h e " t h e r m o s t a t " one may i n f e r t h e e n t r o p y produc- t i o n due t o t h e v i s c o u s f l o w . The u s e of a t h e r m o s t a t i s e s s e n t i a l t o d i s t i n g u i s h t h e g e n u i n e n o n l i n e a r v i s c o u s b e h a v i o r from n o n l i n e a r i t i e s due t o v i s c o u s h e a t i n g .

Some t y p i c a l r e s u l t s a r e d i s c u s s e d n e x t . F i r s t l y , t h e s h e a r f l o w i n - duced d i s t o r t i o n o f t h e s t r u c t u r e of a L e n n a r d - J o n e s l i q u i d a t n = 0 . 7 0 , T=1.0 i s compared w i t h t h a t of a c o r r e s p o n d i n g r - 1 2 s o f t s p h e r e f l u i d

( t h e s o f t s p h e r e t e m p e r a t u r e i s 0 . 2 5 ) . N o t i c e , t h a t t h e Lennard-Jones t r i p l e p o i n t i s a t n = 0 . 8 4 , T=0.72, and t h e s o f t s p h e r e s y s t e m " f r e e z e s "

a t n=0.82. I n F i g . l a , t h e p a i r - c o r r e l a t i o n f u n c t i o n s gs+1/2g+ a r e p l o t t e d f o r t h e ( L e n n a r d - J o n e s ) s h e a r r a t e y = 0 . 2 5 . F o r s m a l l y r 2 , c f . ( 1 . 1 9 ) , t h e s e q u a n t i t i e s c h a r a c t e r i z e t h e s t r u c t u r e " i n t h e s h e a r p l a n e f o r d i r e c t i o n s e n c l o s i n g t h e a n g l e s 450 and 1 3 s 0 w i t h t h e f l o w v e l o c i t y .

A s e x p e c t e d , t h e f i r s t p e a k s o f t h e L e n n a r d - J o n e s c u r v e s a r e somewhat n a r r o w e r and c l o s e r t o t h e o r i g i n a s compared w i t h t h e c u r v e s c o r - r e s p o n d i n g t o t h e p u r e l y r e p u l s i v e r-12 p o t e n t i a l . I n F i g . l b , t h e p e r t i n e n t f u n c t i o n s g + a r e p l o t t e d . N o t i c e t h a t t h e o v e r a l l b e h a v i o r of t h e p a i r - c o r r e l a t i o n f u n c t i o n s i s r a t h e r s i m i l a r f o r b o t h p o t e n - t i a l s , however, g+ i s somewhat s m a l l e r f o r t h e Lennard-Jones c a s e . Ob- v i o u s l y , t h e a t t r a c t i o n between t h e p a r t i c l e s t e n d s t o d e c r e a s e t h e

s h e a r f l o w i n d u c e d d i s t o r t i o n o f t h e s t r u c t u r e of t h e f l u i d . T h i s i s a l s o r e f l e c t e d by t h e s t r u c t u r a l r e l a x a t i o n t i m e T~ a s i t i s i n f e r r e d

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C3-196 JOURNAL DE PHYSIQUE

from the shear viscosity and the shear modulus which have also been calculated in the simulation. One finds

T

* 0.075 and T2 0.12 for the Lennard-Jones and the soft sphere Sluid (inLJ-units). Clearly, in both cases T2Y is small such that the functions g-, go, g4 can be disregarded (linear flow regime).

Fig.1. The pair-correlation functions g 2 0.5g+as functions of r for the r-12 and Lennard-Jones potentialç. ~ï?e values for the temperature and the density are 1.0 and 0.70 respectively, the shear rate is y=0.25

(in LJ reduced units). In the linear f low regime g +O. 5 (squares) and gs-0. 5g+ (crosses) characterize the pair-correlation s '+ function for directions in the shear plane enclosing the angles 45' and 13S0 with the flow velocity. The two curves in the lower part of Fig.1 are the functions g+ for the r-12 and the Lennard-Jones potentials.

In Figs.2 and 3, the pair correlation functions

g

, g , g-, go., and g4

are displayed for the r-12 potential (T

=

0.25, nz 0.30); the shear

rates are y=1.0 and y=4.0. Effects nonlinear in the shear rate show

up clearly. Furthermore, the projections of the coordinates of the

particles on the yz, xy, and xz-planes are shown. Note, that these

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F i g . 2 . The p a i r - c o r ~ ~ $ a t i o n f u n c t i o n s g s , g,, g-, g o , and g a s f u n c - t i o n s o f r f o r t h e r p o t e n t i a l . The t e m p e r a t u r e and d e n s i ? y a r e T=0.25 and n = 0 . 7 0 , t h e s h e a r r a t e i s y = 1 . 0 ( i n s o f t s p h e r e u n i t s ) . T h e t h r e e s q u a r e s on t h e r . h . s . o f F i g . 2 show t h e p r o j e c t i o n s o f t h e c o o r - d i n a t e s o f t h e p a r t i c l e s o n t 0 t h e z y , y x , a n d z x - p l a n e s .

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C3-198 JOURNAL DE PHYSIQUE

-. .

-8- -, r. : :

.

: *$,,>

,",

$'.;.*..;,.$'.";;

. ...

"

.'

-.- -. .

$ . . . h 0 7 * $ * - *1-.* s;..

a -

. :..

r...--S.&-.

.

'

..

*'/

-

$. A..'.. ;.*

'..;.

<S..&

.: .,./: ' -

*a ::;.11 ,A . ! : -,* O:*.

...,

:.$,3 r.

.

8

I>,;f

' = :;

::*; *h--*:

.'

8

$

.

i',;.$

;!

. .. .

f $. -*2 A.. i

. . / $ S .

: .

.*

.. . -.

a'

-

.

.

, S m ..# .$;.

- - .

1.*.

. .. - ". ,

:

*.* .*:.. - ,

F i g . 3 . Same a s F i g . 2 e x c e p t t h a t t h e s h e a r r a t e i s y=4.0.

(10)

a r e " s n a p - s h o t p i c t u r e s " whereas t h e p a i r - c o r r e l a t i o n f u n c t i o n s a r e a v e r a g e s o v e r a l 1 p a i r s of t h e system and o v e r s e v e r a l thousand time s t e p s ( t y p i c a l l y 4000-8000).

S e v e r a l remarks a r e i n o r d e r . F i r s t l y , t h e extrema of t h e d i r e c t i o n a l a v e r a g e g o f g ( 9 d e c r e a s e w i t h i n c r e a s i n g s h e a r r a t e , e . g . t h e f i r s t maximum O ? gs e q u a l s " 2 . 6 , 2 . 3 , and 1.9 f o r y =0.25, 1 . 0 , and 4 . 0 , r e s p e c t i v e l y . The f u n c t i o n g + , r e p r e s e n t a t i v e of t h e s h e a r f l o w i n - duced d i s t o r t i o n i n t h e l i n e a r f l o w regime, i n c r e a s e s w i t h i n c r e a s i n g s h e a r r a t e f o r y 5 2 , b u t it d e c r e a s e s a t h i g h e r s h e a r r a t e s ; e . g . t h e f i r s t minima and maxima of g+ a r e -3 .O4 and 1 . 2 1 f o r y =1 b u t -2.94 and 0.84 f o r Y = 4 . 0 .

The f u n c t i o n s g - , g o , g which a r e c h a r a c t e r i s t i c f o r t h e n o n l i n e a r f l o w regime, s t i l l i n c r e a s e from y = 1 .O t o y=4.0. These f e a t u r e s c a n be u n d e r - s t o o d on t h e b a s i s of t h e t h e o r y p r e s e n t e d i n t h e p r e v i o u s s e c t i o n . Before a d i s c u s s i o n of t h e Stokes-Maxwell r e l a t i o n s (1.19

-

2 1 , 1.23) i s g i v e n , t h e p r o j e c t i o n s of t h e c o o r d i n a t e s on t h e w a l l s of t h e "box"

a r e c o n s i d e r e d . For y = 4 . 0 , t h e p r o j e c t i o n on t h e zy-plane ( t h e flow v e l o c i t y i s r e c a l l e d t o be i n t h e x - d i r e c t i o n ) r e v e a l s a s u r p r i s i n g r e g u l a r i t y a l o n g t h e y - d i r e c t i o n , i . e . a l o n g t h e d i r e c t i o n of t h e g r a - d i e n t of t h e f l o w v e l o c i t y . A t a second g l a n c e , one a l s o d i s c o v e r s a r e g u l a r i t y a l o n g t h e y - d i r e c t i o n i n t h e p r o j e c t i o n u n t o t h e x y - p l a n e

( s h e a r p l a n e ) . A t h i g h s h e a r r a t e s , t h e p a r t i c l e s o b v i o u s l y t e n d t o o r d e r i n p l a n e s p a r a l l e l t o t h e Stream l i n e s . The s p a t i a l d i s t r i b u t i o n w i t h i n t h e s h e a r p l a n e a s s e e n i n t h e p r o j e c t i o n o n t 0 t h e x z - p l a n e

seems t o be r a t h e r random. A t s t i l l h i g h e r s h e a r r a t e s , r u n s were made f o r y = 8 . 0 , t h e o r d e r i n g i n p l a n e s becomes even more pronounced. How- e v e r , f o r y 2 2.0, t h e r e i s a l s o a s t r o n g a n i s o t r o p y i n v e l o c i t y s p a c e , v i z . t h e y and z components ( p e r p e n d i c u l a r t o t h e f l o w d i r e c t i o n ) o f t h e t h e r m a l v e l o c i t i e s a r e r e d u c e d a n a l o g o u s l y t o t h e c o o l i n g of t h e t r a n s v e r s e t h e r m a l m o t i o n s i n a n e x p a n s i o n f l o w .

I n F i g . 4 , t h e Stokes-Maxwell r e l a t i o n s (1.19-21,1,23) a r e t e s t e d f o r t h e r - 1 2 p o t e n t i a l , T=0.25, n=0.70 and y = 1 . 0 . The f u n c t i o n s g + , g-, g o , g 4 a s d e t e r m i n e d by t h e s i m u l a t i o n a r e shown i n t h e ~ . h . s . , th e f u n c t i o n s o b t a i n e d by d i f f e r e n t i a t i n g gs and g+ a c c o r d i n g t o t h e S t o k e s - Maxwell r e l a t i o n s a r e shown on t h e r . h . s . More s p e c i f i c a l l y , i n t h e

t o p g r a p h s , g+ i s compared w i t h - r g l , . The s i m i l a r i t y i s s t r i k i n g , a l - though t h e peaks of t h e g+ c u r v e a r e somewhat w i d e r and s l i g h t l y s h i f - t e d t o l a r g e r r - v a l u e a s compared w i t h t h e c u r v e o b t a i n e d by d i f f e r e n t - i a t i o n . Comparison of t h e d e p t h and h e i g h t o f t h e f i r s t minima and maxima y i e l d s t h e v a l u e s B1

"

0.17 and B = 0.14, r e s p e c t i v e l y , f o r t h e p r o p o r t i o n a l i t y f a c t o r o c c u r r i n g i n

($.

1 9 ) . These v a l u e s compare q u i t e f a v o r a b l y w i t h t h e v a l u e B

-

0.15 i n f e r r e d from t h e r e l a t i o n

y n + = GB1 which f o l l o w s from ( 1 9 ; G i s t h e s h e a r modulus, n + [ 7 , 9 ,

1 2 1

i s t h e non-Newtonian s h e a r v i s c o s i t y . P r a c t i c a l l y t h e same v a l u e f o r B1 h a s been found i n a s i m u l a t i o n w i t h 108 p a r t i c l e s g] The g r a p h i n t h e 2nd row of F i g . 4 shows g-

.

According t o

b.

Z O j

,

i t

s h o u l d be p r o p o r t i o n a l t o g+ which i s d i s p l a y e d above. The agreement i s r e a s o n a b l y s t a t i s f a c t o r y , t h e v a l u e of B 2 i n f e r r e d from a c o m p a r i s o j of t h e f i r s t minima and maxima a r e Bsz 0 . 1 ; t h u s somewhat s m a l l e r t h a n B

.

I n t h e t h i r d row of F i g . 4 , go i s compared w i t h 1 . 5 g+ + rg;, c f .

(1.21) ; one f i n d s B z 0.11. The f u n c t i o n s g- and go which a r e t y p i c a l f o r t h e n o n l i n e a r f?ow regime q u a l i t a t i v e l y behave a s e x p e c t e d on t h e b a s i s o f t h e k i n e t i c e q u a t i o n ( 1 . 8 ) , however, i t seems t h a t t h e v o r - t i c i t y w and t h e d e f o r m a t i o n r a t e t e n s o r y, c f . ( 1 . 1 5 ) a r e somewhat reduced f o r s m a l l d i s t a n c e s . T h i s means,

i n

t h e k i n e t i c e q u a t i o n f o r g @ ) ,

2

and

y

s h o u l d be r e p l a c e d by " s c r e e n e d " v a l u e s .

F i n a l l y , i n t h e l a s t row of F i g . 4 , t h e " c u b i c t l p a i r c o r r e l a t i o n f u n c - tien g4 i s compared w i t h rg:

-

2 g + , c f . ( 1 . 2 3 ) . From a comparison of

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JOURNAL DE PHYSIQUE

10 4

U.827

% I R C a a E U T I C N FCT 1 . W 2 . 4 9 1

( I . S + r - l G + d R ci r

Fig.?. Test of the Stokes-Maxwell relations. The curves on theR.h.s.

of Fig.4 are the pair-correlation functions extracted from the simu- lation; on the r.h.s. are the pertinent functions obtained from gsand g+ according to (1.19-21) and (1.23)

.

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the maxima one infers B

-

0.22. This indicates that the relaxation time r4 is larger than 'r

,

cf. (1.22,24). Furthermore, notice that the "measured" g4- curve does not show, for small r, the deep minum of the curve displaced on the r.h.s. This may be associated to a cubic (fcc) ordering at small distances.

2. DECAY OF A CRYSTALLINE STRUCTURE, RELAXATION OF BOND ORIENTATIONAL ORDER , The anisotropy of the first few coordination shells is one of the most striking features distinguishing, on a microscopic level, a crystalline solid from a liquid. Thus it is suggestive to introduce tensorial ord- er parameters associated with this anisotropy in a dynamic Landau theo- ry p9J or a mean field theory c20l for the liquid-solid phase trans- ition. There is an analogy with the isotropic-nematic phase trans- ition where the ordering of tJhe figure axis of the molecules is des- cribed by a 2nd rank tensor L21-231 ; here order parameters of the rank 4 and 6 are needed to specify the cubic anisotropy of the "bond"

directions in the first coordination shell of a system cornposed of spherical particles.

2.1. CUBIC PAIR-CORRELATION FUNCTIONS, BOND ORIENTATIONAL ORDER PARA- METER

For a cubic system which is the simplest crystalline structure, the lowest nonisotropic terms in the expansion of the pair-correlation func- tion g(r) with respect to spherical (cf.(l.l)) or Cartesian tensors are of rank 4; tensors of rank 6 will also be considered. If the coordi- nate axes are chosen parallel to the (local) cubic axes, the expansion of the pair-correlation function can be written as

g ( ~ ) = g s ( r ) + g 4 ( r ) K 4 ( E ) + g 6 ( r ) K 6 ( P ) +

.

(2.1) As before, gs is the scalar (isotropic) part of g(r). The

K

are fully symmetric (normalized) cubic harmonics 1241

,

viz. : 4,6

Again, the components of the unit vector

2

are denoted by 2 ,

Y ,

?.The expansion functions g (r), g. = 4,6,

...,

referred to as cubic pair-cor- relations, are given

hy

g,(r) = ( 4 ~ ) - ' ( ~ ~ ( g g ( ~ ) d 2 i. (2.4)

The quantity g4 has already been uséd in (1.23).

A

distribution function f(S) describing the anisotropy of the first CO

ordination shell with radius RI can be defined by

Y, = 4 n / R l g s ( r > r d r . 2

The product of the volume VI with the number density n yields the num- ber NI of particles in this shell. Similar to g(r), f(2) can be ex- pan ed with respect to spherical or normalized Cartesian tensors

of rank !L depending on

2

[14,15]

+ W . .

. . .

f ( 2 )

= (4~)-l (2.7)

The expansion coefficient tensors are given by

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C3-202 J O U R N A L DE PHYSIQUE

A s b e f o r e , Greek s u b s c r i p t s r e f e r t o C a r t e s i a n components, t h e sum- m a t i o n c o n v e n t i o n i s u s e d . For a c u b i c s y s t e m , t h e l o w e s t o r d e r p a r a - m e t e r t e n s o r i f of r a n k 4 ; i t can b e w r i t t e n a s Li91

where ( 4 )

_

3 e i i i i 1

e e e - - ( 6 6 +6 6 6 )

H p v ~ ~

;=,

p v K ~ 5 p v ~ x v ? v + ~ ~ . ~ K v x ( 2 . 1 0 ) i s t h e c u b i c " d i r e c t o r " ' t e n s o r c o n s t r u c t e d fjom 3he components of t h e t h r e e m u t u a l l y o r t h o g o n a l u n i t v e c t o r s

c, ,

2 which s p e c i f y t h e l o c a l c u b i c a x e s . The q u a n t i t y

a4 = j ~ , ( ~ ) r ( ~ ) d ~ î (2.11)

i s a c u b i c o r d e r p a r a m e t e r a n a l o g o u s t o t h e Maier-Saupe o r d e r p a r a - m e t e r of n e m a t i c s . For f u l l c u b i c symmetry, a l s o a d i r e c t o r t e n s o r o f

r a n k 6 c a n be i n t r o d u c e d i n a n a l o g y t o ( 2 . 9 , 1 0 ) . Then t h e expansion ( 2 . 7 ) r e d u c e s t o

f

(z)

= ( 4 ~ ) -1 ( 1 + a4K4 ( i ) +a6K6

(2) +. . .

) ; (2.121

a i s def i n e d a n a l o g o u s l y t o (2 . I l ) . The ag of (2.12) a r e r e l a t e d t o

tRe g, of ( 2 . 1 ) by R

a, = V;'~nl ' g I ( r ) r 2 d r , (2.13)

f o r V, s e e ( 2 . 6 ) . O

The 4*h r a n k ' a n i s o t r o p y t e n s o r (2.9) h a s been u s e d i n a dynamic Ginz- burg-Landau t h e o r y by t h e p r e s e n t a u t h o r [19] a n d , i n d e p e n d e n t l y , by Mitus and P a t a s h i n s k i i [20] i n a mean f i e l d t h e o r y . Nelson and Toner

[25] r e f e r r e d t o t h e c o r r e s p o n d i n g o r d e r e d p h a s e a s " c u b a t i c " s i n c e t h e b r e a k i n g of t h e t r a n s l a t i o n a l symmetry h a s n o t been t a k e n i n t o a c c o u n t e x p l i c i t e l y .

2.2 DYNAMIC GINZBURG-LANDAU THEORY

Let au,

-....

be a c u b i c o r d e r p a r a m e t e r t e n s o r ( o f r a n k 4 o r 6 ) d i s - t i n g u i s h i n g t h e o r d e r e d p h a s e from t h e i s o t r o p i c l i q u i d . An e q u a t i o n g o v e r n i n g i t s time e v o l u t i o n can be o b t a i n e d by a combination of a Landau t h e o r y w i t h t h e i d e a s of i r r e v e r s i b l e thermodynamics a s formu- l a t e d i n r e f . Cl91

.

I n t h i s a p p r o a c h , t h e t i m e change o f a . . i s s e t up s u c h t h a t t h e e n t r o p y p r o d u c t i o n a s s o c i a t e d w i t h i t i s p o s i t i v e . T h i s r e q u i r e s an assumption on t h e dependence of l o c a l thermodynamics f u n c t i o n s on t h e o r d e r p a r a m e t e r . I n p a r t i c u l a r , one assumes t h a t t h e s p e c i f i c - f n t r o p y s , t h e s p e c i f i c i n t e r n a 1 e n e r g y u , and t h e s p e c i f i c v o l u m e g ( f i s t h e mass d e n s i t y ) c a n be w r i t t e n a s

S = S + s a , u = U, + u

7

- 1 = y , ' + (2.14)

where t h e q u a n t i t i e s w i t h t h e s u b s c r i p t "a" v a n i s h f o r a . . . = O . The p a r t s of t h e thermodynamic f u n c t i o n s l a b e l l e d by t h e s u b s c r i p t "O", which a r e i n d e p e n d e n t o f t h e a n i s o t r o p y , a r e assumed t o obey t h e u s u a l Gibbs r e l a t i o n -

d s o = T-' ( d u o + P d p - 1 )

-

Use o f (2. 1 4 , 1 5 ) l e a d s t o

-1

X d a

d r = T - ' ( d u + Pdq )+-

m aa.. (2.16)

w i t h t h e thermodynamic p o t e n t i a l

SI

g i v e n by - - i l ; = ç kB - T - 1

m - a (ua +

~ q ) .

(2.17)

The b a s i c assumption made i n t h e f o l l o w i n g i s t h a t (2.17) i s a l s o va l i d i n a n o n e q u i l i b r i u m s i t u a t i o n .

(14)

With the special assumptions

1 2 1 2

u = -

-€(a...)

,

2

7;

=

-;

vaCa

...

) ,

(2.18)

I I

S a

-

+

y

C ( a . . )

L

one obtains the Landau potential

2 1 1

$L =

! A(T,P)(a..) -

7 B(i. .))

- - $ ( a .

.) 4

2 (2.20)

with

A = A [ 1 -T-* 2-

O A&B (c+Pv,) 1 = A,[ 1

-Tx/T] (2.21)

In (2.18)

E

and v are the characteristic energy and the volume as- sociated with thea anisotropy a... The coefficients Ao, B, C of the anisotropy entropy (2.19) are assumed to depend weakly on the temper- ature T and the pressure P. The pseudo critical temperature T is pro- portional to

E

+Pva. For a spatially inhomogeneous system, the simp- les expression for

$ -

occurring in (42) is

1 2

2

9

=

gL

+

so(xa..) (2.22)

where t0 is a (bare) correlation length and *Lis given by (2.20).

For this case, an ansatz which guarantees that the entropy production associated with changes of the anisotropy tensor a... is

aa..

- =

-

-1

ag

a

t 'la

a=. .

* 7 > O

with a (bare) relaxation time coefficient

7

which is assumed to be a weakly dependent function of T and P. Use a of (2.23) with (2.20, 22)

leads to the nonlinear relaxation equation

a -

1 2

a..

+ T A T P . .

- B . .

+ ( a 3

-

= O.

(2.24) This equation is very similar to that one governing the 2nd rank align- ment tensor of a liquid crystal which is the basis of a unified theory for the isotropic and nematic phases [23] .

Due to (2.23), the stationary, spatially homogeneous solutions of (2.24) are equivalent to those obtained by a Landau theory approach: "local"

stability requires -

> o.

In particular, one finds a.. .

=

O and

a . . . # O for T

<

T and T

>

T where the transition temper- ature T > T * is desermined by S

S (2.25)

Here, we are concerned with the pretransitional behavior in the un- ordered phase as it follows from (2.24) in linear approximation, viz:

- a ,, a..

+ t-' (a..-~~~a. .)

-

O

(2.26)

where the relaxation time

T

and the correlation 5 are given by

T

-

T~(I-T*/T)-',

c2 - ci(l-~~/~)-'. (2.27) In the following, a cornputer simulation is discussed which indeed shows a pretransitional slowing down of the relaxation of the local ani-

sotropy, however, the density n rather than the temperature T is varied.

2.3 COMPUTER SIMULATION FOR A STRUCTURAL DECAY AND RELAXATION OF THE CUBIC ORDER PARAMETERS

The simulations to be repox-f5d now have been performed for 1024 parti-

cles interacting with an r potential, the temperature was T

=

0.25

in al1 cases. For densities (n=0.60 to n=0.78) where the equilibrium

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JOURNAL DE PHYSIQUE

F i g . 5 . The p a i r - c o r r e l a t i o n f u n c t i o n s g s , g 4 , g 6 ( d e n o t e d by GO, G4, G6 i n t h e g r a p h s ) f o r t h e r - 1 2 p o t e n t i a l , t h e tempe a t u r e T = 0 . 2 5 and t h e d e n s i t y n = O . l l a s f u n c t i o n s o f t h e time t and r' ( r u n n i n g a s i n - d i c a t e d by t h e a r r o w s ) . The t v a l u e s a r e from O t o 2.9, t h e d i s t a n c e r r a n g e s from 0.8 t o 2.5. The g r a p h s on t h e l e f t and r i g h t hand s i d e s of F i g . 5 show t h e same d a t a s e e n from d i f f e r e n t s i d e s : t h e s p i k e s a t s m a l l t mark t h e f i r s t 5 c o o r d i n a t i o n s h e l l s .

(16)

s t a t e i s known t o b e t h e f l u i d p h a s e , t h e p a r t i c l e s were p l a c e d on b c c l a t t i c e s i t e s . They s t a r t e d o f f w i t h random v e l o c i t i e s c o r r e s p o n d i n g t o t h e p r e s c r i b e d t e m p e r a t u r e , t h e t h e r m o s t a t i n g was done a s d e s c r i b e d b e f o r e . A g r e a t number of m a c r o s c o p i c a v e r a g e s ( e n e r g y , k i n e t i c and p o t e n t i a l p r e s s u r e t e n s o r s , e l a s t i c i t y c o e f f i c i e n t s , ...) i n c l u d i n g t h e bond o r i e n t a t i o n a l a n i s o t r o p y o r d e r p a r a m e t e r s and t h e c u b i c p a i r - c o r - r e l a t i o n f u n c t i o n m e n t i o n e d p r e v i o u s l y have b e e n r e c o r d e d a s f u n c t i o n s of t h e t i m e ( p r e a v e r a g e d o v e r 20 t i m e s t e p s of l e n g t h 0 . 0 0 5 ) .

A s a t y p i c a l example, i n F i g . 5 , t h e s c a l a r p a i r - c o r r e l a t i o n f u n c t i o n g and t h e c u b i c p a i r - c o r r e l a t i o n f u n c t i o n s g 4 and g 6 ( d e n o t e d by G O ,

~ 2 ,

G6 i n t h e g r a p h s ) a r e shown a s f u n c t i o n s of r 2 and t . The r a n g e of t h e r - v a l u e s i s from 0.8 t o 2 . 5 , t r a n g e s from O t o 2.9. The g r a p h s on t h e l e f t and r i g h t hand s i d e of F i g . 5 r e p r e s e n t t h e same d a t a view- e d from d i f f e r e n t s i d e s . The t e m p e r a t u r e and d e n s i t y a r e T=0.25 and n = 0 . 7 4 . The s p i k e s f o r s h o r t t i m e s mark t h e f i r s t 5 c o o r d i n a t i o n s h e l l s o f t h e b c c s t r u c t u r e . F o r l a r g e t i m e s , g s r e d u c e s t o t h e l i q u i 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 , g 4 and g v a n i s h . A 2nd g l a n c e a t F i g . 5 shows t h a t t h e r e i s a r e l a t i v e l y f a s t d e c a y a t s h o r t t i m e s f o l l o w e d by a s l o w e r r e l a x a t i o n p r o c e s s f o r t i m e s to>. 0 . 8 . N o t i c e , t h a t g 4 and g6 h a v e n o t d e c a y e d f u l l y f o r t = 2 . 9 a t t h e d e n s i t y n = 0 . 7 4 ; f o r s m a l l e r d e n s i t i e s t h e d e c a y i s p r a c t i c a l l y c o m p l e t e d a t t h i s t i m e . These f e a - t u r e s a r e more pronounced a t s m a l l d i s t a n c e s and a r e r e f l e c t e d i n t h e c u b i c o r d e r p a r a m e t e r s which a r e e s s e n t i a l l y i n t e g r a l s o f t h e c u b i c p a i r - c o r r e l a t i o n f u n c t i o n s o v e r t h e f i r s t c o o r d i n a t i o n s h e l l , c f . ( 2 . 1 3 ) . The r a d i u s o f t h i s s h e l l was c h o s e n a s R1 = 1 . 2 1 1 3 . For t h e d e n ç i - t i e s s t u d i e d , i n t h e f l u i d p h a s e , t h e number o f p a r t i c l e s i n t h e s h e l l d o e s n o t d e v i a t e from i t s i n i t i a l v a l u e 8 by more t h a n 3 % . I n F i g . 6 a , t h e c u b i c o r d e r p a r a m e t e r t i m e - c o r r e l a t i o n f u n c t i o n s C and C 6 a r e d i s - p l a y e d f o r t h e d e n s i t i e s n = 0 . 7 0 and 0 . 7 6 . These q u a n t i t i e s a r e d e f i n e d

by a R ( t ) = CR(t)aR(o)

( 2 . 2 8 ) where t h e a R a r e r e c a l l e d a s t h e c u b i c o r d e r p a r a m e t e r s , a R ( 0 ) a r e t h e i n i t i a l l a t t i c e v a l u e . A g a i n , n o t i c e t h e c r o s s o v e r froin a f a s t r e l a x a t i o n p r o c e s s t o a s l o w e r one a t t z 0.8. A t t h i s t i m e t h e l * s c a l a r "

q u a n t i t i e s l i k e t h e p o t e n t i a l c o n t r i b u t i o n s t o t h e p r e s s u r e P and t h e i n t e r n a 1 e n e r g y U ( p e r p a r t i c l e ) shown i n F i g . 6 b p r a c t i c a l l y have a l - r e a d y r e a c h e d t h e i r " f l u i d " v a l u e s . I n F i g . 7 , RnC4 and RnC a r e p l o t - t e d as f u n c t i o n s o f t h e t i m e t f o r t h e d e n s i t i e s n=0.60,0.70,0.74,076,078.

N o t i c e t h e s l o w i n g down o f t h e 2nd r e l a x a t i o n p r o c e s s w i t h i n c r e a s i n g d e n s i t y ; t h e f r e e z i n g t r a n s i t i o n i n t o a f c c s t a t e i s e x p e c t e d f o r n"0.82.

The v a r i o u s r e l a x a t i o n r e g i m e s c a n be u n d e r s t o o d a s f o l l o w s . F o r v e r y s h o r t t i m e s , t 5 0 . 1 , C ( t ) d e c r e a s e s p r o p o r t i o n a l t o - t 2 due t o t h e f r e e - f l i g h t o f p a r t i c f e s . F o r t r 0 . 1 t i l l t r 0 . 8 which c o r r e s p o n d s t o 2nwE

-

where a~ i s t h e E i n s t e i n f r e q u e n c y , one o b s e r v e s t h e f a s t d e c a y a s s o c i a t e d w i t h f i l l i n g t h e a v a i l a b l e f r e e volume. For n 2 0 . 7 0 , c o l - l e c t i v e p r o c e s s e s d e t e r m i n e t h e d e c a y o f t h e c u b i c p a i r - c o r r e l a t i o n f u n c t i o n s and o r d e r p a r a m e t e r s f o r t

>

0 . 8 . The p r o c e s s i s s l o w e r f o r C6 t h a n C

.

T h i s i s due t o t h e f a c t t h a t g 4 h a s o p p o s i t e s i q n s i n t h e f i r s t fwo c o o r d i n a t i o n s h e l l s w h e r e a s g6 i s p o s i t i v e i n b o t h . Hence a r e l a t i v e r a d i a l m o t i o n o f t h e p a r t i c l e s l e a d s t o a p a r t i a l c a n c e l l a t i o n o f g and c o n s e q u e n t l y o f C 4 w h e r e a s C6 d e c a y s m a i n l y due t o " r e o r i e n t a ? i o n p r o c e s s e s .

The e f f e c t i v e r e l a x a t i o n t i m e T of C6 i n t h i s t i m e r a n g e i s compa- t i b l e w i t h a p r e t r a n s i t i o n a l b e k a v i o r o f t h e form ( n > 0 . 7 0 )

T6 = ~ ~-n/nx) ( 1-Y ( 2 . 2 9 )

w i t h t h e r e f e r e n c e r e l a x a t i o n t i m e T~ p r a c t i c a l l y i n d e p e n d e n t of n ,

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C3-206 JOURNAL DE PHYSIQUE

t h e r e f e r e n c e number d e n s i t y n x ?- 0.8 and t h e p s e u d o - c r i t i c a l exponent y w i t h l < y < 2 . N o t i c e t h a t t h e Landau t h e o r y ( w i t h t h e v a r i a b l e n r a t h e r t h a n T) y i e l d s y = l , c f . ( 2 . 2 7 ) . I t s h o u l d be mentioned t h a t , a t n = 0 . 7 0 , T~ i s a b o u t 6 t i m e s l a r g e r t h a n t h e Maxwell r e l a x a t i o n t i m e

T i n f e r r e d from t h e v i s c o s i t y Q and t h e s h e a r modulus G a c c o r d i n g t a ~ = G T

.

From n=0.70 t o n = 7 8 , ~ i n c r e a s e s by a f a c t o r o f a b o u t 20.

c l e a r l y ? t h e decay o f t h e c u b i c g r d e r p a r a m e t e r s shows d r a m a t i c p r e - t r a n s i t i o n a l s l o w i n g down. So f a r , t h i s phenomenon h a s been demons- t r a t e d f o r one p a r t i c u l a r i n t e r a c t i o n p o t e n t i a l and t h e decay of one s p e c i f i c c r y s t a l l i n e s t r u c t u r e ; e x t e n s i o n s t o o t h e r i n t e r a c t i o n po- t e n t i a l s and o t h e r ( c u b i c ) s t r u c t u r e s a r e s t r a i g h t f o r w a r d . F u r t h e r - more, t h e r e l a x a t i o n of t h e c u b i c o r d e r p a r a m e t e r s c a n a l s o be probed by t u r n i n g on and s w i t c h i n g o f f o r i e n t i n g f i e l d s w i t h t h e d e s i r e d cu- b i c symmetry. These s i m u l a t i o n e x p e r i m e n t s y i e l d i n f o r m a t i o n on t h e f r e e z i n g r o c e s s which i s complementary t o t h a t o b t a i n e d from quench s t u d i e s $6,271

.

F i g . 6 . The c u b i c o r d e r p a r a m e t e r time c o r r e l a t i o n f u n c t i o n s C 4 , C g , and t h e p o t e n t i a l c o n t r i b u t i o n s t o t h e p r e s s u r e P and t h e a v e r a g e e n e r g y U ( p e r p a r t i c l e ) a s f u n c t i o n s of t h e t i m e t f o r t h e d e n s i t i e s n=0.70 and n=0.76. The d a t a a r e f o r t h e r - 1 2 i n t e r a c t i o n p o t e n t i a l , t h e t e m p e r a t u r e i s 0.25.

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Fig.7. The quantities RnC4 and RnC6 as functions of the time

t

for the r-l2 interaction, the temperature 0.25 and the densities n=0.60, 0.70, 0.74, 0.76, 0.78.

Two questions require further investigations. Firstly, is there a finite range of temperatures and densities where a cubic bond orient- ational phase, as suggested in Ref.1251 exists between the fluid and a true solid phase with long range positional ordering

?

Secondly, is there a metastable bcc phase intermediate the flui? and the fcc phase as suggested by the arguments put forward in Ref.

i

282 even for poten- tials where the calculations of Ref.u9lindicate a-direct fluid-fcc transition

?

CONCLUDING

REMARKS

In this article, theoretical ideas and computer simulation results have

been reported for the anisotropy of the pair-correlation function in

two types of nonequilibrium situations viz. a steady shear flow and the

decay of

an

unstable crystalline structure. In a certain sense, shear-

induced melting combines these two phenomena. A nonequilibrium mole-

cular dynamics simulation has recently been made [30]

;

an analysis of

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C3-208 JOURNAL DE PHYSIQUE

changes i n t h e s t r u c t u r e as f u n c t i o n of t h e a p p l i e d s h e a r r a t e a l o n g t h e l i n e s i n d i c a t e d h e r e promises some f u r t h e r i n s i g h t i n t o t h e u n d e r - l y i n g p r o c e s s e s . I n p a r t i c u l a r , t h e f i n d i n g s d i s p l a y e d i n Fig.4 i n - d i c a t e t h a t one may e x p e c t a r e e n t r a n t o r d e r e d ( l a y e r e d ) s t r u c t u r e a t h i g h s h e a r r a t e . T h e o r e t i c a l work on t h e i n f l u e n c e of a C o u e t t e s h e a r

flow on f r e e z i n g and m e l t i n g çomewhat a k i n t o t h e t h e o r y of t h e i n - f l u e n c e o f a s h e a r f l o w on t h e i s o t r o p i c - n e m a t i c p h a s e t r a n s i t i o n p r e - s e n t e d i n Ref. 1311 i s i n p r o g r e s s

F2J.

With due m o d i f i c a t i o n s , c f . Ref. b 3 1

,

t h e o r e t i c a l and computer simu- l a t i o n r e s u l t s o b t a i n e d f o r s i m p l e l i q u i d s and s o l i d s c a n be a p p l i e d t o c o l l o i d a l l i q u i d s and c r y s t a l s where an i n c r e a s i n g number o f i n t e r - e s t i n g e x p e r i m e n t a l - r e s u l t s (on n o n e q u i l i b r i u m b e h a v i o r ) a r e becoming a v a i l a b l e 12, 34-361

.

ACKNOWLEDGEMENTS

1 t h a n k P. N o z i è r e s and t h e members o f t h e t h e o r y c o l l e g e o f t h e ILL f o r t h e h o s p i t a l i t y shown t o me. Many t h a n k s a r e due t o Howard Hanley, N.B.S., B o u l d e r , f o r having c o n v i n c e d me of t h e g r e a t p o t e n t i a l and of t h e v i r t u e s - ' o f n o n e q u i l i b r i u m m o l e c u l a r dynamics. The s e r v i c e s of t h e computer c e n t r e of t h e I.L.L. which were e s s e n t i a l f o r a f a s t program- me development and t e s t i n g a r e g r a t e f u l l y acknowledged. F u r t h e r m o r e ,

1 t h a n k J . Hayter f o r t h e 30 g r a p h i c s r o u t i n e s u s e d t o p r e p a r e f i g . 5 .

REFERENCES

1. H. S. Green, Handbuch d e r Physik

g

(1960)

R . E i s e n s c h i t z , S t a t i s t i c a l Theory of I r r e v e r s i b l e P r o c e s s e s , Oxford U n i v e r s i t y P r e s s , Oxford 1958

2 . N.A. C l a r k , B . J . Ackerson, P h y s . R e v . L e t t . f i , (1980)2844 3 . S. Hess, Phys. Rev. A22 (1980) 2844

4. W. T. A s h u r s t and W . C ~ o o v e r , Phys.Rev. &,(1975)658

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6.

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D. J . Evans, Phys.Rev.

g ,

(1981)1988

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8. H . J . M . Hanley, D . J . Evans and S. Hess, J.Chem.Phys.E (1983)1140 9. S. Hess and H.J.M. Hanley, P h y s . L e t t .

98A

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10. D. J. Evans, H . J . M . Hanley and S . Hess, P h y s i c s T o d a y , x (1984)26 11. S. Hess, P h y s i c a

3,

(1983)79

Phys.Rev.&, (1982) 614

1 2 . S . Hess and H . J . M . Hanley, 1 n t . J . Thermophys.4 (1983)97 13. H . J . M . Hanley, J.C. R a i n w a t e r , N.A. C l a r k and B . C . Ackerson,

J. Chem. P h y s . 2 (1983) 4448

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20. A. C . Mitus and A. 2 . P a t a s h i n s k i i , Phys.Lett.=, 199 (1982) 2 1 . M. J. S t e p h e n and J.P. S t r a l e y , Rev.Mod.Phys.s, 617 (1974) 2 2 . P. G. d e Gennes, The P h y s i c s o f L i q u i d C r y s t a l s , Clarendon P r e s s ,

Oxford 1974

23. S. Hess, Z . N a t u r f o r s c h . = , 728 (1975) 1224

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Von Der Lage and H.A. Bethe, Phys.Rev.71, (1947) 612 A. Hüller and J . W . Kane, J. Chem.Phys.g, 3599 (1974)

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R. Nelson and J. Toner, Phys.Rev.=, (1981)363

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Hsu and A. Rahman, J. Chem. P h y s . 3 (1979) 5234 71, (1979) 4974

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