THE THEORY OF SIMPLE CLASSICAL FLUIDS : UNIVERSALITY IN THE SHORT RANGE STRUCTURE

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THE THEORY OF SIMPLE CLASSICAL FLUIDS : UNIVERSALITY IN THE SHORT RANGE

STRUCTURE

Yaakov Rosenfeld

To cite this version:

Yaakov Rosenfeld. THE THEORY OF SIMPLE CLASSICAL FLUIDS : UNIVERSALITY IN THE SHORT RANGE STRUCTURE. Journal de Physique Colloques, 1980, 41 (C2), pp.C2-77-C2-81.

�10.1051/jphyscol:1980212�. �jpa-00219804�

THE THEORY OF SIMPLE CLASSICAL FLUIDS : UNIVERSALITY IN THE SHORT RANGE STRUCTURE Yaakov Rosenfeld,

Nuclear Research Center-Negev, P.O. Box 9001, Beer-Sheva, Israel

Abstract.- A theory of simple classical fluids is presented in which both the static structure (on the "pair" level), and the thermodynamics, of all systems describable by spherically symmetric pair potentials, can be calculated by a unified approach. The theory is based on the diagramatic expansion of the pair distribution function that leads to a modified hypernetted chain (HNC) integral equation. It consists of the approximation that the bridge functions (i.e. the sum of all elementary graphs, assumed zero in the HNC approximation) constitute the same (universal) family of curves, irrespective of the assumed pair potential.

Using the parametrized computer simulation data for hard spheres as input in the integral equation, it was found possible to virtually duplicate a large body of computer simulation data compiled for a variety of quite disparate interparticle potentials (the one and two component plasma in particular). The statement of universality enables to obtain the potential of mean force at small separations directly from the solutions of the integral equation, and the resulting enhance- ment factors for nuclear reaction rates (in the dense plasma) are in excellent agreement with Jancovici's recent calculations (by an indirect method) for equal charges, and Salpeter's ion-sphere prediction for mixtures.

A theory for classical fluids is presented in which both the static structure (on the

"pair" level), and the thermodynamics, of all system describable by spherically symme- tric pair potentials, can be calculated by a unified approach. The theory is based on the diagramatic expansion of the pair distribu- tion functions that leads to modified hyper- netted -chain (HNC) integral equations. It consists of the approximation that the brid- ge functions (i.e. the sum of all elementary graphs, assumed zero in the HNC approximat- ion) constitute the same (universal) family of curves, irrespective of the assumed pair potentials. Using the parametrized computer simulation data for hard spheres as input in the integral equations, it was A u n d possi- ble to virtually duplicate a large body of

computer simulation data, compiled for a variety of quite disparate interparticle potentials (the one and two-component plasma in particular). T he statement of universality enables to obtain the poten- tial of mean force, at small separations, directly from the solutions of the integral equations. The resulting enhancement fac- tors for nuclear reaction rates (in the dense plasma) are in excellent agreement with Jancovici's recent calculations (by

an indirect method) for equal charges, and Salpeter's ion-sphere predictions for mix- tures. Consider a mixture containing No particles of type a, interacting via the potentials u ag ( v ) - Let N = ENo , V_= total volume, 1(1 = N , ifia = Nct_ , Xa= Na.

V V N

JOURNAL D E PHYSIQUE Colloque C2, supplément au n° 3, Tome 41, mars 1980, page C2-77

Résumé.- Une théorie des fluides classiques simples est présentée dans laquelle à la fois la structure statique (au niveau des "paires")» et la thermodynamique de tous les systèmes pouvant être décrits par des potentiels de paires à symétrie sphérique peut être calculée par une approche unifiée. La théorie est fondée sur l'expansion diagrammatique de la fonction de distribution de paires conduisant à une équation intégrale HNC modifiée. Elle consiste en l'approximation que les

"bridge functions" (c'est-à-dire la somme de tous les graphes élémentaires, annulés dans l'approximation HNC) constituent la même (universelle) famille de courbes, quel que soit le potentiel de paires adopté. En utilisant les simulations numériques paramétrisées du modèle des sphères dures comme donnée'initiale dans l'équation in- tégrale il devient possible de reproduire un large éventail de données numériques calculées pour des potentiels d'interaction de particule à particule très diffé- rents ( les plasmas à une et deux composantes en particulier). L'hypothèse d'uni- versalité permet d'obtenir le potentiel de force moyenne à courte distance directe- ment à partir de solutions de l'équation intégrale ; les factures d'améliorations des réactions nucléaires (dans le plasma dense) qui en remettent sont en bon accord avec les résultats récents de Jancovici (par une méthode indirecte) pour des char- ges identiques, et avec le modèle de la sphère ionique de Salpeter pour les mélan- ges.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980212

JOURNAL DE PHYSIQUE

Consider a s p e c i f i c q component c a s e w i t h a given s e t X I , X 2 ,

...,

Xq. The g r a p h i c a l ana- l y s i s of 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 s /1/

g a g (v ) y i e l d s t h e following e x a c t e q u a t i o n s f o r t h e t o t a l c o r r e l a t i o n f u n c t i o n s , h a g ( v )

= gaf3 ( v ) - 1 , and t h e d i r e c t c o r r e l a t i o n func- t i o n s C a 6 ( v ) :

e f f

ha 6 ( v ) + 1 = exp

[Z-

^{U a B (v)/kBT}

+

h a 6 ( v ) -

c a 6 ( v )

]

( I )

-.

h a 6 ( k )

-

^{CaB(k) =}

+

^h ( k ) , C a 6 ( k )

aY ^{( 2 )}

where

Eq. ( 2 ) i s t h e Orenste'n-Zernike r e l a t i o n f o r m i x t u r e s , w r i t t e n i n e m s of F o u r i e r t r a n s - forms. B a B ( v ) d e n o t e s , t h e f u n c t i o n s r e p r e s e n -

f

t e d by minus t h e sum of a l l elementary gra- phs w i t h P-layer f-bonds, which have a p a r t i - c l e of t y p e a and a p a r t i c l e of t y p e 6 a s r o o t p o i n t s , while f i e l d p o i n t s can belong t o p a r t i c l e s of any type. Note t h a t B ( v )

a 6 can be a l s o r e p r e s e n t e d by a s u b s e t of t h e elementary graphs, t h e " b a s i c " set ( c o n s i s - t i n g of a l l elementary graphs w i t h a t l e a s t t r i p l y connected f i e l d p o i n t s ) p o s s e s s i n g i n s t e a d of f bonds o n l y h bonds ( I ) .

Y 6 Y 6

Given t h e p o t e n t i a l

u:if(v) ,

eqs. (1) and ( 2 ) r e p r e s e n t t h e h y p e r n e t t e d c h a i n (HNC) equa- t i o n s f o r t h e unknowns h a g ( v )

,

C U B ( v )

.

^With

t h e assumption u e f f ( v ) = U a 6 ( v ) i . e . B a 6 ( v ) a 6

= 0, t h e s e e q u a t i o n s c o n s t i t u t e t h e u s u a l HNC approximation. With a g i v e n ( n o t neces- s a r i l y t h e e x a c t ) s e t of f u n c t i o n s B ( v )

,

a B we r e f e r t o eqs. (1)

-

^{( 3 )} a s t h e modified HNC scheme (MHNC)

.

Various i n t e g r a l e q u a t i o n s t h a t r e s u l t from widely used approximations i n t h e t h e o r y of l i q u i d s , can be c a s t i n t h e MHNC form ( 2 ) . I n p a r t i c u l a r , t h e Percus- Yevick ( P Y ) e q u a t i o n s a r e obtained w i t h B" (v) = y:E(v)

-

¹

-

lnyE:(v),

a B where

y a 6 ( v ) = g a 6 ( v ) e x p p a 6 ( v ) / k B ~ ] 2 exp

Eq.(5) d e f i n e s t h e p o t e n t i a l s of mean f o r c e , H a 6 ( v ) . From t h e way t h e MHNC e q u a t i o n s have been w r i t t e n , t h e e x a c t " b r i d g e functions1', Ba 6 ( v ) , a c t u a l l y p l a y t h e r o l e of p e r t u r b i n g p o t e n t i a l s i n t h e u s u a l HNC scheme, poten-

t i a l s t h a t a r e t o bc determined s e l f c o n s i s - t e n t l y v i a t h e f u n c t i o n s h ( v ) , by an i t e -

a 6

r a t i v e procedure. L e t t h e symbol'( 1 denote t h e s e t of a l l a6 p a i r s . L e t us denote t h e s o l u t i o n of e q s . (1)-(3) w i t h a g i v e n s e t CBa6 ( v ) 1 by' I h a g ( v ) ;' C B a B ( v ) 11, and l e t C C h ( i ) 1 r e p r e s e n t t h e b r i d g e f u n c t i o n s ,

ba6

C B ~ $ ) ( v ) 1, o b t a i n e d by t h e summation of t h e i n f i n i t e s e t of b a s i c diagrams w i t h t h e h y 6 ( v ) bonds chosen from a given s e t 1h::iv) 1. The i t e r a t i v e s e l f c o n s i s t e n c y scheme may t a k e t h e f o l l o w i n g form :

The o v e r a l l q u a n t i t a t i v e s i m i l a r i t y between t h e HNC and " e x a c t " computer s i m u l a t i o n r e - s u l t s . f o r t h e s t r u c t u r e , t o g e t h e r w i t h t h e h i g h l y connected n a t u r e of t h e w b a s i c " d i a - grams, s u g g e s t s t h a t t h e s e r i e s (6) i s f a s - t l y convergent. We make t h e following pro- p o s i t i o n /2,3/ :

A s i t s t a n d s , t5is p r o p o s i t i o n i s o f no p r a c t i c a l s i g n i f i c a n c e s i n c e even one i t e - r a t i o n r e q u i r e s t o sum an i n f i n i t e number of diagrams. A way o u t i s suggested a s f o l l o w s : A w e l l founded n o t i o n , on which t h e p h y s i c a l understanding of simple c l a s - s i c a l f l u i d s i s based, i s manifested p i c t o - r i a l l y i n t h e p o s s i b i l i t y t o s c a l e b o t h g

( v ) and S (k) ( t h e s t r u c t u r e f a c t o r )

,

f o r q u i t e d i s p a r a t e systems ( p o t e n t i a l s v a r y i n g between, say v-l, and v - m ) SO t h a t t h e y a r e n e a r l y congruent ( 4 )

.

In. o t h e r words, t h e r e e x i s t s a " f i r s t o r d e r " u n i v e r s a l i t y of s t r u c t u r e

ha (v) ; $

,

T 1 rz independent of

provided we c o n s i d e r t h e whole s e t a s fun- c t i o n of d e n s i t y an$ temperature. This f i r s t o r d e r u n i v e r s a l i t y provides t h e b a s i s f o r t h e s u c c e s s f u l a p p l i c a t i o n o f t h e v a r i a - t i o n a l method w i t h one u n i v e r s a l r e f e r e n c e system (e.9. t h e hard-sphere s y s t e m ) , f o r

a v a r i e t y of p o t e n t i a l s i n c l u d i n g t h e coul- /7/. A g e n e r a l scheme t o e x t r a c t t h e b r i d g e omb plasma, w i t h good r e s u l t s f o r t h e t h e r - f u n c t i o n s from t h e s i m u l a t i o n d a t a may con- modynamics /5/. The d e v i a t i o n s from (8) a r e , s i s t of : (i) assuming a form f o r ' {B ( v ) )

a B however, t h e main o b j e c t of i n t e r e s t i n t h e w i t h some f r e e parameters, (ii) s o l v i n g t h e theory of c l a s s i c a l f l u i d s . These d e v i a t i o n s MHNC e q u a t i o n s (1) - ( 3 ) t o f i n d t h e c o r r e s - a r e r e l a t i v e l y s m a l l , of t h e same o r d e r of ponding'( g a g ( v ) 1, (iii) a l t e r i n g t h e f r e e magnitude a s t h e d e r i v c t i o n s h z C ( v ) -hEtaj:). parameters u n t i l b e s t f i t f o r t h e comguter Thus i f we m a i n t a i n p r o p o s i t i o n ( 7 ) . i . e . d a t a i s achieved. I n p a r t i k u l a r , thermody- t h e "one i t e r a t i o n " assumption, we e x p e c t namic c o n s i s t e n c y between t h e c o m p r e s s i b i l i - t h a t t h e accuracy o f ( h::) ( v ) 1 can be a c h i e - t y e q u a t i o n of s t a t e v i a ' {C ( v ) }and t h e ved a l s o i f we choose t h e b r i d g e f u n c t i o n s energy e q u a t i o n of s t a t e v i a (g aP ( v ) ]

a 6

from a u n i v e r s a l s e t a p p r o p r i a t e t o one (any should be imposed. Even w i t h o u t r e f e r e n c e one) p a r t i c u l a r c h o i c e of t h e p o t e n t i a l s t o b r i d g e f u n c t i o n s a s a key q u a n t i t y , t h i s

' {UaB(v) 1: scheme p r o v i d e s a numerically sound proce-

e x a c t d u r e f o r o b t a i n i n g ' {SaB ( k ) 1 from' ( g a B ( v ) 3 ,

' {BaB ( v ) ; $, T

=

independent of b e t t e r i n f a c t t h a n a l l p r e v i o u s methods ( 9 ) used. A p a r t i c u l a r such f i t w i l l reproduce

{UaB ( v ) 1 e x a c t

Iga 6 1 and y i e l d a l s o ' IC ( v ) 1 , b u t i n a B

The approximation of u n i v e r s a l i t y a s embo- view of the relation h a g ( v ) = C a B ( v ) ⁺ H a B ( v ) d i e d i n ( 8 ) p r o v i d e s , i n t h e c o n t e x t of t h e

+

B a B ( v )

,

only t h e sum Ha6(v)+ B a B ( v ) can v a r i a t i o n a l scheme, reasonably a c c u r a t e r e s u _be determined f o r small v , and not, ,bath fun- I t s f o r t h e thermodynamics. We i t e r a t e t h i s ction separately. This fitting scheme is n o t i o n of u n i v e r s a l i t y , and e x p e c t t h a t ( 9 ) not sensitive at all to the values of the i n t h e c o n t e x t of t h e 14HNC scheme w i l l g i v e fitting functions, C B ( v ) at small v ,

-

t h e corresponding f i r s t o r d e r c o r r e c t i o n s t o since a finite perturbation aB on an effecti- t h e s t a t e m e n t (8)

.

^{I t} t u r n s o u t , however, v e l y i n f i n i t e ( s i n c e g ( v ) < c 1) p o t e n t i a l t h a t t h e s e " c o r r e c t e d " r e s u l t s a r e n e a r l y

has no e f f e c t on the structure, For values i n d i s t i n g u i s h a b l e from t h e b e s t computer si-

of beyond the first peak of m u l a t i o n d a t a p r e s e n t l y a v a i l a b l e , f o r a l l

i s of o r d e r

1

h,;(v). I n t h e c o n t e x t of t h e p h y s i c a l systems considered. PlHNC scheme 2 t h i s corresponds t o a weak and How t h e b r i d g e f u n c t i o n s look l i k e ? I n t h e _long range p e r t u r b i n g whose e f f - absence (indeed t h e n o n f e a s i b i l i t y ) of any ect on _{( v )} is very small, within the noi-

a B

diagram sUNnation that be meaningfu1 s e of p r e s e n t day s i m u l a t i o n s ( a b o u t 0.01, f o r a dense ^' f l u i d , we focus a t t e n t i o n on t h e e x a c t i.e. % )

.

^{We thus arrive at the} conclusion computer s i m u l a t i o n d a t a f o r ,

'

and the t h a t from t h e s t a n d p o i n t of i t s s t r u c t u r a l thermodynamics. These "exact" r e s u l t s s u f f e r consequences, the important region for which from two (among o t h e r s ) i n t r i n s i c l i m i t a t i o n s B a B ( v I is to be specified, in any theory, : ( i ) g a g ( v ) i s given o n l y in the range

Ocvck

_{i s} t h e r e g i o n of t h e f i r s t peak. Moreover,

(where L' i s t h e volume of t h e b a s i c simu- 2 _in t h a t narrow B ~ ~has the ( ~ ) uni- l a t i o n cube) t h a t u s u a l l y covers Only the v e r s a l p r o p e r t y of being e f f e c t i v e l y a r e - f i r s t few peaks. This p r o h i b i t i s t h e unambi- pulsive potential. This last result is ob- gous d e t e r m i n a t i o n of S ( k ) o r C ( v )

.

This pro- tained by

published HNC and blem i s t r e a t e d more o r l e s s s a t i s f a c t o r i l y results for a large variety of sys- by j o i n i n g t h e t a i l s o f , {gE:aCF t o t h e ~ 0 1 ~ - tems. At t h i s point universality (9)

t i o n of s m e very p l a u s i b l e and even a family of s t r a i g h t

(PY. HNC,etc)/6/. ' ( i i ) ' I n t h e r e g i o n of v e r y lines should do a fine job of fitrlng - the

strong repulsion where g ( v )

2

it d a t a /8/. The statement of uni-

i s numerically impossible t o c a l c u l a t e H ( v ) versality ( 9 ) allows the of

I

the data via g ( v )

.

^{The im-} _H ( v ) a t small v prov ded FTe know

the 'exact

portance this quantity stems the fact b r i d g e f u n c t i o n s f o r a t least one system.

t h a t exp I H a B ^{( 0 )}

I

i s t h e f i r s t cgder approx- _There

is only one f l id system for which imation f o r t h e enhancement f a c t o r s f o r nut- one can regorously obtain the values of l e a r r e a c t i o n s r a t e s i n dense i o n i z e d m a t t e r

C2-80 JOURNAL DE PHYSIQUE

H a R ( v ) a t s m a l l v i n terms of computationa- l l y f e a s i b l e thermodynamic q u a n t i t i e s , and t h a t i s t h e system composed of hard spheres with d i a m e t e r s a l r a 2 , . . . a I n p a r t i c u l a r /9/, H a g ( 0 ) = l~Ea whereq;: d e n o t e s t h e ex- c e s s chemical p o t e n t i a l f o r type a p a r t i c l e s kBT and aa 6 oB. The computer s i m u l a t i o n d a t a f o r hard spheres have been parametrized /9/

a s c o r r e c t e d v e r s i o n s of t h e a n a l y t i c s o l u - t i o n of t h e PY e q u a t i o n s , w i t h a n accuracy t h a t e n a b l e s t o i n f e r e t h e corresponding b r i - dge f u n c t i o n s ' { ~ z g ( v ; o l , . . . , ~ ; $1.

For a q-component system t h e s e r e p r e s e n t Si a q-parameter f a m i l y of c u r v e s . For t h e sake of comparison we a l s o c o n s t r u c t t h e c o r r e s - ponding b r i d g e f u n c t i o n f a m i l y from t h e ana- l y t i c PY r e s u l t s v i a eq. ( 4 )

.

^A most i n t e r e s - t i n g o b s e r v a t i o n i s t h e f a c t t h a t e x c e p t f o r a r e l a b l i n q of t h e parameters a t h e two

HSInexac@and B H S ,PY " I

f a m i l i e s B a r e n e a r l y l n d e n t i c a l /2/.

I n o t h e r words, a s e t ( o l , . . , a ) i n one f a - mily corresponds, w i t h a high accuracy t o q some s e t ( a;,

. . . ,

a ^{I )} i n t h e o t h e r .

That means t h a t i n c o r p o r a t i n g t h e PY b r i d g e q f u n c t i o n s i n t h e MHNC scheme, we can d u p l i - c a t e t h e computer s i m u l a t i o n d a t a f o r hard s p h e r e m i x t u r e s , f o r t h e p a i r s t r u c t u r e and thermodynamics, i n c l u d i n g a d i r e c t determi- n a t i o n of H a B ( o ) . According t o t h e c o n j e c t u - r e of u n i v e r s a l i t y , we should u s e i n such a scheme ( f o r any p o t e n t i a l s ) t h e e x a c t hard

PIHNC e q u a t i o n s reproduces both t h e s t r u c - t u r e and t h e e q u a t i o n of s t a t e a s o b t a i n e d by computer s i m u l a t i o n s t o w i t h i n - t h e i r n o i s e . This s o l u t i o n a l s o provides a p r e d i - c t i o n f o r H a B ( o ) , i n f a c t t h e o n l y d i r e c t p r e d i c t i o n f o r t h i s q u a n t i t y a v a i l a b l e a t p r e s e n t .

Unlike t h e h a r d s p h e r e s , t h e p o s s i b i l i t y of o b t a i n i n g H ( 0 ) f o r a dense plasma ( p o s i -

a B

t i v e i o n s immersed i n a compensating uni- form charge charge background) r e l i e s on s p e c i f i c p h y s i c a l c o n s i d e r a t i o n s . For t h e i n t e r a c t i o n s UaB(v) z,zBe2

- r

- -

zzaz6-

LBT k B ~ v (v/a)

(where a = ( 3 / 4 1 ~ 4 ~ / ~ )

,

t h e dominant s t r o n g coupling ( 'I > > I ) c o n t r i b u t i o n t o t h e ex- c e s s f r e e energy i s expected t o have t h e following form :

F~~ ( X I , . . . , x q ) = -E

. <

^Z>'I

<

z ~ / ~ > T lJ

NkBT ^' (1 0)

where,

and t h e "Madelung" c o n s t a n t i s w e l l appro- ximated by t h e ion-sphere m o d e l / l l / , E:' = 9 / 1 0 . A regorous consequence of (10) i s

I

(11) J a n c o v i c i /7/ made use of Monte C a r l o d a t a s p h e r e ' b r i d g e f u n c t i o n s . Our l a t e s t observa- _/12/ to improve Salpeterls ion-sphere pre- t i o n i m p l i e s t h a t t h e MHNC c a l c u l a t i o n s can diction for the case of the one-component be performed w i t h the1 a n a l y t i c i n p u t from PY, plasma. For I'

,,

_{both results are very}

t h u s making t h e whole procedure f r e e f r o m

close and agree very well with our direct any "noisy" i n p u t .

1

p r e d i c t i o n s /2/ using t h e MHNC scheme w i t h We performed t h e MHNC c a l c u l a t i o n s f o r a

PY b r i d a e f u n c t i o n s . Our d i r e c t p r e d i c t i o n s l a r g e v a r i e t y of p o t e n t i a l s . The f o l l o w i n g

s i n g l e component systems we considered /2/ : hard s p h e r e s , Lennard-Jones, i n v e r s e f i f t h power ( v - ~ ) p o t e n t i a l a p p l i c a b l e t o t h e he- l i u m ground s t a t e problem, Coulomb ( i . e . t h e c l a s s i c a l one component plasma), Yukawa, charged hard s p h e r e s , and an o s c i l a t o r y po- t e n t i a l proposed f o r l i q u i d m e t a l s . The c a l - c u l a t i o n s f o r b i n a r y m i x t u r e s i n c l u d e

/ l o /

^:

hard-spheres, Lennard-Jones, Coulomb ( t h e two component plasma). For each system con- s i d e r e d a t a g i v e n temperature and d e n s i t y , a s i n g l e b r i d g e f u n c t i o n , from t h e PY hard sphere s e t , c o u l d , b e found, such t h a t a thermodynamically c o n s i s t e n t s o l u t i o n of t h e

<

f o r a b i n a r y mixture

/ l o /

w i t h Z 1 Z 2 = 2 a g r e e w e l l w i t h (11). These r e s u l t s p r o v i - de a s e v e r e t e s t f o r b o t h t h e accuracy of t h e t h e o r y and i t s d i a g r a m a t i c i n t e r p r e t a - t i o n .

I n view of t h e s t a t e m e n t of u n i v e r s a l i t y and i t s a p p a r e n t c o n f i r m a t i o n by computer s i m u l a t i o n s , t h e f o r m u l a t i o n of a proce- d u r e f o r a n a p r i o r i c a l c u l a t i o n of both t h e s t r u c t u r e ( i n c l u d i n g t h e o t h e r w i s e i n a c c e s s i b l e H (v << 1 ) ) and t h e thermo- dynamics of any p h y s i c a l l y concievable p a i r p o t e n t i a l s U a B ( v )

,

can be c a r r i e d o u t i n many ways. For one component systems,

f o r which t h e h a r d s p h e r e b r i d g e f u n c t i o n s c o n s t i t u t e a o n e p a r a m e t e r f a m i l y o f c u r v e s , t h i s i s e a s i l y a c h i e v e d by i m p o s i n g thbrmo- dynamic c o n s i s t e n c y i n o r d e r t o o b t a i n ' t h e a p p r o p r i a t e v a l u e o f t h i s p a r a m e t e r as f u n c - t i o n o f d e n s i t y and t e m p e r a t u r e . An e s s e n - t i a l l y s i m i l a r p r o c e d u r e c a n b e u s e d f o r m i x t u r e s b u t it i n v o l v e s more t e c h n i c a l d e t a i l s .

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