THEORIES OF ENTROPIES OF LIQUID METALS ; PHONONS VERSUS PACKING

(1)

HAL Id: jpa-00220252

https://hal.archives-ouvertes.fr/jpa-00220252

Submitted on 1 Jan 1980

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

THEORIES OF ENTROPIES OF LIQUID METALS ; PHONONS VERSUS PACKING

I. Ohkoshi, I. Yokoyama, Y. Waseda, W. Young

To cite this version:

I. Ohkoshi, I. Yokoyama, Y. Waseda, W. Young. THEORIES OF ENTROPIES OF LIQUID MET-

ALS ; PHONONS VERSUS PACKING. Journal de Physique Colloques, 1980, 41 (C8), pp.C8-599-C8-

602. �10.1051/jphyscol:19808152�. �jpa-00220252�

(2)

THEORIES OF ENTROPIES OF LIQUID METALS ; PHONONS VERSUS PACKING

I . Ohkoshi, I . Yokoyama, Y . ~ a s e d a * and W.H. Young

**

Department of Mathematics and Physics, National Defense Academy, Yokosuka, Japan.

* ~ e s e a r c h I n s t i t u t e of Mineral Dressing and Metallurgy, Tohoku Uniuersity, Sendai, Japan.

*schooZ of Mathematics and Physics, University o f East AngZia, Norwich, UK.

Abstract.- I t i s known t h a t t h e entropy of a l i q u i d metal can be l a r g e l y explained i n terms of atomic packing. We show t h a t t h e Percus-Yevick phonon theory w i l l a l s o s u f f i c e and a connection between t h e two methods i s e s t a b l i s h e d .

1. I n t r o d u c t i o n A t t h e B r i s t o l Conference, one of us gave a t a l k

[

1

'1

i n which t h e e n t r o p i e s of l i q u i d metals were described i n terms of t h e packing p r o p e r t i e s of t h e i o n s . Such an approach i n e v i t a b l y focusses p r i m a r i l y on t h e p r i n c i p a l peak of t h e s t r u c t u r e f a c t o r , a ( k ) . From t h i s region of k-space, t h e packing and thereby t h e entropy was deduced.

Afterwards, t h e speaker was reminded by Prof.

March of t h e PY ( ~ e r c u s and Yevick) phonon t h e o r y

[ 2

]

which seemed capable of d e s c r i b i n g success-

problem remains o f t h e connection between t h e two methods. The work below i s addressed p r i m a r i l y t o t h i s m a t t e r .

2. The packing method We f i r s t b r i e f l y review t h e packing method. Suppose we a r e given a measured a ( k )

.

Provided we avoid very low k ' s

[ 1, 4 , 5

1

^iti s u s u a l l y p o s s i b l e t o d e s c r i b e it approximately by a h a r d sphere form [ 6 , 7

1 .

^An

e f f e c t i v e d i m e t e r may thus be obtained, one method of unequivocally determining t h i s being t o match t h e h e i g h t s of t h e f i r s t peaks. The entropy t h e n f u l l y such a property a s t h e entropy. This i s follows from

indeed t h e case, a s we i n d i c a t e below. But t h e S(packing method) = Sgas + Sn + Selec (1) phonon method would appear t o be mainly concerned

Here with much lower k-values t h a n those around t h e

f i r s t peak of a ( k ) . How t h e n can t h e two methods

which i s t h e i d e a l gas expression and Sn i s a be reconciled?

I n an attempt t o answer t h i s question, Gray, well-understood 1 8 ] f u n c t i o n o f t h e packing Y o k o y m and Young

1

³] c a l c u l a t e d t h e entropy of

l i q u i d sodium using t h e PY d e s c r i p t i o n . 'hey found t h a t though t h e independent phonon c o n t r i - bution i n e v i t a b l y a r i s e s from t h e lower-k region, a c o n t r i b u t i o n of o r d e r 5

-

10% a r o s e from phonon-phonon i n t e r a c t i o n s . The l a t t e r involve phonon p a i r s with q u i t e l a r g e wavenumber d i f f e r - ences and a d e s c r i p t i o n of them does indeed involve t h e p r i n c i p a l peak of a ( k )

.

Nevertheless, t h e bulk of t h e entropy s t i l l a r i s e s from t h e independent phonons and t h e b a s i c

f r a c t i o n TI. The term Selec a r i s e s from thermal e x c i t a t i o n of t h e conduction e l e c t r o n s , i s small f o r simple metals, and i s adequately approximated below by i t s Sommerfeld form.

I f , a s suggested above, matching t a k e s p l a c e a t t h e f i r s t peak, t h e n t h e subsequent peak h e i g h t s of t h e measured a ( k ) a r e overestimated by t h e theory ( f i g . 3 of r e f . [ 1 1. I n o t h e r words, t h e e f f e c t i v e diameter obtained from t h e observed f i r s t peak aldne i s a l i t t l e t o o b i g and t h e entropy i s , t h e r e f o r e , corresponaingly t o o small.

Nevertheless, t h i s simple procedure l e a d s t o lower

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19808152

(3)

C8-600 JOURNAL DE PHYSIQUE

bounds which a r e reasonably a c c u r a t e ( f i g . 7 of t h e concluding remarks of s e c . 2 ) thus suggests

C1l).

t h a t t h e PY method succeeds whenever t h e packing 3 . The PSI d e s c r i p t i o n Ifere we imagine t h e method succeeds. This p r o p o s i t i o n i s now i n v e s t i - l i q u i d t o nossess 3 N normal modes of density-

f l u c t u a t i o n fonn. We comment on t h i s assumption i n our concluding discussion but f o r t h e p r e s e n t accept i t . Thus, t h e zeroth order p i c t u r e i s of l o n g i t u d i n a l phonons with :-vectors occupying an enlarged Debye sphere of r a d i u s kg = 3 l l 3 k n where k = (61?/~2)1/3 i s t h e u s u a l value.

D

The zeroth order phonon spectrum t a k e s t h e f0.m C 9 - j

and i n t e n s o f t h i s ane o b t a i n s an entropy con- t r i b u t i o n of

I n t h i s ex?ression, x = hw ( k ) / k g . :&en phonon i n t e r a c t i o n s a r e introduced, a l e a d i n g order c o r r e c t i o n S1 i s obtained ( f o r d e t a i l s , see r e f .

[ 3 and t h e t o t a l entropy from t h i s p o i n t of view reads

S(Py) = 30 + S 1 + Selec ( 5 ) As Gray e t a l . ' s f i g . 3 shows, an E i n s t e i n

d e s c r i p t i o n i s reasonably a p p r o p r i a t e f o r a l l except t h e lowest k ' s and t h e l a t t e r , of course, c o n t r i b u t e w l t t l low 'weight. &cause of t h i s weighting, it i s important t h a t t h e s i z e of a ( k ) i s adequately descriSed f o r k kg.

:k s m a r i z e , i n t a b l e 1, some r e s u l t s obtained by t h e o r e s e n t authors using eq. ( 5 ) and measured s t r u c t u r e f a c t o r s [IS]. The i n t e r e s t r e a d e r who r e f e r s t o Gray e t a l . should note

gated.

Table 1

Entropies ( i n u n i t s of Nk B ) of some l i q u i d metals

4. Relationship between methods F i r s t , l e t u s note t h a t t o h i g h accuracy, t h e summand of eq. ( 4 ) can be expanded f o r small x t o give 1

-

En X .

It i s then p o s s i b l e t o w r i t e eq. ( 5 ) i n t h e form S(PY) = PY

+ 'pack ⁺'elec

~ ( e x p t ) 7.84 9.14 10.26 11.13 9.01 8.62 9.13 11.19

where Na K Rb Cs Mg AP.

I n Pb

Eq. ( 7 ) depends on a ( k ) and Q but not

e x p l i c i t l y on T o r M. I n view of eqs. ( 1 ) and (61, t h e d i f f e r e n c e between t h e two methods of

' I

0.46 0.46 0.48 0.46 0.45 0.47 0.43 0.48

c a l c u l a t i o n hinges only on t h e d i f f e r e n c e between 'elec S(PY)

0.05 7.43

0.07 8.81

0.08 9.79

0.09 10.84

0.12 8.95

0.11 8.63

0.07 8.70

0.12 11.10 T ( K )

378 343 313 303 953 943 433 613

PY and S

'pack 9'

SO 6.92 8.28 9.23 10.29 8.38 8.05 8.20 10.50

I f we next s p e c i f y a ( k ) t o be of hard sphere form, eq. ( 7 ) can be e v a l u a t e d a s a f u n c t i o n of n only and comparison w i t h Sn c m be made ( f i g . 1).

Under normal conditions t h e curves a r e c l o s e t o g e t h e r ; f o r exsmple, f o r any given n i n t h e range (0.32, 0.471, which encompasses a l a r g e m a j o r i t y of conceivable a p p l i c a t i o n s , t h e t h a t we s e t t h e i r S1 = 0 and s o S1 = S1). It d i f f e r e n c e i s l e s s t h a n 0.4 Mcg.

w i l l be seen t h a t t h e c a l c u l a t e d values a r e a c c u r a t e and t h a t t h e method apparently l e a d s t o lower bounds. The numerical evidence ( t a b l e 1 and

(4)

It follows from t h e above a n a l y s i s t h a t i f a measured a ( k ) i s reasonably well represented by a s i n g l e hard sphere form over t h e ranee of k between k O - and t h a t of t h e p r i n c i p a l peak, then t h e e n t r o p i e s obtained by t h e PY and packing methods w i l l d i f f e r l i t t l e . Such a ( k ) behaviour i s , i n f a c t , t h e b a s i s of the roughly equal q u a l i t i e s of t h e PY data of t a b l e 1 and t h e packing r e s u l t s of r e f .

[

1

1.

Actually, t h e PY entropy f o r any given case i s r a t h e r higher and, t h e r e f o r e , nearer t o experiment than t h a t obtained by t h e packing method. To see why, note t h a t a l l systems of t a b l e 1 a r e near t h e i r melting p o i n t s . It follows t h a t ( i ) we a r e i n t e r e s t e d i n t h e r i g h t p a r t of f i g . 1 and ( i i ) t h e measured a ( 0 ) w i l l be lower than t h a t f o r hard spheres f o r any r e a l i s t i c a l l y chosen packing f r a c t i o n

[

1, 4, 5 ]

.

^{Let us}

choose 11, a s i n r e f . [ 1

1,

t o y i e l d t h e observed f i r s t peak h e i g h t . By ( i ) , we o b t a i n a higher P Y than packing entropy. But, by ( i i ) , we expect (and f i n d ) t h a t t h e hard sphere s t r u c t u r e f a c t o r corresponding t o must be lowered a

f u r t h e r i n c r e a s e i n entropy i s thereby obtained.

A t higher temperatures, t h e l e f t p a r t of f i g . 1 w i l l apply and f o r t h i s reason alone t h e PY entropy can be expected t o be lower than t h e packing r e s u l t . However, a t s u f f i c i e n t l y high T, t h e measured a ( k ) r i s e s above t h e hard sphere form a t very low k so it i s not p o s s i b l e t o .oredict with c e r t a i n t y what happens a t k % ko. So f a r , we have not made'any d e t a i l e d c a l c u l a t i o n s on r e a l systems a t elevated temperatures.

5 . s c u s s o n The ?acking method has achieved a measure of acceptance over t h e p a s t decade and i t s value and usefulness seem t o be e s t a b l i s h e d

[ I , 10

1.

On t h e o t h e r hand t h e ultimate v a l i d i t y of t h e PY method appears t o be more u n c e r t a i n .

The .fundamental question i s whether a given l i q u i d can be adequately described by 3 N normal v i b r a t i o n a l m o d e l s . The work of Bratby, Gaskell and March

[

11

1

suggests so f o r l i q u i d metals but not f o r l i q u i d argon. This i s because of t h e r a t h e r d i f f e r e n t c h a r a c t e r s of t h e interatomic f o r c e s involved.

I n t h i s connection we emphasise t h a t ou,r c a l c u l a t i o n i n s e c t i o n 4 was intended f o r hard spheres with, however, temperature and d e n s i t y dependent diameters. Such a p r e s c r i p t i o n provides t h e same formal r e s u l t f o r t h e entropy a s one would o b t a i n f o r a b s o l u t e l y i n v a r i a n t diameters

[ 1 2

1,

but t h e s a e c i f i c h e a t s , f o r example, would d i f f e r by t e r n s depending unon aq/aT.

&en i f t h e b a s i c hypothesis i s accented, t h e choice of t h e 3 N nodes remains a problem. ?'he PY s e l e c t i o n , a s we have seen, c o n s i s t s e n t i r e l y of l o n g i t u d i n a l phonons and indeed 3 N well- defined d e n s i t y f l u c t u a t i o n s appear t o e x i s t i n Rb [ 1 3

1.

Ifowever, r e c e n t l y [ 1 4

1,

t h e

40

(5)

C8-602 JOURNAL DE PHYSIQUE

p r e s e n c e of t r a n s v e r s e v i b r a t i o n s h a s been e s t a b - [ 1 0 1 A s h c r o f t , N. W., S t r o u d , D . , S o l i d S t a t e l i s h e d i n a few l i q u i d m e t a l s y s t e m s . Perhaps, i n P h y s i c s 33, F. S e i t z , D . T u r n b u l l and an improved t h e o r y , t h e phonons a r e of E i n s t e i n

c h a r a c t e r and t h e p o l a r i z a t i o n i s unimportant.

Under such circumstances t h e s i z e b u t not t h e c h a r a c t e r of t h e h i g h e r f r e q u e n c i e s would be i m p o r t a n t .

I n t h e p r e s e n t work we have l e f t such matters a s i d e , however. Our aim has been t o c l a r i f y t h e r e l a t i o n s h i p between t h e PY method, i n i t s unmodified form, and t h e packing method and t h i s was a c h i e v e d i n s e c . 4.

&k~o~l-e_d-gements I . 0 . and I .Y. a r e g a t e f u l t o Prof. Satoh of t h e N a t i o n a l Defense Academy f o r h i s i n t e r e s t i n t h e p r e s e n t work. W.H.Y. acknowledges s u p p o r t from t h e Science Research Council and i s g r a t e f u l t o Prof. Cusack and D r S i l b e r t f o r h e l p - f u l d i s c u s s i o n s .

R e f e r e n c e s

---

[ 1 ) ^Young,^W. ^H., L i q u i d Metals 1976, Conf.

S e r i e s No.

32,

R . Evans and D. Greenwood ( ~ d s . ) ( I n s t . o f W y s . , B r i s t o l and ond don) 1977; P 1.

1_2

1

^Percus,^{J .}^{K . ,} Yevick, G. J . , Phys. 8ev.

110

- -

1-

³

-1

Gray, P . , Yokoyama, I . , Young, W . H., J .

~ h y s . F.

10

(1980) 197.

1:

⁴] ^Faber,^{T .}^E.,I n t r o d u c t i o n t o t h e Theory o f L i q u i d Metals (Cambridge U n i v e r s i t y P r e s s , L o n b n ) 1972.

[ 5

3

^Evans, ^{R . ,} S h i m c h e r , W., J . Phys. C

11

(1978) 2437.

1 6

1

T h i e l e , E., J. Chem. Phys. 2 (1963) 474.

117

1

^Wertheim,^M. S . , Phys. Rev, L e t t .

10

(1963) 321.

[

8

1

^Carnhan,^N. ^F.,S t a r l i n g , K. E., J. Chem.

phys

. 51

(1969 635.

1 9

1

E g e l s t a f f , P . , Rep. Progr. Phys. ?_9_ (1966)

H. B r e n r e i c h ( ~ d s . ) ( ~ c a d e m i c P r e s s , New York) 1978; p. 1.

Ir11

1

^Bratby,^{P . ,} G a s k e l l , T . , March, N. H . , Phys.

and Chem. of L i q s . 2 ⁽¹⁹⁷⁰⁾53.

[ 1 2 ] Edwards, D. J . , J a r z y n s k i , J . , J. Phys. C 2

(1972) 1745.

[ ^{1 3}

?_

^Copley,^{J .}^{R .}^{D . ,} Lovesey, S. W., L i q u i d Metals 1976, Conf. S e r . No

30,

R . Evans and D. Greenwood ( ~ d s . ) ( 1 n s t . o f Phys., B r i s t o l and on don) 1977; TI. 575.

] I 1 4 ] J a c u c c i , G., McDonald, I . R., Mol. Phys. 2

(1980) 515.

L15

1

^Waseda,^{Y .} 1979 The s t r u c t u r e of Non

C r y s t a l l i n e M a t e r i a l s , L i q u i d s and Amorphous S o l i d s , (New York : McGraw-Hill)

Waseda, Y . and Jacob, K.T., 1980 P h i l . Mag.

t o appear.