STRUCTURES OF AMORPHOUS MATERIALS AND SPECIFIC VOLUME VARIATIONS VERSUS THE TEMPERATURE

(1)

HAL Id: jpa-00222447

https://hal.archives-ouvertes.fr/jpa-00222447

Submitted on 1 Jan 1982

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are published 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.

STRUCTURES OF AMORPHOUS MATERIALS AND SPECIFIC VOLUME VARIATIONS VERSUS THE

TEMPERATURE

J. Sadoc, R. Mosseri

To cite this version:

J. Sadoc, R. Mosseri. STRUCTURES OF AMORPHOUS MATERIALS AND SPECIFIC VOLUME VARIATIONS VERSUS THE TEMPERATURE. Journal de Physique Colloques, 1982, 43 (C9), pp.C9-97-C9-100. �10.1051/jphyscol:1982918�. �jpa-00222447�

(2)

JOURNAL DE PHYSIQUE

Colloque C9, supplément au n°12, Tome 43, décembre 1982 page C9-97

S T R U C T U R E S OF A M O R P H O U S M A T E R I A L S A N D S P E C I F I C V O L U M E V A R I A T I O N S V E R S U S T H E T E M P E R A T U R E

+ . * J.F. Sadoc and R. Mosseri

+Physique des Solides, V.P.S., Bâtiment 510, 91405, Orsay, France

*Physique des Solides, C.N.R.S., 1 Place A. Briand, 92190 Meudon-Bellevue, France

Résumé. - Une description systématique de la structure amorphe est présentée et utilisée pour expliquer les variations de volume spécifique dans les com- posés amorphes lors des variations de température.

Abstract. - A systematic structural description of amorphous materials is presented and used, to explain the variation of the specific volume of amorphous (or glassy) compounds versus temperature.

Introduction. - Thermal variation of the specific volume of elements or simple compounds generally exhibit a small increase with T in the solid state, a discon- tinuity at the melting point and an important increase for the liquid specific volume at higher temperatures. Nevertheless if the compound is in an amorphous state the specific volume variation curve is continuous and can be divided in two regions.

At low temperature the specific volume is a few percent higher than in the crystalline state. The two curves showing the variation of the crystal and the glass specific volume are approximately parallel.

In the second region, at higher temperatures, the.glass specific volume curve is in continuity with the undercooled liquid one. The narrow transition zone between

the two parts is generally explained in terms of a glass transition.

In this paper we interpret this behaviour with pure topological (structural) arguments.

Amorphous structures described as regular structures in curved spaces

In various papers (1 )(2 )(3 ) we have presented a new description for amorphous structures: amorphous structures are supposed to appear if a local order induced by local chemical binding is uncompatible with periodicity requirement (for example in the case of a 5-fold local symetry). But a motive of a given local order can be arranged regularly in a non-euclidean space having a constant curvature (spherical or hyperbolic space). A map of the curved space onto the euclidean space introduces topological defects in the regular structure and leads to a non- periodic structure. These defects can be of different types, for example internal surfaces or disclination lines associated with elastic distortions. These defects can be also described in terms of added volume, as they change the density while they allow a decrease of the space curvature.

The density of close packed structure

We limit our presentation to close packing structure but we think that it can be extented to other structure. It is well know that the crystalline close packing structure (f.c.c.) have a packing fraction p = 0.74. Nevertheless, since there is octahedral and tetrahedral holes in this structure we can imagine a more dense structure in which there is only tetrahedral holes. As the tiling of euclidean Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982918

(3)

C9-98 JOURNAL DE PHYSIQUE

space w i t h r e g u l a r t e t r a h e d r a i s impossible t h e r e i s n o t any p e r i o d i c s t r u c t u r e w i t h t h i s property. But such a r e g u l a r s t r u c t u r e can be described i n a s p h e r i c a l curved space ( 3 ) . I n t h i s case a l l holes are t e t r a h e d r a and the l o c a l coordina- t i o n polyhedronis a r e g u l a r icosahedron. This s t r u c t u r e i s an a p p r o p r i a t e s t a r t i n g

p o i n t t o r t h e d e s c r i p t i o n of mono-atomic amorphous m e t a l l i c s t r u c t u r e s . The packing f r a c t i o n of t h i s s t r u c t u r e i s equal t o 0.774 which i s g r e a t e r than t h e f.c.c. value

( 2 ) . But t h i s s t r u c t u r e i s defined i n a curved space. Decrease o f t h e curva- t u r e r e q u i r e s defects which lower the d e n s i t y . I n dense random packing o f hard spheres i t i s found e x p e r i m e n t a l l y ( 5 ) and t h e o r e t i c a l l y ( 2 ) t h a t t h e packing f r a c t i o n i s 0.63. I f atoms i n t e r a c t w i t h a more r e a l i s t i c p o t e n t i a l t h a n t h e hard sphere p o t e n t i a l , a1 lowing extension and compression o f t h e i n t e r a t o m i c distance, then p = 0.70 i s a more accurate value. For i n s t a n c e t h i s value i s observed a f t e r r e l a x a t i o n o f the hard sphere packing w i t h a Lennard-Jones p o t e n t i a l ( 6 ) .

S p e c i f i c volume v a r i a t i o n vs temperature i n terms o f t h e curved space d e s c r i p t i o n There are numerous s p e c i f i c volume data concerning metals. Here we use the copper as an example, b u t t h e copper r e s u l t s can be extented t o o t h e r metals. The speci- f i c volume v a r i a t i o n given by d i f f e r e n t authors ( 7 ) ( 8 ) i s presented on f i g . l a f o r both the l i q u i d and t h e f.c.c. copper. The data are normalized t o t h e atomic volume a t T = OK. With t h i s u n i t V ( 0 ) = 1/0.74 f o r the c r y s t a l l i n e s p e c i f i c volume a t 0 K t a k i n g i n t o account t h e f.c.c. packing f r a c t i o n p = 0.74. The dashed curve shows t h e e x t r a p o l a t i o n o f t h e s p e c i f i c volume o f the l i q u i d down t o low temperature. As y e t observed by Kauzmann i n i t s o r i g i n a l paper the l i q u i d appears t o s t r i v e f o r a s m a l l e r s p e c i f i c volume than t h e c r y s t a l a t temperatures w e l l above 0 K.

The e x t r a p o l a t e d s p e c i f i c volume f o r t h e l i q u i d copper tends a t 0 K t o the value v ~ ( 0 ) = 1/0.769 ( w i t h t h e same u n i t as Vc(0)).

I t i s s t r i k i n g t h a t t h i s value i s very close t o t h e s p e c i f i c volume o f t h e s t r u c t u r e d e f i n e d as t h e i d e a l dense s t r u c t u r e i n curved space v1 = 1/0.774 ( t a k i n g i n t o account t h e above mention packing f r a c t i o n p = 0.774). T h i s behaviour observed f o r copper i s a l s o present f o r argon o r some o t h e r simple metals.

-

The l i q u i d s p e c i f i c volume

The behaviour of t h e l i q u i d s p e c i f i c volume, and o f i t s e x t r a p o l a t i o n t o low tempe- r a t u r e can be explained using the h y p o t h e t i c a l s t r u c t u r e above defined i n a curved space. I n t h i s s t r u c t u r e , we suppose an increase o f t h e number o f d e f e c t s when the temperature increases. This d e f e c t d e n s i t y v a r i a t i o n i s responsible f o r the regu- l a r v a r i a t i o n o f the s p e c i f i c volume from 0 K t o t h e l i q u i d s t a t e temperature. But i n t h i s d e s c r i p t i o n t h e s t r u c t u r e remains i n a curved space. I f we suppose t h a t t h e defects can change the c u r v a t u r e (which i s t h e case f o r d i s c l i n a t i o n s o r i n t e r - n a l surfaces), an increase o f t h e temperature i s associated w i t h a decrease o f the curvature. For a given temperature To t h e number o f d e f e c t s i s s u f f i c i e n t t o achie- ve a complete decrease o f the c u r v a t u r e a n d t h e a c t u a l s t r u c t u r e belongs t o t h e euclidean space. For temperature over To t h e increase o f t h e number of d e f e c t s does n o t change t h e c u r v a t u r e b u t correspond t o a change o f the volume a l l o w i n g d i f f u s i o n process and l e a d i n g t o a n u s u a l l i q u i d behaviour. Under To t h e change o f t h e volume i s associated w i t h a decrease o f t h e space curvature,over To t h e "added volume"

can be c a l l e d " f r e e volume". N o t i c e t h a t i s i t o n l y over To t h a t the described s t r u c t u r e can be p h y s i c a l l y observed.

- The glass ( o r amorphous) s p e c i f i c volume

Looking a t an h y p o t h e t i c a l amorphous copper we can e a s i l y p r e d i c t i t s s p e c i f i c volume v a r i a t i o n s u s i n g s t r u c t u r a l arguments. Suppose the s t r u c t u r e t o be a dense random packing o f sphere. As we s a i d above t h i s s t r u c t u r e can be described by mapping a curved s t r u c t u r e onto t h e euclidean space. The packing f r a c t i o n ( p = 0.70 w i t h s o f t i n t e r a c t i o n ) o f t h i s s t r u c t u r e leads t o a reasonable value o f the s p e c i f i c VO-

lume of t h e amorphous s t a t e a t T = 0 K t o v ~ ( 0 ) = 1/0.70. As pure amorphous copper i s n o t e x p e r i m e n t a l l y observed t h e r e i s no s p e c i f i c volume measurements b u t t h i s value i s o f t h e o r d e r o f what i s observed i n most amorphous m e t a l l i c a l l o y s .

(4)

The thermal variation of the volume i n the solid s t a t e does not depend on topological change,but only on the asymmetry of the pair interaction between atoms.It i s the reason why we suppose the same variation f o r amorphous and the c r y s t a l l i n e

s p e c i f i c volume. From these hypotheses i t r e s u l t s t h a t the amorphous s p e c i f i c volume s t a r t s from VG ( 0 ) = 1/0.70 a t T = 0 K and slowly increases l i k e the c r y s t a l - l i n e s p e c i f i c volume.

Comparison between the liquid and the glass s p e c i f i c volume

The experimental liquid copper s p e c i f i c volume versus temperature extrapolated f o r the undercooled l i q u i d , i s ploted on the f i g . l b . The estimated variation f o r the hypothetical amorphous copper i s a l s o presented. These two curves showing the l i - quid and the amorphous ( o r glassy) behaviour intercept f o r the temperature To.

Indeed a t t h i s temperature the amorphous s o l i d s t a t e and the l i q u i d s t a t e are described by the same model: the curved space model w i t h exactly the optimum number of defects allowing a complete decrease of the curvature to zero.

Conclusions and discussions

In numerous papers the density variations of the liquid and amorphous materials a r e explained by an extrapolation of the liquid density t o low temperature, w i t h a change in the slope of the curve when the extrapolated density reaches the c r y t a l l i n e one ( 9 ) ( 1 0 ) . This ideal case occuring only a t an extremely low cooling s t a t e . Consequently i n a l l these models an ideal amorphous s t r u c t u r e with a density equal t o the c r y s t a l l i n e s t r u c t u r e i s (more o r less i m p l i c i t l y ) supposed. In other terms i t is possible t o go continuous~y from a real amorphous s t r u c t u r e t o the c r y s t a l l i n e one.

In the present model we a l s o consider ideal amorphous s t r u c t u r e s . B u t two ideal s t r u c t u r e s are defined. The f i r s t one i s a perfect regular structure (with the amorphous local order) defined i n a space of a given curvature. The second one, i s the ideal s t r u c t u r e obtained by mapping the l a s t one onto the euclidean space with an optimum (minimum) density of defects. These structures a r e completely d i f - f e r e n t from the c r y s t a l l i n e one, as they have not the same local order. The struc- t u r e defined in euclidean space a l s o d i f f e r from a crystal because i t contains ne- cessarily defects. I t can be considered as the basic s t r u c t u r e t o which a l l real s t r u c t u r e s can be compared. The existence of such s t r u c t u r e i s related t o a very low cooling r a t e . However such a cooling r a t e would not prevent c r y s t a l l i z a t i o n . Consequently real amorphous s t r u c t u r e s contain a number of quenched defects greater than the optimum value resulting from the f i n i t e cooling r a t e .

Extrapolation of the liquid s t r u c t u r e t o very low temperature leads t o the ideal s t r u c t u r e defined in curved space. A similar approach has been already used t o explain q u a l i t a t i v e l y the density variation in a Si-H compound versus the H content (11). The i n t e r c e p t of the two curves showing glass and liquid s p e c i f i c volume variations allow the definition of an ideal glass t r a n s i t i o n temperature To independantly of the crystal l i n e s o l i d s t a t e . In a recent paper Jackle (12)gives two formulations of the glass: glass as a quenched liquid or glass as a disordered so- l i d . The present description associates these two formulations since the two models describing the glass s t r u c t u r e and the liquid s t r u c t u r e a r e identical a t the temperature To. Notice t h a t t h i s approach i s , i n a topological sense, close t o the Edwards(l3,14)approach using dislocation a s a parameter which i s frozen i n t h e amorphous s t a t e . Here the parameter i s related t o an other kind of defect ( f o r example i t can be disclination with a parameter q defined as the length of disclinaiion i n a unit volume). Nevertheless the Edwards approach using a dislocation model f o r the liquid and the amorphous s t a t e implicitly suppose the continuity between the c r y s t a l l i n e and the amorphous s t a t e i n contrast with the present model. Following our hypothesis i t i s easy t o understand why the " q " parameter i s frozen: i t i s due to absolute necessity f o r the s t r u c t u r e to belong t o the euclidean space which requires a minimum number of defects.

This description of the amor~hous and the liquid s t a t e gives an answer t o the Kauzmann paradox, "which would imply the existence of some End of s t a t e of h-igh

(5)

C9-100 JOURNAL DE PHYSIQUE

order for the liquid at low temperature which differs from the normal crystaZZine state" ( W . K . )

.

We have i n t h i s paper used t h i s approach t o e x p l a i n t h e s p e c i f i c volume behaviour o f a s i m p l e metal i n l i q u i d and amorphous s t a t e . B u t i t i s p o s s i b l e t o extends ^{i t} t o o t h e r p r o p e r t i e s ( s p e c i f i c heat, compressibi 1 i t y , e t c . .

.

⁾o r t o o t h e r elements o r compounds.

Fig.1-a) V a r i a t i o n o f t h e copper s p e c i f i c volume f r o m experimental r e s u l t s f o r t h e l i q u i d ( R ) and t h e c r y s t a l l i n e ( c ) s t a t e .

b ) t x t r a p o l a t i o n t o low temperature o f t h e l i q u i d v a r i a t i o n . T h i s v a r i a t i o n corresponds t o an under c o o l e d l i q u i d ( u ) . k s t i m a t i o n o f t h e s p e c i f i c volume v a r i a - t i o n f o r an i d e a l g l a s s y copper ( g ) . The u n i t f o r volume i s t h e volume o f a s p h e r i - c a l atom o f copper. p i s t h e p a c k i n g f r a c t i o n .

BIBLIOGRAPHY

(l)M.Kleman, J.F.Sadoc, J . de Phys. L e t . 40 (1979) L 569.

(2)J.F.Sadoc, J. Non C r y s t . S o l . 44 ( 1 ~ 8 1 F 1 .

(3)J.F.Sadoc, R.Mosseri, P h i l .Mag. B, 45 ( 1 5 8 2 ) , 4 6 7 .

(4)H.S.M. L o x e t e r , Regular Polytopes, Dover p u b l i c a t i o n s New York (1973).

(5)J.L.Finney, Proc. R. Soc. A 319 (1970) 4/9.

( 6 ) F . L a n ~ o n , L - B i l l a r d , J.Laugier, A.Chamberod, J. Phys. F.: Pet.Phys. 12 (1982)259 (7)K.Bornemann and Sauerwald, Z. M e t a l l . 14 (1922) 145.

(8)A. t l Mehairy and R.G. Ward, Trans. Met. Soc. AIFIE Z27 (1963) 1228.

(9)W. Kauzmann, Chem. Rev. 43 (1348) 219.

(lO)C.T. Moynihan, A.J. E a s t e l and M.A. De B o l t , J. o f t h e Am. Cer. Soc. 59 (1976)12 (ll)J.F.Sadoc, R. Mosseri, Jour. de Phys. C4, sup.nO 10, 42 (1981) C4-189.

(12)J. JSckle, P h i l . Mag. B, 44 (1981) 533.

(L3)S.F. Edwards, Polymer 17 (1976) 933.

(14)N. R i v i e r , J o u r . de Phys. C6, sup.nV 8, 39 (1978) C6-984.