THE STRUCTURE OF AMORPHOUS SOLIDS - A PERSPECTIVE VIEW

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THE STRUCTURE OF AMORPHOUS SOLIDS - A

PERSPECTIVE VIEW

P. Gaskell

To cite this version:

JOURNAL

DE PHYSIQUE

Colloque C8, suppldment au n012, Tome 46, ddcembre 1985 page C8-3

THE STRUCTURE OF AMORPHOUS SOLIDS

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A PERSPECTIVE VIEW

P.H. Gaskell

Cavendish Laboratory, University o f Cambridge, MadingZey Road, Cambridge CB3 OHE, U . K .

Resume - On passe en revue des donn6es sur la structure d'un certain nombre de solides amorphes repr6sentatifs

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m6taux purs et semiconducteurs, alliages amorphes et verres d'oxydes

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et on se propose de sugggrer quels sont les modbles les plus adkquats. Historiquement, les empilements denses algatoires ou les r6seaux aleatoires ont 6t6 consid6res comme les modbles les mieux adaptks, surtout pour les solides amorphes de composition chimique simple

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m6taux purs et semiconducteurs, oxydes stoechiom6triques. Pour des matkriaux plus complexes, diffkrentes informations indiquent que les modbles al6atoires ne sont viables qu'au prix de certaines complications comme par exemple, l'existence d'un ordre chimique, l'existence dTunit6s structurales locales ou peut 8tre 5 moyenne distance. On discute l'exemple particulier des verres de m6tasilicates alcalins et on montre qu'il est difficile de constuire un modble al6atoire avec les contraintes idoines qui rende compte de l'ensemble des r6sultats exp6rimentaux

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polydrbdes de coordination d6finis aussi bien pourles cations alcalins que pour les atomes de silicium, tendence 3 la formation de chaines d1unit6s SiOj--, densit6 mesurge et en m8me temps fonctions de distribution de paires exp6rimentales. La question se pose alors de savoir si les modbles, bases sur des empilements periodiques on non et dont l'ordre est bris6 par des dgfauts, ne seraient pas plus appropri6s. On suggbre une strat6gie pour repondre 3 cette question.

Abstract - Structural data for a number of typical amorphous solids

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elemental metals and semiconductors, amorphous alloys and oxide glasses

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are surveyed and an attempt made to suggest the most appropriate models. Historically, random close-packed structures or random networks heve been considered to be the most appropriate, especially for chemically simple amorphous solids

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elemental metals and semiconductors, stoichiometric oxides. For more complex materials, there are a number of indications that random models are only tenable with some qualification

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to allow chemical ordering, local and perhaps medium-range structural units, for instance. The particular example of alkali metasilicate glasses is discussed and it is shown that it is difficult to devise a suitably constrained random model which incorporates all the known experimental facts

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defined coordination polyhedra for the alkali cations as well as the silicon atoms,

a

preference for Si03-- units connected in chains, the measured density, as well as the experimental pair distribution functions. Questions are therefore raised: do random models represent the most general paradigm for the structure of amorphous materials, or are defective ordered models based on periodic or aperiodic packing more appropriate? An attempt is made to suggest a strategy to answer such questions.

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INTRODUCTION

The i n v i t a t i o n t o present t h e opening t a l k of t h e t h i r d conference on t h e S t r u c t u r e of Non-Crystalline M a t e r i a l s r e p r e s e n t s a g r e a t honour, f o r which I am g r a t e f u l . There a r e d e f i n i t e r e s p o n s i b i l i t i e s involved i n giving an i n t r o d u c t o r y l e c t u r e , however: a requirement t o maintain a degree of balance and t o give a s t a t e s m a n l i k e account of t h e s e v e r a l p o s i t i o n s occupied by t h e proponents of various ' t h e o r i e s ' of t h e s t r u c t u r e of g l a s s e s . Secondly, a need t o e x t r a c t a coherent view of t h e r e s u l t s of a number of s t r u c t u r a l techniques and, t h i r d l y , t o p r e s e n t an account which allows each of u s working over a wide range of m a t e r i a l types t o f e e l t h a t h i s / h e r c o n t r i b u t i o n may be included i n an o v e r a l l scheme of s t r u c t u r a l o r g a n i s a t i o n . For t h e s e reasons I include both t h e backwards look and an attempt t o s e e over t h e parapet a l i t t l e way i n t o t h e f u t u r e

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i n what I d e s c r i b e a s a p e r s p e c t i v e view

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t r y i n g t o p i c t u r e r e a l i t y i n a l l i t s dimensions

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n o t n e c e s s a r i l y t h r e e , a s we s h a l l s e e l a t e r i n t h e conference.

To begin w i t h a d i g r e s s i o n . I n 1974, when t h e f i r s t conference i n t h i s s e r i e s was being planned under t h e auspices of t h e Society of Glass Technology one f e l t t h a t 'our approach t o an understanding of t h e s t r u c t u r e of t h e s e s o l i d s is evolving more r a p i d l y now than a t any e a r l i e r period i n t h e h i s t o r y of t h e s u b j e c t ' / I / . I n f a c t evolution was s o r a p i d t h a t one wondered whether t h e s t r u c t u r e of g l a s s e s might be solved before t h e conference was held i n 1976. There was no need t o worry: whole new f a m i l i e s of g l a s s e s have been developed or extended s i n c e then. The s u b j e c t h a s continued t o grow a t such a r a t e t h a t fragmentation i s now a more worrying f e a t u r e : t h r e e i n t e r n a t i o n a l conferences i n t h e space of f i v e weeks, g l a s s y metals, amorphous semiconductors, h a l i d e g l a s s e s have l a r g e l y gone t h e i r s e p a r a t e ways. The frequency of t h i s meeting has been doubled and t h e l a r g e number of papers o f f e r e d t o t h e conference r e p r e s e n t s an eloquent testimony t o t h e i n t e r e s t i n t h e s u b j e c t .

What a r e t h e reasons f o r t h i s i n t e r e s t ? F i r s t l y , t h e m a t e r i a l s we study

are

important with new i n d u s t r i e s evolving and o l d i n d u s t r i e s r e l i s h i n g i n a h i s t o r y reaching back t o t h e dawn of c i v i l i z a t i o n . Secondly, t h e s u b j e c t encapsulates a c r u c i a l , l a r g e l y unsolved mystery. Thirdly, t h e answer t o t h e mystery l i e s a t t h e h e a r t of any understanding of t h e n a t u r e of t h e transformations between l i q u i d s , g l a s s e s and c r y s t a l s .

I1

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APPROACHES TO THE STRUCTURE OF GLASSES (AND AMORPHOUS SOLIDS)

I n i t i a l l y , we l i m i t d i s c u s s i o n t o g l a s s e s quenched from t h e melt, r a t h e r than from t h e vapour. ( I t is not a s e r i o u s r e s t r i c t i o n s i n c e u l t r a r a p i d quenching by Q-switched l a s e r techniques allows a-Si t o be produced from t h e melt whereas i t had h i t h e r t o been prepared by vacuum d e p o s i t i o n ) . With t h i s r e s t r i c t i o n , we a r e f o r c e d t o recognize t h e c e n t r a l i t y of t h e g l a s s t r a n s i t i o n between t h e supercooled l i q u i d and g l a s s y s t a t e s . The importance of Turnbull's c r i t e r i a /2/ r e l a t i n g t o t h e avoidance of c r y s t a l l i z a t i o n , developed f u r t h e r by Davies /3/ and Uhlmann /4/,

becomes evident. Moreover, t h e apparently continuous, homogeneous n a t u r e of t h e transformation between t h e super-cooled l i q u i d and g l a s s y s t a t e s s u g g e s t s a need t o r e l a t e t h e s t r u c t u r e of t h e g l a s s a t room temperature t o a s e r i e s of l i q u i d s t a t e s t r u c t u r e s and, by i m p l i c a t i o n , a s i m i l a r i t y between t h e s t r u c t u r e s of g l a s s e s and l i q u i d s .

This then l e a d s n a t u r a l l y t o our f i r s t concepts of t h e s t r u c t u r e of g l a s s e s a s being a disordered arrangement of atoms

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i n accord with t h e s i m p l e s t views of l i q u i d s -

but with t h e p o s s i b i l i t y of i n t e r a t o m i c bonding a t l o c a l l e v e l l e a d i n g t o networks which may have a high v i s c o s i t y o r a l a r g e temperature c o e f f i c i e n t ( - arl/aT) l e a d i n g t o i n h i b i t i o n of nucleation.

I n simple s t o i c h i o m e t r i c g l a s s e s l i k e Si02, t h e concept of t h e (continuous) random network /5/ seems t h e most a p p r o p r i a t e r e a l i z a t i o n of t h e s e notions. For l e s s open s t r u c t u r e s - t h e a l k a l i s i l i c a t e s , f o r example

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a l k a l i ions a r e i n s e r t e d i n t o s u i t a b l e spaces i n t h e network with l o c a l charge n e u t r a l i t y maintained by

I n close-packed systems such a s t h e m e t a l l i c a l l o y s , a s i m i l a r conceptual r a t i o n a l i z a t i o n is p o s s i b l e : close-packing of metal atoms l e a d s t o l o c a l t e t r a h e d r a l o r d e r i n g - a s opposed t o t e t r a h e d r a l - o c t a h e d r a l ordering i n cubic c l o s e packing and t h e view of t h e random close-packed s t r u c t u r e , a s s o c i a t e d with i d e a s exposed by Bernal /6/, emerges a s t h e model f o r l i q u i d and g l a s s y monoatomic metals. Binary, t e r n a r y a l l o y s e t c a r e accommodated e i t h e r by ' s t u f f i n g ' t h e a d d i t i o n a l components i n i n t e r s t i t i a l c a v i t i e s , a s i n t h e model proposed by Polk / 7 / , o r a s random mixtures of t h e two elements.

Disorder or randomness appears i n another g u i s e i n t h a t it is assumed t h a t t h e s t r u c t u r e corresponds not only t o t h e mean chemical composition but a l s o includes f l u c t u a t i o n s around t h a t mean. A p a r t i c u l a r example

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a g l a s s of d i s i l i c a t e composition might t h e r e f o r e be represented by a network containing d i s i l i c a t e groups =Si-0- with one non-bridging oxygen ion (NBO)

and

s i g n i f i c a n t f r a c t i o n s of S i atoms with two or more N B O ' s , and t h o s e with none ( f i g . 1 ) .

Fig. 1

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Schematic r e p r e s e n t a t i o n of 'chemical' and 'compositional' o r d e r i n a g l a s s of d i s i l i c a t e composition. a ) The d i s i l i c a t e anion with 3 bridging oxygens (Q3

i n NMR terminology). b) A

p o s i t i o n a l l y - d i s o r d e r e d but compo- s i t i o n a l l y - o r d e r e d network. P o s i t i o n a l l y and compositionally- disordered networks a r e shown i n c ) and d ) : c ) shows segregated Q~ and

Q* regions whereas d ) r e p r e s e n t s a random mixture of Q4-Q1 anions.

These two viewpoints thus lead t o t h e notion of disordered p o s i t i o n a l and compositional c o r r e l a t i o n s a s expressed, f o r i n s t a n c e , i n t h e s e p a r a t e Bhatia- Thornton s t r u c t u r e f a c t o r s /8/ o r , what is e a s i e r t o v i s u a l i z e , t h e f o u r i e r t r a n s - form

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the reduced r a d i a l compositional and p o s i t i o n a l c o r r e l a t i o n f u n c t i o n s / 9 / .

I n any l o g i c a l development of t h e s u b j e c t , t h i s simple - s i m p l i s t i c , even

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s e t of concepts, r e p r e s e n t s t h e c o r r e c t s t a r t i n g point: Occam's r a z o r , a f t e r a l l

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explanatory p r i n c i p l e s should not be needlessly m u l t i p l i e d . C e r t a i n l y , f o r simple g l a s s e s , t h e successes of t h e random models a r e based u l t i m a t e l y on observation not dogma

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t h e i d e a s do r e p r e s e n t and explain a l a r g e proportion of experimental f i n d i n g s . A s t h e s u b j e c t has evolved however, an i n c r e a s i n g number of f a c t s have emerged which f i t i n t o t h e e x i s t i n g framework only a f t e r modification and refinement and even then uncomfortably. I t is perhaps time t o review t h e s i t u a t i o n a t l e a s t , and p r e s e n t t h e a l t e r n a t i v e s f o r examination and t o suggest a s t r a t e g y f o r f u r t h e r progress.

C8-6 JOURNAL

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d e l i b e r a t e l y and t h e i m p o r t a n t i n v e s t i g a t i o n s of P h i l l i p s , Boolchand and o t h e r s a r e d e s c r i b e d , by t h e l a t t e r , e l s e w h e r e i n t h i s volume.

111.1 E l e m e n t a l amorphous t e t r a h e d r a l s e m i c o n d u c t o r s One of t h e more i m p o r t a n t c h a r a c t e r i s t i c s o f t h i s group o f m a t e r i a l s is t h e a v a i l a b i l i t y o f good e x p e r i m e n t a l 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 s - d e r i v e d from X-ray / 1 0 / , / 1 1 / and n e u t r o n /12/ s t r u c t u r e f a c t o r s i n t h e c a s e o f a-Ge. T h i s l a t t e r m a t e r i a l h a s b e e n e x t e n s i v e l y i n v e s t i g a t e d and s e v e r a l CRN models, and some amorphous c l u s t e r models have been d e v i s e d . P e r h a p s t h e f a i r e s t summary o f t h e s i t u a t i o n h a s been g i v e n by E t h e r i n g t o n e t a l . /12/ who compared e x p e r i m e n t a l d a t a w i t h p r e d i c t e d r d f s f o r most models and c o n c l u d e d t h a t no model a d e q u a t e l y r e p r o d u c e s e x p e r i m e n t a l d a t a t o w i t h i n e x p e r i m e n t a l a c c u r a c y . N o n e t h e l e s s CRN models a r e u s u a l l y r e g a r d e d as g i v i n g t h e most i n f o r m a t i v e r e p r e s e n t a t i o n o f s t r u c t u r a l p r o p e r t i e s and t h e y a l s o p r o v i d e a s u c c e s s f u l b a s i s f o r t h e c a l c u l a t i o n of e l e c t r o n i c p r o p e r t i e s and v i b r a t i o n a l s p e c t r a . A r e c e n t a d d i t i o n t o t h e l i t e r a t u r e is t h e computer-generated model r e p o r t e d by Wooten, Winer and Weaire /13/ i n which a p e r i o d i c 216-atom c e l l w i t h t h e diamond c u b i c s t r u c t u r e is modified by i n t e r c h a n g i n g atoms f o l l o w e d by some r e l a x a t i o n ( w i t h changes i n t o p o l o g y a l l o w e d ) . The p r o c e s s is s t o p p e d when t h e f i t t o t h e e x p e r i m e n t a l r d f is judged t o be s a t i s f a c t o r y . The f i t o b t a i n e d is good b u t a s y e t o n l y a l i m i t e d d e s c r i p t i o n h a s been g i v e n of t h e d e t a i l s of t h e more s u c c e s s f u l s t r u c t u r e s : t h e p r e s e n c e o f 7 - f o l d r i n g s a p p e a r s t o b e a c r u c i a l i n g r e d i e n t .

I t is a l s o worth n o t i n g t h a t no a d e q u a t e model e x i s t s f o r a-C. I n d e e d t h e r e l a t i v e i m p o r t a n c e of t r i g o n a l l y - and t e t r a h e d r a l l y - b o n d e d carbon h a s y e t t o b e unambiguously e s t a b l i s h e d . One of t h e major o b s t a c l e s t o p r o g r e s s is t h e v a r i a b i l i t y o f t h e s t r u c t u r e of a-C a s a f u n c t i o n of p r e p a r a t i o n c o n d i t i o n s

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t h e ' i d e a l ' amorphous s t a t e of t h i s s y s t e m h a s y e t t o b e e s t a b l i s h e d .

111.2 E l e m e n t a l amorphous m e t a l s Data e x i s t s f o r amorphous Fe, Co and C r /14/ p r e p a r e d by c o n d e n s a t i o n on He-cooled s u b s t r a t e s and examined below t h e i r c r y s t a l l i z a t i o n t e m p e r a t u r e s (- 100 K ) . A more r e c e n t s t u d y h a s been made by L a u r i a t /15/ of t h e s t r u c t u r e of a-Fe p r e p a r e d by d e c o m p o s i t i o n of Fe(C0)tj. The r e s u l t i n g powder h a s some c a r b o n - c o n t a i n i n g i m p u r i t i v e s

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i t is s u g g e s t e d t h a t l e s s t h a n 1 % carbon is c o n t a i n e d i n t h e bulk a-Fe, t h e remainder (- 7%) b e i n g a s u r f a c e i m p u r i t y . The X-ray and n e u t r o n - d e r i v e d p a i r f u n c t i o n s a r e e s s e n t i a l l y i d e n t i c a l and t h e peaks a r e c o n s i s t e n t w i t h a p o l y t e t r a h e d r a l s t r u c t u r e ( f i g . 2 ) . Note i n p a r t i c u l a r t h e a b s e n c e o f a J2 d i s t a n c e which would c o r r e s p o n d t o t h e d i a g o n a l o f a n o c t a h e d r o n .

F i g . 2 - X-ray r d f f o r a-Fe a f t e r L a u r i a t /15/.

can be shown t o give an adequate r e p r e s e n t a t i o n of experimental d a t a , none a r e exact and t h e d i f f e r e n c e s do not allow p r e f e r r e d r i n g s t a t i s t i c s e t c t o be e s t a b l i s h e d . Amorphous c l u s t e r models a r e about equally s u c c e s s f u l . I n 1982 i t was p o s s i b l e t o w r i t e I . . . our knowledge of t h e s t r u c t u r e of v i t r e o u s s i l i c a , a m a t e r i a l f i r s t

s t u d i e d e x t e n s i v e l y over 50 y e a r s ago, is e s s e n t i a l l y incomplete.' /16/.

For B203, t h e weight of evidence has s h i f t e d s o t h a t i t now seems t h a t t h e l o c a l s t r u c t u r e contains a high proportion of boroxol r i n g s

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a s o r i g i n a l l y suggested on t h e b a s i s of 1 . R and Raman d a t a by Krogh-Moe / I T / . Apart from s e v e r a l modelling s t u d i e s / l a / , t h e 1 1 ~ NMR d a t a of J e l l i s o n e t a 1 /19/ i n d i c a t e s t h e presence of two d i f f e r e n t oxygen s i t e s which a r e c o n s i s t e n t with bridging oxygens within t h e boroxol r i n g and t o e x t e r n a l B03 t r i a n g l e s . (There might be some d i s c u s s i o n over t h e e x t e n t t o which t h e boroxol r i n g r e p r e s e n t s l o c a l o r medium-range ordering.)

I n amorphous alumina, l i k e a-C, questions a r i s e over t h e coordination s t a t e of A 1

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t e t r a h e d r a l or octahedral or a mixture. EXAFS and EELS near edge s t u d i e s y i e l d t h e A1-0 bond l e n g t h from which t h e r e l a t i v e proportion of 4- and 6-coordinated A 1 can be estimated /20/.

111.4 Binary amorphous a l l o y s The e x t e n t of p o s i t i o n a l and compositional o r d e r i n t h e s e m a t e r i a l s appears t o be very v a r i a b l e . For many a l l o y s Bhatia-Thornton p a r t i a l s t r u c t u r e f a c t o r s have been obtained and i n d i c a t e chemical ordering of t h e atomic s p e c i e s - f o r o t h e r s t h e o r d e r i n g i s r e l a t i v e l y i n s i g n i f i c a n t . P o s i t i o n a l o r d e r may be diagnosed i n an approximate manner by measuring t h e breadth, AQ, of t h e f i r s t d i f f r a c t i o n peak and, a s with line-broadening i n c r y s t a l s , an e f f e c t i v e c o r r e l a t i o n l e n g t h

5

= 2*/AQ may be obtained. Values range from about 8

8

i n t h e l i q u i d t o 12

1

i n g l a s s y NiglBlg /21/ (which, a s shown below, appears t o be well-ordered on t h e b a s i s of p a r t i a l s t r u c t u r e f a c t o r s ) t o about 18

1 f o r s e v e r a l

magnesium-based a l l o y s (Mg-Cu, MgNi and MgCa). I n some a l l o y s of t h i s type, an e f f e c t i v e "ordering p o t e n t i a l " can be c a l c u l a t e d /22/, and t h i s is p a r t i c u l a r l y important a s t h e degree of chemical o r d e r can then be q u a n t i f i e d .

Accurate p a r t i a l s a r e a v a i l a b l e f o r t h e t r a n s i t i o n metal-metalloid g l a s s e s and t h e g r e a t e r information content of t h e s e d a t a allows more r e f i n e d i n t e r p r e t a t i o n . Broadly t h e evidence a v a i l a b l e from neutron s c a t t e r i n g and NMR d a t a i n d i c a t e s almost complete chemical ordering: B is surrounded p r e f e r e n t i a l l y by N i i n a-Ni81Blg with a f i r s t neighbour s h e l l of 9 atoms and approximately t r i g o n a l p r i s m a t i c c o o r d i n a t i o n (6 a t t h e v e r t i c e s and a f u r t h e r 3 atoms capping square f a c e s ) / 2 3 / .

Elsewhere, I have argued t h a t t h i s apparently secure experimental f a c t

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a well-defined l o c a l s t r u c t u r e f o r t h e m e t a l l o i d p r e s e n t s considerable d i f f i c u l t i e s 2 I t has not yet been e s t a b i l i s h e d t h a t t h e r e a r e any cogent reasons f o r t h e s t a b i l i t y of an i n d i v i d u a l t r i g o n a l p r i s m a t i c u n i t

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on c h e m i c a l grounds ( i . e . s t r o n g d i r e c t i o n a l bonding a s i n oxides) or from s p a c e - f i l l i n g c o n s i d e r a t i o n s ( u s i n g a c e n t r a l f o r c e f i e l d ) . The most p l a u s i b l e reason f o r t h i s p a r t i c u l a r l o c a l s t r u c t u r a l arrangement may be t h a t i t is a p a r t of a much l a r g e r o r g a n i z a t i o n a l scheme which extends t o 10

1

a t l e a s t and thus can be considered t o be a medium-range s t r u c t u r a l e f f e c t . B r i e f l y , t h e argument is t h a t f o r a well-defined

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Fig. 3

-

A t r i g o n a l p r i s m a t i c u n i t c o n s i s t i n g of 9 metal atoms ( l a r g e c i r c l e s ) surrounding a m e t a l l o i d (small f i l l e d c i r c l e s ) must be e x t e n s i v e l y connected t o o t h e r prisms by edge and vertex-sharing i n t y p i c a l m e t a l l i c g l a s s e s with about 20 a t . $ of t h e metalloid. I f t h e neighbouring t r i g o n a l prisms a r e packed e f f i c i e n t l y , only t e t r a h e d r a l and half o c t a h e d r a l i n t e r s t i c e s e x l s t . The p o s s i b l e p o s i t i o n s of neighbouring l o c a l s t r u c t u r a l u n i t s a r e s e v e r e l y c o n s t r a i n e d - a s t h e s h e e t of prisms i n d i c a t e s . Note t h a t a s i m i l a r s h e e t of prism e x i s t s above and below t h e plane linked by capping atoms such a s t h e one shaded. The r e s u l t is t h a t f o r a well-defined t r i g o n a l prism t o e x i s t a t a l l , t h e medium-range s t r u c t u r e is a l s o defined w i t h i n r e l a t i v e l y narrow l i m i t s .

This e s s e n t i a l l y s p e c u l a t i v e argument is (more s e c u r e l y ) supported by t h e o s c i l l a t i o n s i n t h e 8-B p a r t i a l p a i r d i s t r i b u t i o n f u n c t i o n ( r e p r e s e n t i n g t h e c o r r e l a t i o n s between t h e c e n t r e s of t r i g o n a l prismatic c l u s t e r s ) which extend t o a t l e a s t 12

4 .

This and o t h e r experimental d a t a , and t h e f a c t t h a t a random sphere packing model of Lamparter e t a l , /21/ does not reproduce t h a t d a t a , have prompted t h e examination by Dubois, Gaskell and Le CSer of a domain model f o r TM-M a l l o y s based on packed domains of p o s i t i o n a l l y - c o r r e l a t e d atoms with c a r e f u l l y p r e s c r i b e d i n t e r f a c i a l s t r u c t u r e s /25/. B r i e f l y t h e argument is t h a t an a c c e p t a b l e i d e a l s t r u c t u r e should be based on l o c a l s t r u c t u r a l p r i n c i p l e s which a r e maintained f o r a l l atoms

-

even i n t h e boundaries between domains. The r e s u l t s of t h i s model a r e i n good agreement with experiment.

Another s u c c e s s f u l model is t h e , o s t e n s i b l y random, molecular dynamics s i m u l a t i o n of Beyer and Hoheisel /26/. Experimental r d f s a r e well represented - what is not s o c l e a r is what t h e f i n a l s t r u c t u r e looks l i k e . A preliminary examination (3-.M. Dubois and t h e a u t h o r ) suggests t h a t t r i g o n a l prisms e x i s t but with no evidence a s y e t of a d i s c e r n a b l e medium-range o r d e r /24/.

111.5 Alkali s i l i c a t e g l a s s e s A random model f o r t h e s e g l a s s e s s u g g e s t s t h e following c h a r a c t e r i s t i c s .

a ) P o s i t i o n a l d i s o r d e r

-

absence of geometrically-ordered medium-range u n i t s b u t with t h e p o s s i b l i t y o f l o c a l o r g a n i z a t i o n a s i n o t h e r g l a s s e s .

d i s t r i b u t i o n of a l k a l i c a t i o n s .

c ) Chemical d i s o r d e r

-

e s p e c i a l l y t h e presence of a v a r i e t y of a n i o n i c groups. Recent evidence has thrown some l i g h t on t h e assumptions which underly t h e s e c h a r a c t e r i s t i c s .

111.5.1 The l o c a l s t r u c t u r e of network-modifying i o n s I n common parlance, t h e silicon-oxygen groups form I - , 2- o r 3-dimensional networks, depending on composition, and t h e o t h e r atoms 'modify' t h e c o n n e c t i v i t y by breaks i n t h e chain of Si-0-Si bonds. The Si-0 bond is o f t e n regarded a s t h e s t r o n g e s t i n t e r a c t i o n -

l a r g e l y covalent, perhaps - whereas t h e modifier, M-0 l i n k i s considered t o be weak

- o f t e n i o n i c . The s i l i c o n environment i s , i n f a c t , found t o be l a r g e l y u n d i s t o r t e d but t h e environment of many modifier atoms has remained a matter f o r s p e c u l a t i o n u n t i l r e c e n t l y . EXAFS and n e u t r o n - s c a t t e r i n g (with i s o t o p i c s u b s t i t u t i o n ) now allows a much l e s s ambiguous i n t e r p r e t a t i o n of t h e experimental d a t a .

The r e s u l t s a r e i n t e r e s t i n g . Greaves e t a 1 /27/ have shown t h a t t h e environment of Na i n Na2Si205 g l a s s is r e l a t i v e l y well-defined with a coordination number of 5 a s i n t h e d i s i l i c a t e c r y s t a l . Examination of the Ca edges of a number of g l a s s e s and c r y s t a l s again shows a 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 d i s t r i b u t i o n s f o r c r y s t a l s and g l a s s e s - even with s u b s i d i a r y shoulders on t h e f i r s t peak - e q u i v a l e n t t o those i n c r y s t a l s . The coordination number is 7 and t h e s t a t i c breadth only a l i t t l e l a r g e r than t h a t of t h e c r y s t a l . The X-ray absorption near edge s t r u c t u r e (XANES) of Ca-containing g l a s s e s reported by Geere e t a1 /28/ shows t h a t t h e symmetry of t h e f i r s t - n e i g h b o u r s h e l l is l a r g e l y i n s e n s i t i v e t o t h e c o n c e n t r a t i o n and t o t h e occurrence of phase-separation.

More r e c e n t s t u d i e s of t h e environment of Ti i n K20.Ti02.2Si02 g l a s s e s have been r e p o r t e d by Yarker e t a 1 /29/ using a combination of neutron s c a t t e r i n g (with i s o t o p i c s u b s t i t u t i o n of T i ) EXAFS and XANES. The r e s u l t s confirm 5-fold coordination f o r Ti with a well-defined environment. The Ti-0 peak has two components a t 1.65 and 1.96 1( with weights 0.75 and 4.1 and with breadths given by a = 0.006

1

and 0.102

1

r e s p e c t i v e l y . The environment of T i is thus extremely well-defined

-

o values have been c o r r e c t e d f o r termination broadening

-

an e q u i v a l e n t f i g u r e f o r t h e Si-0 f i r s t peak i s o = 0.05

1).

There is evidence f o r a square planar c o n f i g u r a t i o n of oxygen atoms with one s h o r t bond perpendicular t o t h e plane, s i m i l a r t o t h e l o c a l o r g a n i z a t i o n observed i n c-Na2.TiSi05.

111.5.2 S p a t i a l d i s t r i b u t i o n of modifier atoms I n a number of g l a s s e s , t h e modifier atoms appear t o be randomly d i s t r i b u t e d . There a r e a l s o a number of cases where d i r e c t and i n d i r e c t methods suggest some degree of c l u s t e r i n g . The most r e c e n t r e s u l t by a d i r e c t method is contained by t h e work of Yarker e t a 1 /29/ on a-K20.Ti02.2Si02. By using a double d i f f e r e n c e method t h e T i - T i d i s t r i b u t i o n can be obtained and, although i n e v i t a b l y n o i s y , a f i r s t - n e i g h b o u r Ti-Ti peak is observed a t

3.4

1

( f i g .

4).

Fig.

4

-

Three d i f f e r e n t e s t i m a t e s of t h e T i - T i d i s t r i b u t i o n , T ( r ) , f o r v i t r e o u s K20.Ti02.2Si02 ( a f t e r Yarker e t a 1 /29/). Note t h e p o s i t i o n of t h e f i r s t peak a t about 3.4

1

suggesting Ti-Ti c l u s t e r i n g a t smaller d i s t a n c e s than t h e mean value c a l c u l a t e d f o r a s p a t i a l l y

2

4

10

random d i s t r i b u t J o n , shown by

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C a l c u l a t i o n s (by t h i s a u t h o r ) suggest t h a t a purely random d i s t r i b u t i o n of T i atoms i n a g l a s s of t h i s composition and d e n s i t y would lead t o a mean Ti-Ti d i s t a n c e of 6.1

1 (which corresponds t o t h e p o s i t i o n of a second

T i - T i peak i n t h e experimental d a t a ) . This r e s u l t is c o n s i s t e n t with e a r l y X-ray work by Milberg and P e t e r s /30/ who observed a composition-&dependent peak a t 3.8

-

3.9

1 i n t h a l l i u m - s i l i c a t e

g l a s s e s which they a s c r i b e t o T1-T1 p a i r s c l u s t e r i n g a t a p r e f e r r e d d i s t a n c e . L a t e r work on T1-containing g l a s s confirmed t h e presence of a defined environment f o r t h e

~ 1 ' ion. Panek and Bray /31/ have used 2 0 5 ~ 1 NMR t o show T1-T1 p a i r i n g even i n T1-dilute g l a s s e s . Recent EPR d a t a of Kawazoe and Takagi /32/ gave evidence f o r t h e presence of two types of ~ 1 + s i t e , t h e proportions of each being composition-dependent. These i n v e s t i g a t i o n s a r e important s i n c e i t is considered t h a t T 1 can probe t h e a l k a l i s i t e s i n g l a s s e s which a r e otherwise d i f f i c u l t t o examine.

111.5.3 V a r i a b i l i t y i n anion type The e x t e n t of chemical ordering i n s i l i c a t e s is

a matter of some debate. Recent Magic Angle 2 9 ~ i NMR r e s u l t s on glassy sodium s i l i c a t e s /33/ s u g g e s t s t h a t a t s t o i c h i o m e t r i c compositions

-

Si02, Na2Si205. Na2Si03

-

only one S i s i t e is observable ( f i g . 5 ) . Intermediate composition can be represented by a composition-dependent weighted sum of c o n t r i b u t i o n s a s c r i b e d t o S i atoms with 4,3 or 2 bridging oxygens (Qq, Q3, Q2 atoms). The i n d i v i d u a l s p e c i e s e x i s t only within a p p r o p r i a t e ranges: thus f o r compositions between Si02 and Na2Si05, only t h e Qq and Q3 s p e c i e s a r e observed and Q3 and Q2 from d i s i l i c a t e t o met; -.i I i c a t e compositions.

o i CHEMICAL SHIFT (PPM) Fig. 5

-

2 9 ~ i s p e c t r a obtained Magic Angle NMR f o r a s e r i e s sodium s i l i c a t e g l a s s e s : a ) 0-33 mol % Na20;

b ) 33-50 mol % Na20 a f t e r Dupree a 1 /33/.

This evidence implies a degree of chemical o r d e r i n g i n t h e g l a s s

-

a r e s u l t which i s

not a n t i c i p a t e d from t h e usual p i c t u r e s of t h e Zachariasen random network and is a t variance with t h e 'complex anion' models proposed by a number of a u t h o r s 1341.

111.5.4 Constrained randomness i n a model f o r a l k a l i s i l i c a t e s The d i s c u s s i o n above has i n d i c a t e d t h a t random models f o r a l k a l i s i l i c a t e s may need t o i n c o r p o r a t e a d d i t i o n a l c o n s t r a i n t s a s s o c i a t e d w i t h , t h e packing p a t t e r n of atoms which produce a n i d e n t i f i a b l e

M +

environment. Furthermore, t h e n a t u r e of t h e a n i o n i c s p e c i e s may be more c l o s e l y defined and r e l a t e d t o t h e s p e c i e s p r e s e n t i n t h e c r y s t a l than had been imagined previously.

This f a c t s u g g e s t s t h a t t h e packing of oxygen atoms is s i m i l a r i n t h e amorphous and c r y s t a l l i n e m e t a s i l i c a t e phases. I n c-Li2SiO3, / 3 5 / t h e oxygen s u b l a t t i c e is a d i l a t e d hcp s t r u c t u r e with S i and L i atoms occupying h a l f of t h e t e t r a h e d r a l i n s t e r s t i c e s and Si-0, Li-0 chains running p a r a l l e l t o t h e hexad a x i s - each Si-0-Si chain being surrounded by s i x Li-0-Li c h a i n s ( f i g . 6 ) . I n a model f o r t h e g l a s s , t h e hcp oxygen s u b - l a t t i c e is r e t a i n e d but t h e c h o i c e of t e t r a h e d r a l i n t e r s t i c e s is more random. I t cannot be completely random s i n c e chains must predominate t o be c o n s i s t e n t with t h e 2 9 ~ i NMR d a t a , hence c o n s t r a i n t s a r e b u i l t i n t o p r e s e r v e two-fold c o n n e c t i v i t y where p o s s i b l e ( t h r e e - f o l d c o n n e c t i v i t y is excluded). F u r t h e r , each Si-0 c h a i n is surrounded by s i x Li-0 chains a s i n t h e c r y s t a l . F i n a l l y , t h e r a t i o of S i . a n d L i atoms is c o n t r o l l e d : t h e type of atom is s e l e c t e d by a random number g e n e r a t o r , s i t e s being chosen i n o r d e r , working from t h e bottom of t h e model along each { 0 0 1 ] plane and t h e n t o t h e next plane above. Where a S i atom is s e l e c t e d but cannot be placed i n a given l o c a t i o n ( t o preserve Si-Si avoidance, s a y ) i t is then s t o r e d and placed i n t h e next a v a i l a b l e s i t e .

Fig. 6

-

The hexagonally c l o s e packed oxygen s u b - l a t t i c e corresponding t o an i d e a l i s e d m e t a s i l i c a t e s t r u c t u r e . S i l i c o n atoms a r e shown by t h e small c i r c l e s occupying t e t r a - h e d r a l i n t e r s t i c e s ( a t two l e v e l s ) forming c h a i n s running p a r a l l e l t o t h e hexad a x i s . Each Si-0 c h a i n is surrounded by s i x p a r a l l e l Li-0 c h a i n s (medium-sized c i r c l e s ) .

-avoiding random Si-0 and Li-0 c h a i n s . I n e v i t a b l y Si-0 c h a i n s t e r m i n a t e and new ones must t h e r e f o r e o r i g i n a t e . A

m a j o r i t y of S i atoms however have t h e Q~ c o n f i g u r a t i o n . The model is t h e n r e l a x e d using a modified Keating p o t e n t i a l with s t r e t c h i n g , Vs, and bending terms Vb:

The sum is over t h e i s t r e t c h i n g and j bending c o o r d i n a t e s ; a and @ a r e s t r e t c h i n g and bending f o r c e c o n s t a n t s , r l i is t h e v e c t o r connecting atoms 1 and i , d and 8 a r e t h e e q u i l i b r i u m bond l e n g t h and bond a n g l e r e s p e c t i v e l y . Some s o p h i s t i c a t i o n is

C8-12 _{JOURNAL DE PHYSIQUE}

.; GJ r

.,

.

? Na-Si

Fig. 7 - a ) , b) Experimental p a r t i a l 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 , ppdfs, For c-NazSiOj. c ) , d ) Computed ppdfs f o r a model of c-Na2Si03 obtained by energy- minimization of a s t a r t i n g s t r u c t u r e based on t h e hcp l a t t i c e shown i n f i g . 5. Parameters i n t h e Keating p o t e n t i a l energy f u n c t i o n were varied t o produce t h e q u a l i t y of f i t shown here.

Choice of accurate parameters a , e t c is v i t a l f o r success and t o ensure consistency with t h e c r y s t a l l i n e s t r u c t u r e , a t l e a s t , an hcp oxygen s u b l a t t i c e was

' s t u f f e d ' with an ordered a r r a y of S i , L i (Na) atoms and t h e r e s u l t i n g c r y s t a l s t r u c t u r e relaxed and compared with X-ray d a t a f o r c-Li2Si03 and Na2Si03. Constants were then a d j u s t e d t o g i v e an optimal f i t . This by no means guarantees t h a t t h e c o n s t a n t s a r e a p p r o p r i a t e even t o a c r y s t a l l i n e m a t e r i a l of d i f f e r e n t symmetry. s t i l l l e s s t o a g l a s s but i t is d i f f i c u l t t o s e e how a b e t t e r choice can be made. Experimental and c a l c u l a t e d p a r t i a l 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 f o r c r y s t a l l i n e Na2Si03 a r e shown i n f i g . 7.

With t h i s choice of parameters f o r t h e i n t e r a t o m i c p o t e n t i a l , p a r t i a l d i s t r i b u t i o n f u n c t i o n s have been c a l c u l a t e d f o r t h e random model and a r e given i n f i g . 8.

Fig. 8 - Computed ppdfs f o r random c h a i n models of a ) , b) v i t r e o u s Li2Si03

c ) , d ) Na2S03. Parameters i n t h e p o t e n t i a l energy f u n c t i o n a r e i d e n t i c a l t o t h o s e used i n c a l c u l a t i n g t h e ppdfs f o r t h e c r y s t a l l i n e s i l i c a t e s . F(Q1 I I I I I I I

2

-

-

Fig. 9 - C a l c u l a t e d t o t a l

-

reduced X-ray i n t e r f e r e n c e f u n c t i o n , F ( Q ) f o r t h e random

0 _{l i} c h a i n model of a-Na2Si03 ( f u l l _{n e compared with t h e}

'-/'

experimental d a t a of Yasui e t a 1 /36/.

-

J O U R N A L D E PHYSIQUE

Fig. 70

-

Volume p e r oxygen atom, Vo, computed f o r c r y s t a l l i n e ( c ) and g l a s s y ( g ) models f o r Li2Si03 and Na2Si03 a s a f u n c t i o n of t h e non-bridging 0-0 s t r e t c h i n g c o n s t a n t ao-0 i n t h e Keating p o t e n t i a l energy funEtion. For v a l u e s of oo-,-J which produce good agreement with experimental d a t a ( a r r o w s ) , t h e computed value f o r Vo f o r t h e g l a s s a r e s e r i o u s l y i n e r r o r .

For v a l u e s of ao-0 which produce good agreement between experimental and c a l c u l a t e d

Vo values f o r c-Li2Si03 and c-Na2Si03, t h e b e s t agreement with t h e corresponding p a r t i a l 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 and t h e lowest s t r a i n energy, t h e v a l u e s c a l c u l a t e d f o r Vo f o r t h e random model do

not

a g r e e with experimental d a t a f o r t h e g l a s s . The e x t e n t of disagreement is r e l a t i v e l y l a r g e ; f o r Na2Si03 t h e experimental value of Vo is 26.2 1-3 whereas t h e value of Vo o b t a i n e d from t h e model is 23.0 1 - 3 , an e r r o r of about 13%. For Li2Si03, Vo(exp)

=

21

1 - 3 ,

t h e c a l c u l a t e d v a l u e of Vo =

19.6 1-3 an e r r o r of about 8%. Note t h a t i n both cases t h e c a l c u l a t e d value of Vo

i s smaller than t h e experimental d a t a ( t h e d e n s i t y is h i g h e r ) . The d e n s i t y is a l s o equal t o o r h i g h e r t h a n t h a t of t h e c r y s t a l l i n e form.

\ \ \ \

??:li

Ik

easy: Fig. d i s t a n c e s bending 11 p a r a l l e l

-

a t a ) t h e Expansion t o oxygen t h e hexad atom of a x i s ( l a r g e S i - S i is

-

-

-*

c i r c l e s ) only i s required and t h i s f o r c e

/ c o n s t a n t is small. Cooperative r o t a t i o n

1 _{of t h e s i l i c o n oxygen t e t r a h e d r o n is}

p o s s i b l e s o t h a t t h e s t i f f e r S i bond a n g l e is not d i s t o r t e d . b) Where t h e

wy

c h a i n runs i n a d i r e c t i o n normal t o t h e hexad a x i s , i n a random model,

+-

c o o p e r a t i v e motion is no longer p o s s i b l e and bending of t h e ( s t i f f e r ) S i a n g l e becomes necessary t o allow e l o n g a t i o n .

b

This does not mean t h a t a random model cannot be constructed. An a l t e r n a t i v e model could, f o r example, be constructed i n t h e same s p i r i t s t a r t i n g from a dense random- packed c l u s t e r of oxygen atoms with S i , L i atoms i n s e r t e d i n t o t e t r a h e d r a l i n t e r s t i c e s . Since t h e packing f r a c t i o n of random sphere models is lower than hcp (0.63 compared t o 0.73), t h e values of Vo would be expected t o be somewhat higher than observed above. However, u n t i l i t can be proved otherwise, t h e r e must be doubts about t h e a b i l i t y of t h i s model t o s a t i s f y t h e c o n s t r a i n t s r e p r e s e n t i n g c o o r d i n a t i o n , anionic composition (and packing d e n s i t y ) . For t h i s d i s c u s s i o n , t h e p o i n t is t h a t randomness is not n e c e s s a r i l y t h e hallmark of an acceptable model. A

personal view would be t h a t a s u c c e s s f u l model would probably r e q u i r e more o r d e r r a t h e r than l e s s and t e s t s a r e being done on a domain model s i m i l a r t o t h a t f o r t h e a - t r a n s i t i o n metal-metalloid a l l o y s . A model f o r t h e g l a s s based on t h e m e t a s i l i c a t e c r y s t a l has already shown some s u c c e s s /36/.

I V

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ALTERNATIVES TO RANDOMNESS

The i l l u s t r a t i o n s given i n t h e previous s e c t i o n i n d i c a t e t h a t f o r many amorphous m a t e r i a l s i t may be most a p p r o p r i a t e t o r e p r e s e n t t h e s t r u c t u r e of t h e ' i d e a l ' g l a s s by a random model. For o t h e r s , d e p a r t u r e s from randomness a r e such t h a t , o p e r a t i o n a l l y a t l e a s t - more ordered models may be p r e f e r a b l e .

Before proceeding f u r t h e r , two p o i n t s should be examined.

I V . l - The ' i d e a l g l a s s The low temperature s t a t e of a g l a s s is u s u a l l y c h a r a c t e r i z e d by i t s e f f e c t i v e g l a s s temperature, or f i c t i v e temperature, T f . I f we choose t o d e f i n e an i d e a l g l a s s a s t h e c l o s e s t approach t o a t o t a l l y disordered

s o l i d then, Q d e f i n i t i o n , t h e i d e a l g l a s s corresponds t o quenching a t t h e h i g h e s t cooling r a t e s , evaporation a t high r a t e s onto l i q u i d He s u b s t r a t e s e t c . - c o n d i t i o n s which lead t o t h e h i g h e s t values of Tf. A d e f i n i t i o n more c o n s i s t e n t with t h e normal understanding of p e r f e c t d e f e c t - f r e e s o l i d s is t h a t s t a t e r e p r e s e n t i n g t h e lowest energy c o n f i g u r a t i o n a v a i l a b l e t o t h a t phase. This d e f i n i t i o n is adopted h e r e and corresponds t o t h e ground s t a t e of amorphous packing represented by t h e lowest e x t r a p o l a t e d value of Tf corresponding t o i n f i n i t e l y slow quenching r a t e s . Thus Tf-To, where To is t h e c h a r a c t e r i s t i c temperature e n t e r i n g i n t o t h e Fulcher-Vogel equation f o r v i s c o s i t y , q = Aexp(B/(T - To)) l e a d i n g t o i n f i n i t e v i s c o s i t y ( s o l i d - l i k e p r o p e r t i e s ) f o r T STo.

JOURNAL

DE

PHYSIQUE

IV.3

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Constrained randomness I t is important, i f obvious, t o recognize t h a t t h e e s s e n t i a l l o g i c a l c r i t e r i o n f o r choosing a random, no preference, model is t h a t experimental f a c t s do n o t i n d i c a t e any preference. Where experimental f a c t s suggest otherwise

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l o c a l chemical ordering, c o r r e l a t e d domains - then a b s o l u t e randomness

is untenable. There a r e then two a l t e r n a t i v e s ; one is t o r e f i n e random models t o i m p l i c i t l y or e x p l i c i t l y include t h e non-random elements, o r t o abandon t h e p r i n c i p l e and examine d e f e c t i v e but i n t r i n s i c a l l y ordered models.

Extension of t h e concept of t h e random s t r u c t u r e

-

what might be c a l l e d constrained randomness - has already been included i m p l i c i t l y i n a l l s u c c e s s f u l models of g l a s s e s : models b u i l t without c o n s t r a i n t s r a r e l y , i f e v e r , r e p r e s e n t r e a l g l a s s e s . First-neighbour bond l e n g t h s a r e normally c l o s e l y constrained and c e r t a i n bond angles may be a l s o . Chemical s h o r t range o r d e r is included i n models f o r t r a n s i t i o n metal-metalloid a l l o y s . Longer-range o r g a n i z a t i o n leading t o medium-range domains can be incorporated, i n p r i n c i p l e , by adjustment of t h e length s c a l e corresponding t o t h e o n s e t of randomness - a s f o r example, i n models f o r a-B2O3 based on boroxol u n i t s connected randomly, o r a model f o r t r a n s i t i o n metal-metalloid g l a s s e s with randomly packed t r i g o n a l prisms.

I n c r e a s i n g l y , i t would seem, a s more complex, close-packed systems a r e i n v e s t i g a t e d

-

e s p e c i a l l y t h o s e near an edge of a glass-forming r e g i o n , t h e c o n s t r a i n t s become more dominant and t h e p r i n c i p l e of randomness more tenuous.

IV.4

-

Defective ordered s t r u c t u r e s "'Disorder' is not mere chaos: i t implies d e f e c t i v e

order.

To think about a disordered s t a t e we must have i n mind an i d e a l of order from which i t f a l l s s h o r t . I t is much e a s i e r t o c h a r a c t e r i z e disordered systems i n terms of t h e i r d e v i a t i o n s from t h i s i d e a l than i t is t o d e f i n e a p e r f e c t l y disordered system on which some p a r t i a l degree of order is t o be imposed. The concept of d i s o r d e r is p r i m i t i v e and i n t u i t i v e ; i t belongs with s t a t i s t i c a l terms such a s 'random1, ' s t o c h a s t i c ' , ' u n p r e d i c t a b l e ' which can only be defined w i t h i n a s p e c i f i c context of what is a l r e a d y known or can be taken f o r granted." The quotation is t h e opening paragraph of Ziman's "Models of Disorder" /37/ and s e r v e s t o i l l u s t r a t e t h e philosophy underlying models f o r an amorphous m a t e r i a l which may have a 'clearly-expressed preference' f o r a p a r t i c u l a r p e r f e c t l y ordered c o n f i g u r a t i o n but which n e v e r t h e l e s s f a i l t o achieve i t . F a i l u r e t o reach t h e ordered s t a t e may be t h e r e s u l t of k i n e t i c and/or t o p o l o g i c a l c o n s t r a i n t s . I n such cases i t may be more p r o f i t a b l e t o d i s c u s s t h e s t a t e of t h e disordered m a t e r i a l i n terms of d e f e c t s i n t h e ordered s t a t e . I n t h e following two s e c t i o n s t h e p r i n c i p l e s underlying t o p o l o g i c a l l y - and k i n e t i c a l l y - ' f r u s t r a t e d ' s t r u c t u r e s a r e discussed (although i t may be d i f f i c u l t t o d i s e n t a n g l e t h e two types of c o n s t r a i n t s

-

elements of each a r e always p r e s e n t ) .

IV.5

-

Topological c o n s t r a i n t s Formation of t h e s t r u c t u r e of an amorphous t e t r a h e d r a l semiconductor by s e q u e n t i a l a d d i t i o n of atoms n e c e s s a r i l y l e a d s t o a pentagonal r i n g a s t h e lowest energy c o n f i g u r a t i o n of 5 connected atoms. One bond energy is saved and t h e r i n g contains only a small angular mismatch (7.5O). S i m i l a r l y , s e q u e n t i a l close-packing of atoms l e a d s t o a minimum energy c o n f i g u r a t i o n of f o u r atoms a s a t e t r a h e d r o n and f u r t h e r low energy s t r u c t u r e s by c o n t i n u a t i o n of t h i s p o l y t e t r a h e d r a l packing. Five t e t r a h e d r a almost complete a loop around a common a x i s with again only a 7.5O angular mismatch.

Continuation of such s t r u c t u r e s t o give s p a c e - f i l l i n g s t r u c t u r e s without breaks of symmetry is impossible. The s t r u c t u r e may, however, tend towards i t and i t i s

p r o f i t a b l e t o examine t h e a c t u a l s t r u c t u r e from t h e view point of t h e p e r f e c t s t r u c t u r e - i n t h i s c a s e t h e f u l l y ordered polytope i n a higher-dimensional space /38/. Mapping t h i s s t r u c t u r e i n t o ordinary 3-space n e c e s s a r i l y involves d e f e c t i v e regions.

amorphous a l l o y s based on a decoration of pentagonal r i n g s t o g i v e 5-fold r i n g s of t r i g o n a l prisms ( f i g 1 2 ) /24/.

Fig. 1 2 - Pentagonal r i n g s based on a packing of capped t r i g o n a l prisms l i n k e d a ) with t e t r a h e d r a b ) h a l f octahedra.

The s t a b i l i t y of such models t o c r y s t a l n u c l e a t i o n d e r i v e s both from t h e r e l a t i v e l y low energy penalty compared t o t h e c r y s t a l : f o r small p a r t i c l e s , p o l y t e t r a h e d r a l s t r u c t u r e s a r e , both e n e r g e t i c a l l y and e n t r o p i c a l l y s t a b i l i z e d and t h e energy penalty with r e s p e c t t o c r y s t a l s may only be s i g n i f i c a n t f o r almost macroscopic p a r t i c l e s .

IV.6 K i n e t i c c o n s t r a i n t s Amorphous s t r u c t u r e s a l s o have a d i s t i n c t preference a tendency t o form t h e ordered, c r y s t a l l i n e l a t t i c e . I n t h i s case, we can assume t h a t t h e necessary f r u s t r a t i o n must be e s s e n t i a l l y k i n e t i c i n o r i g i n (even i f t o p o l o g i c a l f e a t u r e s a r e involved i n t h e s p e c i f i c a t i o n of k i n e t i c parameters). The r e s u l t may be a m a t e r i a l which, a t one extreme, might be a quenched l i q u i d with c r y s t a l l i n e i n c l u s i o n s and could not t h e r e f o r e be described a s a g l a s s . A t t h e o t h e r extreme might be a p a r a c r y s t a l l i n e o r highly m i c r o c r y s t a l l i n e m a t e r i a l with a domain s i z e comparable t o a small number of i n t e r a t o m i c d i s t a n c e s . Often, i n t h e p a s t , t h i s type of model has been considered only i n terms of a s e r i e s of m i c r o c r y s t a l l i t e s , a r b i t r a r i l y o r i e n t a t e d , with unspecified i n t e r f a c e s

-

p o s s i b l y involving some more disordered 'connecting t i s s u e ' . Such models have been r i g h t l y dismissed - on experimental grounds, because some i n t e r f a c e s must be highly e n e r g e t i c thus l e a d i n g t o i n i t i a t i o n of g r a i n growth - but p r i n c i p a l l y because t h e p a r t i c l e s must be s o small t h a t the i n t e r f a c i a l s t r u c t u r e and t h e r e s u l t i n g s t r a i n spreading throughout t h e model must be s p e c i f i e d and q u a n t i f i e d . " U n t i l a p r o t a g o n i s t of t h e s e models produces an a c t u a l three-dimensional s t r u c t u r e t h a t conforms t o t h e assumptions he makes we must be doubtful whether it can be done a t a l l , nohow." / 4 0 / .

C8-18 JOURNAL DE PHYSIQUE

R e l a t i v e l y few attempts have been made t o t a c k l e t h e s e problems i n t h e manner advocated by Ziman /40/. A personal view is t h a t success must depend on a low energy i n t e r f a c e s o t h a t models based on twinning, s t a c k i n g f a u l t s ( p o s s i b l y ) seem l i k e l y contenders. A model based on m u l t i p l e twinning o r i g i n a l l y designed f o r a-Ge

/39/ and then extended t o Si02 /16/ provides c l o s e f i t s t o both s e t s of r d f s and i n t e r f e r e n c e f u n c t i o n s

-

arguably n o t a s good a s a CRN model f o r Ge and b e t t e r than most CRN models f o r Si02. A more r e c e n t approach i s t h a t of Dubois e t a 1 / 2 5 / ,

whose model is based on domains separated by i n t e r f a c e s where 'chemical twinningt planes change o r i e n t a t i o n . S t r a i n is generated by i n t e r f a c i a l mismatch but s i n c e each i n t e r f a c e atom is completely bonded, t h e s t r a i n is small and spreads throughout t h e s t r u c t u r e . Formation of a domain s t r u c t u r e on a s u f f i c i e n t l y small s c a l e is

thought t o be t h e r e s u l t of e i t h e r i n t r i n s i c compositional f u n c t i o n s i n t h e supercooled l i q u i d or f l u c t u a t i o n s generated by e x s o l u t i o n from growing domains. The e s s e n t i a l point is t h a t t h e l o c a l composition a t t h e i n t e r f a c e between two

c l u s t e r s w i l l be d i f f e r e n t from t h a t i n t h e i n t e r i o r of a domain. A composition f l u c t u a t i o n can t h u s be incorporated i n t o t h e s t r u c t u r e by a l o c a l v a r i a t i o n i n o r i e n t a t i o n of t h e chemical twinning plane s e e a l s o /44/. Examples of m u l t i p l e twinned c r y s t a l growth a r e seen i n t h e p r e c i p i t a t i o n of S i from a hyper-eutectic A1-Si a l l o y /45/.

V - SUMMARY

The evidence presented i n t h i s paper suggests t h a t t h e e x i s t e n c e of a unique s t r u c t u r a l d e s c r i p t i o n f o r a l l g l a s s e s may be a mirage. Almost c e r t a i n l y t h e r e a r e m a t e r i a l s - t h e good glass-formers with open s t r u c t u r e s , elemental amorphous metals and semiconductors where a random network or dense random-packed model may be t h e most a p p r o p r i a t e model. Equally t h e r e a r e o t h e r s , close-packed a l l o y s with d i s i m i l a r elements, glass-forming m a t e r i a l s a t t h e edge of t h e i r s t a b i l i t y range, f o r which t h e concept of randomness introduces d i f f i c u l t y and a r t i f i c i a l i t y . I t may be p r e f e r a b l e , t h e r e f o r e , t o abandon randomness a s a paradigm i n favour of d e f e c t i v e ordered models. Intermediate cases w i l l occur where "constrained- randomnesstt may r e p r e s e n t an a p p r o p r i a t e d e s c r i p t i o n .

For t h e f u t u r e , i t seems important t h a t t h e concept of v a r i a b i l i t y i n g l a s s s t r u c t u r e s should be recognized and developed.

The c o n s t r a i n t s on randomness should be i n v e s t i g a t e d f u r t h e r with t h e accent on t h e s p e c i f i c a t i o n of t h e n a t u r e and magnitude of t h o s e c o n s t r a i n t s .

I t should no longer be assumed t h a t a random model c o n s i s t e n t with t h e known c o n s t r a i n t s is even p o s s i b l e , a s t h e d i s c u s s i o n of t h e a l k a l i s i l i c a t e s i l l u s t r a t e s . Further experimental work i s needed t o d e f i n e t h e l o c a l s t r u c t u r e . I t has been shown how an adequate q u a n t i t a t i v e d e f i n i t i o n of l o c a l s t r u c t u r a l parameters - which a r e o f t e n a l l we

can

quantify - may a l s o lead t o bounds being placed on a c c e p t a b l e medium-range s t r u c t u r e s . For many g l a s s e s , such l o c a l parameters e i t h e r do not

--

e x i s t or a r e incomplete - even i n simple g l a s s e s l i k e t h e a l k a l i s i l i c a t e s and b o r a t e s

-

hence t h e importance of experimental s t u d i e s of p a r t i a l s by EXAFS, neutron s c a t t e r i n g e t c .

e s s e n t i a l l y d i s c o n t i n u o u s , h e t e r o g e n e o u s ( i n t h e most g e n e r a l s e n s e o f t h e word) n u c l e a t i o n p r o c e s s we might imagine t h a t under c o n d i t i o n s which f a v o u r a n e v o l u t i o n a r y p r o g r e s s i o n t o w a r d s t h e i d e a l g l a s s , t h e e n e r g y o f a c o n s t r a i n e d random model would d e c r e a s e a s would f l u c t u a t i o n s i n bond l e n g t h and a n g l e s

-

t h e m a t e r i a l becomes more o r d e r e d . T o p o l o g i c a l l y - f r u s t r a t e d models a l s o become more o r d e r e d b u t p e r h a p s w i t h a r e d i s t r i b u t i o n o f t h e p e n t a g o n a l domains and d e f e c t r e g i o n s . K i n e t i c a l l y - f r u s t r a t e d models must, by d e f i n i t i o n , p r o g r e s s t o a more o r d e r e d s t r u c t u r e , b u t h e r e , s u r e l y , t h e changes s h o u l d be s e e n by h i g h r e s o l u t i o n e l e c t r o n microscopy

-

t h e s t r u c t u r e s h o u l d r i p e n a s t h e domains grow l e a d i n g e v e n t u a l l y t o a m i c r o c r y s t a l l i n e s t r u c t u r e w i t h c h a r a c t e r i s t i c c h a n g e s i n p r o p e r t i e s . To d a t e , o n l y a s t a r t h a s been made i n a n s w e r i n g most o f t h e s e q u e s t i o n s .

A f i n a l q u o t a t i o n (on e v o l u t i o n ) . " N a t u r a l s e l e c t i o n t h r o u g h random m u t a t i o n must have p l a y e d a n i m p o r t a n t p a r t i n t h o s e p r o c e s s e s , b u t t o r e g a r d t h a t a s t h e s o l e e x p l a n a t i o n o f e v o l u t i o n a r y change is i m p r o b a b l e , u n p r o v a b l e and d o g m a t i c , t h e f a c t

is t h a t m a t t e r h a s a n i n n a t e t e n d e n c y t o a s s e m b l e i t s e l f i n t o more and more complex forms" / 4 7 / .

ACKNOWLEDGMENTS

T h i s p a p e r h a s b e n e f i t e d from i l l u m i n a t i n g d i s c u s s i o n s w i t h D r . J - . M. Dubois. Generous s u p p o r t from P i l k i n g t o n B r o t h e r s PLC is g r e a t e f u l l y acknowledged.

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