KINETICS OF GLASS FORMATION AND DEVITRIFICATION BEHAVIOR

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KINETICS OF GLASS FORMATION AND DEVITRIFICATION BEHAVIOR

D. Uhlmann

To cite this version:

D. Uhlmann. KINETICS OF GLASS FORMATION AND DEVITRIFICATION BEHAVIOR. Journal

de Physique Colloques, 1982, 43 (C9), pp.C9-175-C9-190. �10.1051/jphyscol:1982933�. �jpa-00222462�

JOURNAL DE PHYSIQUE

CoZZoque C9, supple'ment au n012, Tome 43, dkcembre 1982 page C9-175

K I N E T I C S O F G L A S S F O R M A T I O N A N D D E V I T R I F I C A T I O N B E H A V I O R

Department of Materials Science and Engineering, Massachusetts Institute of Technology

,

Cambridge, Massachusetts, U. S. A.

[email protected] La formation du v e r r e e s t consideree sous un a s p e c t c i n e t i q u e . La c i n e t i q u e de n u c l e a t i o n , a u s s i bien homogene qulh&t&rogene, a i n s i que l a c r o i s - sance du c r i s t a l s o n t i n i t i a l l e m e n t s u i v i e s . Les informations obtenues dans ces domaines s o n t reunies avec des t r a i t e n e n t s de c r i s t a l l i s a t i o n pendant un refroidissement c o n s t a n t a f i n d 1 6 v a l u e r des v i t e s s e s c r i t i q u e s de r e f r o i d i s - sement n e c e s s a i r e s

a

l a formation de v e r r e s de d i f f 6 r e n t s materiaux. Les pa- rametres du materiau contribuant

a

l a formation e t au proc6d6 de c r i s t a l l i s a - t i o n 2 chaud du v e r r e sont consideres. Le t r a i t e m e n t c i n e t i q u e e s t a u s s i u t i - l i s P pour d e c r i r e d ' a u t r e s phenomenes comme l e s e f f e t s d 1 h @ t 6 r o g 6 n 6 i t @ s de nucleation s u r l a formation du v e r r e e t l ' u t i l i s a t i o n d'experiences de DTA pour evaluer l e s b a r r i e r e s de l a nucleation du c r i s t a l .

Abstract

.-

The process of g l a s s formation i s viewed from a k i n e t i c perspective.

I n i t i a l a t t e n t i o n i s d i r e c t e d t o t h e k i n e t i c s o f nucleation, both homogeneous and heterogeneous, and of c r y s t a l growth. Information obtained i n t h e s e a r e a s i s combined with treatments of c r y s t a l l i z a t i o n during continuous cooling t o e v a l u a t e t h e c r i t i c a l cooling r a t e s required t o form g l a s s e s of various m a t e r i a l s . Consideration is given t o t h e material parameters which a r e

conducive t o g l a s s formation, and t o t h e process of c r y s t a l l i z a t i o n on reheat- ing of g l a s s . The k i n e t i c treatment i s a l s o used t o d e s c r i b e o t h e r phenomena such a s t h e e f f e c t s of nucleating h e t e r o g e n e i t i e s on g l a s s formation and t h e use of DTA experiments t o e v a l u a t e t h e b a r r i e r s t o c r y s t a l nucleation.

1. Introduction.- A l i q u i d may s o l i d i f y i n e i t h e r of two ways: i t may form a c r y s t a l l i n e s o l i d , i n which t h e molecules a r e r e g u l a r l y arranged on a l a t t i c e ; o r i t can form an amorphous s o l i d , c a l l e d a g l a s s , i n which t h e molecular a r r a y i s c h a r a c t e r i z e d by t h e absence of long-range o r d e r . I t i s generally believed ( a l t h o u g h n o t proven, t o t h e a u t h o r ' s knowledge) t h a t c r y s t a l l i n e s o l i d s r e p r e s e n t t h e thermo- dynamically s t a b l e s t a t e of matter a t low temperatures; but t h e formation of such s o l i d s r e q u i r e s t h e nucleation and growth of a new phase, and t h e s e processes do not t a k e place w i t h i n f i n i t e r a p i d i t y . I f t h e cooling r a t e is s u f f i c i e n t l y r a p i d , r e l a t i v e t o t h e k i n e t i c s of c r y s t a l l i z a t i o n , t h e l i q u i d phase can be cooled inde- f i n i t e l y without t h e occurrence of d e t e c t a b l e c r y s t a l 1 in i t y and a g l a s s w i l l be formed.

Once formed, amorphous s o l i d s can p e r s i s t f o r long periods of time i n t h e i r thermo- dynamically unstable s t a t e : r e c a l l t h e b i l l i o n - y e a r - o l d g l a s s e s returned from t h e s u r f a c e of t h e moon, where they had been preserved a t temperatures near ambient i n t h e absence of mineralizing agents such a s water. The very e x i s t e n c e of amorphous s o l i d s provides a prima f a c i e v i o l a t i o n of t h e t h i r d law of thermodynamics i n t h e form o r i g i n a l l y advanced by Nernst, and seems t o r e q u i r e i t s r e s t r i c t i o n t o systems

'presently on sabbatical leave with Dspartement des MatGriaux, Ecole Polytechnique Fgdgrale de Lausanne, Suisse.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982933

C9-176 JOURNAL DE PHYSIQUE

i n i n t e r n a l thermodynamic e q u i l i b r i u m , as suggested bySimon ( 1 ) . Further, glasses a r e s u b j e c t t o the paradox noted by Kauzmann ( 2 ) , viz., t h a t i f s u f f i c i e n t time were allowed f o r t h e amorphous system t o reach equilibrium,extrapolation o f t h e e q u i l i b r i u m p r o p e r t i e s ( f r o m above t h e glass t r a n s i t i o n ) would suggest t h e existence of amorphous phases having s m a l l e r s p e c i f i c volumes and e n t r o p i e s than the corresponding c r y s t a l s .

I n a d d i t i o n t o p r o v i d i n g us w i t h such i n t r i g u i n g fundamental problems, glasses a r e a l s o o f considerable t e c h n o l o g i c a l importance. The p a s t two decades have seen impressive progress across a broad f r o n t , r a n g i n g from t h e e f f e c t i v e completion of the f l o a t glass r e v o l u t i o n and t h e extensive e x p l o r a t i o n o f semiconducting glasses and amorphous s o l a r c o l l e c t o r s , t o t h e i n t r o d u c t i o n o f glass o p t i c a l waveguides as e s s e n t i a l p a r t s of o p t i c a l communications systems and glassy p a s s i v a t i o n l a y e r s i n i n t e g r a t e d c i r c u i t devices.

I t i s now w i d e l y recognized t h a t glass formation i s n o t r e s t r i c t e d t o t h e f a m i l i a r s i l i c a t e s , o r even t o oxide m a t e r i a l s more g e n e r a l l y . Glassy polymers such as poly- carbonate and polymethyl methacrylate are i m p o r t a n t a r t i c l e s o f commerce; metal a1 l o y glasses are r e c e i v i n g wide a t t e n t i o n ; and simple organic glasses, fused s a l t glasses, and aqueous s o l u t i o n glasses have a l l been e x t e n s i v e l y i n v e s t i g a t e d .

While glasses can be formed i n a wide v a r i e t y o f ways, ranging from flame h y d r o l y s i s and vacuum vapor d e p o s i t i o n t o shock wave treatment of c r y s t a l s and e l e c t r o l y t i c d e p o s i t i o n (see Ref. 3 f o r extensive discussion), the c o o l i n g o f a m e l t i s by f a r t h e most important. Since a c o o l i n g m e l t can c r y s t a l l i z e a t temperatures below t h e l i q u i d u s , t h e process o f glass formation seems b e s t viewed from the perspective o f whether d e t e c t a b l e c r y s t a l l i n i t y w i l l develop a t a given c o o l i n g r a t e . This, i n turn, d i r e c t s a t t e n t i o n t o the processes o f c r y s t a l n u c l e a t i o n and growth, whose k i n e t i c s e s t a b l i s h a time s c a l e f o r t h e c o o l i n g process.

The present paper w i l l be concerned w i t h t h e k i n e t i c c o n d i t i o n s r e q u i r e d t o form glasses o f various m a t e r i a l s , and i n p a r t i c u l a r w i t h k i n e t i c treatments which consider b o t h n u c l e a t i o n and c r y s t a l growth. Other approaches t o the i s s u e o f glass formation have been discussed elsewhere (4, e.g.). S p e c i f i c a t t e n t i o n w i l l be d k e c t e d t o r e c e n t a p p l i c a t i o n s o f t h e k i n e t i c a n a l y s i s t o areas such as c r y s t a l l i - z a t i o n on r e h e a t i n g a glass, the r o l e of n u c l e a t i o n t r a n s i e n t s i n glass formation, and t h e measurement o f n u c l e a t i o n r a t e s .

2. Nucleation and C r y s t a l Growth.- The steady s t a t e r a t e o f homogeneous n u c l e a t i o n

1!0 d i s p l a y s a complex temperature dependance:

Here N: i s t h e number o f molecules p e r u n i t volume i n t h e l i q u i d ; v i s t h e frequency o f atom t r a n s p o r t a t t h e nucleus-matrix i n t e r f a c e ; K1 i s a geometric f a c t o r

1 ~ T T

(K1 = - f o r s p h e r i c a l n u c l e i ) ; u i s t h e c r y s t a l - l i q u i d s u r f a c e f r e e energy;

AGv i s t h e d i f f e r e n c e i n Gibbs f r e e energy p e r u n i t volume between l i q u i d and 3 c r y s t a l ; and k i s 601 tzmann's constant.

A t small undercoolings, AT, below t h e m e l t i n g p o i n t , TE, AGv increases l i n e a r l y w i t h undercool i ng, as

where AHv i s t h e h e a t o f f u s i o n per u n i t volume. For metals, t h i s expression provides a good approximation c v e r a wide range o f undercooling ( 5 ) ; b u t f o r most oxides and simple organics, t h e expression due t o Hoffman ( 6 )

-

which considers the d i f f e r e n c e i n heat c a p a c i t y between c r y s t a l and l i q u i d

-

represents a b e t t e r approximation:

I t i s o f t e n assumed t h a t t h e temperature dependence o f t h e frequency factor,^

,

i s t h e same as t h a t o f t h e v i s c o s i t y , r! :

For simple organics, t h e c o e f f i c i e n t b may w e l l be taken as t h e Stokes-Einstein v a l ue:

where a. i s a molecular diameter. For complex oxides, such as t h e f a m i l i a r s i l i c a t e s , r e c e n t work ( 7 ) has suqclested a p r o p o r t i o n a l i t y f a c t o r l a r g e r by abouta f a c t o r o f 4 0 :

40 kT

b.- 3 ( 6 )

31ra~

The o v e r a l l v a r i a t i o n o f n u c l e a t i o n r a t e w i t h temperature r e f l e c t s b o t h t h e v a r i a t i o n o f v i s c o s i t y w i t h temperature

-

i t s e l f r a t h e r complex save f o r t h e c l a s s i c network 1 iq u i d s (Si02, Ge02, Na2!.A1203-3Si02), which e x h i b i t Arrhenian behavior

-

and the v a r i a t i o n o f exp (-K/TAG:). I t would t h e r e f o r e be remarkable t o observe, over any extensive range o f temperature, an Arrhenian temperature dependence of Iv.

When n u c l e a t i n g h e t e r o g e n e i t i e s a r e present i n a sample, one must a l s o consider t h e i r e f f e c t on the n u c l e a t i o n r a t e . This i s u s u a l l y e f f e c t e d using t h e fami 1 ia r s p h e r i c a l cap model o f t h e nucleus, w i t h r e s u l t :

'3

where nv i s the number of h e t e r o g e n e i t i e s per u n i t volume, A i s t h e area per hetero- geneity, and 4 can be expressed :

2+cos0) ( 1 -cos9) 2

@ = ( 4 ( 8 )

Here 0 i s t h e c o n t a c t angle between n u c l e a t i n g s u b s t r a t e and c r y s t a l nucleus.

I n t r e a t i n g t h e process o f heterogeneous n u c l e a t i o n , i t should be noted t h a t the number o f a c t i v e h e t e r o g e n e i t i e s i s depleted as such n u c l e a t i o n occurs. T h i s deple- t i o n can be approximated i n terms o f i t s e f f e c t on nv ( 8 ) :

where n and n: t are t h e concentrations o f n u c l e a t i n g h e t e r o g e n e i t i e s a t time t and i n i t i a l Y y ; and V i s t h e volume o f t h e sample. This equation assumes t h a t each hete- r o g e n e i t y can provide a s i n g l e n u c l e a t i o n event.

With one n o t a b l e exception (9, l o ) , experimental data on homogeneous n u c l e a t i o n are g e n e r a l l y i n good accord w i t h c l a s s i c a l n u c l e a t i o n theory, Eqn. (1) above (11, 12, e.g.). Both t h e temperature dependence and the pre-exponential constant are i n rea- sonable accord w i t h p r e d i c t i o n s o f t h e theory. This i s i l l u s t r a t e d by t h e data on a n o r t h i t e (Ca0.A1203.2Si02) i n F i g . 1. As p r e d i c t e d by Eqns(1)

+

(3) c ( 4 ) , t h e I n Iv VS(AT:T?)-' r e l a t i o n i s a s t r a i g h t l i n e o f negative slope. Here Ur = AT/TE and Tr = T/TE. The i n t e r c e p t i n Fig. 1 y i e l d s a pre-expotential f a c t o r o f

l o z 7

sec -1 poise, w h i l e theory p r e d i c t s

l o z 9 - lo3'

sec-I poise.

JOURNAL DE PHYSIQUE

F i g . 1. Logarithm ( I v q ) V S .

( A T T ) r e l a t i o n f o r a n o r t h i t e . A f t e r Ref. 11.

I n c o n t r a s t t o t h e agreement between experiment and c l a s s i c a l theory shown i n Fig.1, as w e l l as t h a t found f o r a v a r i e t y o f o t h e r m a t e r i a l s , t h e r e s u l t s on L i 0.2Si0 a r e i n c o n s i s t e n t

-

i n t h e magnitude o f t h e n u c l e a t i o n r a t e as w e l l as i n i t s $.empera$ure dependence

-

w i t h t h e p r e d i c t i o n s o f c l a s s i c a l theory (9, 10). The o r i g i n o f t h i s discrepancy i s n o t c l e a r a t t h e present w r i t i n g . Experiments by Yinnon and Uhlmann on t h e h e a t i n g r a t e dependence o f t h e c r y s t a l l i z a t i o n temperature o f L i 2 0 * 2 S i 0 glass i n d i c a t e "normal" behavior ( i . e . behavior i n accord w i t h c l a s s i c a l n u c l e a t ~ o n Zheory employed i n the a n a l y s i s ) , w i t h a reasonable n u c l e a t i o n b a r r i e r . While s t u d i e s o f homogeneous n u c l e a t i o n i n glass

-

forming systems are few i n number, d e t a i l e d s t u - dies o f the k i n e t i c s o f heterogeneous n u c l e a t i o n a r e almost non-existent. I t i s known t h a t h e t e r o g e n e i t i e s which p r o v i d e a good match i n l a t t i c e parameter w i t h t h e c r y s t a l being nucleated ( w i t h i n perhaps 5 p c t . ) serve as p o t e n t n u c l e a t i n g c a t a l y s t s . The s i t u a t i o n seems l e s s c l e a r - c u t , however, f o r h e t e r o g e n e i t i e s which p r o v i d e i n t e r - mediate matches i n l a t t i c e parameter (perhaps 10-15 % ) . I n such cases, n u c l e a t i o n on a d v e n t i t i o u s h e t e r o g e n e i t i e s ( c h a r a c t e r i z e d by smaller c o n t a c t angles ? ) can s t i l l p l a y a s i g n i f i c a n t r o l e . More i n t e r e s t i n g , and p o t e n t i a l l y more important, than such observations on second-phase i m p u r i t i e s i s t h e r o l e o f d i s s o l v e d i m p u r i t i e s i n pro- moting c r y s t a l n u c l e a t i o n . There are s c a t t e r e d r e p o r t s i n t h e l i t e r a t u r e o f various t r a n s i t i o n metal and r a r e e a r t h ions having a s i g n i f i c a n t e f f e c t i n t h i s regard;

b u t much a d d i t i o n a l work, t h e o r e t i c a l as w e l l as experimental, i s needed t o provide the r e q u i s i t e i n s i g h t .

For s h o r t c r y s t a l l i z a t i o n times, t h e steady-state concentrations o f s u b - c r i t i c a l em- bryos may n o t be developed, and t h e n u c l e a t i o n r a t e may be time-dependent. Such time- dependent n u c l e a t i o n r a t e s a r e o f t e n approximated by t h e expression o f Zeldovich (13), which should be good f o r s h o r t times :

Here Iv t and are t h e n u c l e a t i o n r a t e s per u n i t volume a t time t and i n steady- s t a t e , the la!ter given by Eqn. (1); and t h e t r a n s i e n t time T i s given, t o order

-

o f

-

magnitude accuracy, as : -c 2 (n*) /nS 2 v

where n* and n a r e t h e number o f molecules i n t h e c r i t i c a l nucleus and on the sur- face of t h e c r i ? i c a l nucleus. The t r a n s i e n t time, ^T

,

1 ik e t h e time f o r o v e r a l l crys- t a l l i z a t i o n of a sample, should thus scale w i t h t h e v i s c o s i t y .

Turning now t o t h e growth o f c r y s t a l s once nucleated, i t should be noted t h a t t h i s process was discussed a t l e n g t h by t h e present author i n a r e c e n t review ( 4 ) . As no- t e d t h e r e , t h e n a t u r e o f t h e c r y s t a l - l i q u i d i n t e r f a c e on an atomic scale i s expected t o have a d e c i s i v e i n f l u e n c e on t h e morphology and k i n e t i c s o f growth. According t o

J a c k s o n ' s c r i t e r i a (15, 1 6 ) , m a t e r i a l s w i t h s m a l l e n t r o p i e s o f f u s i o n (AS <2R) shoud have c r y s t a l - l i q u i d i n t e r f a c e s which a r e rough on an atomic s c a l e , w h i l e Those w i t h l a r g e e n t r o p i e s o f f u s i o n (ASM>4R) s h o u l d have smooth i n t e r f a c e s .

F o r s m a l l dS m a t e r i a l s , t h e macroscopic i n t e r f a c e s s h o u l d b e non-faceted i n b o t h c r y s t a l l i z a t i o n and m e l t i n g ; t h e growth r a t e a n i s o t r o p y s h o u l d be s m a l l ; t h e k i n e t i c s o f m e l t i n g and c r y s t a l l i z a t i o n , c o r r e c t e d f o r t h e v a r i a t i o n o f v i s c o s i t y w i t h tempe- r a t u r e , s h o u l d be equal a t equal d e p a r t u r e s f r o m e q u i l i b r i u m ; and t h e growth r a t e s h o u l d be d e s c r i b e d by t h e normal growth r e l a t i o n :

F o r l a r g e -AS m a t e r i a l s , t h e macroscopic i n t e r f a c e s should be f a c e t e d i n c r y s t a l l i z a - t i o n and non-faceted i n m e l t i n g ; t h e growth r a t e a n i s o t r o p y s h o u l d be l a r g e ; a t equal d e p a r t u r e f r o m e q u i l i b r i u m , m e l t i n g s h o u l d t a k e p l a c e more r a p i d l y t h e n c r y s - t a l l i z a t i o n ; and t h e f r a c t i o n o f growth s i t e s on t h e i n t e r f a c e s h o u l d i n c r e a s e s i - g n i f i c a n t l y w i t h i n c r e a s i n g u n d e r c o o l i n g .

To d e s c r i b e c r y s t a l g r o w t h i n such s m o o t h - i n t e r f a c e m a t e r i a l s , t w o s i m p l e models have been proposed. They r e p r e s e n t l i m i t i n g cases which f o c u s a t t e n t i o n on growth a t d i f - f e r e n t t y p e s o f s i t e s .

( a ) Growth a t s t e p s i t e s p r o v i d e d by screw d i s l o c a t i o n s : Here t h e f r a c t i o n of growth s i t e s i n c r e a s e s l i n e a r l y w i t h u n d e r c o o l i n g , r e f l e c t i n g t h e d i s l o c a t i o n s win- d i n g i n t o t i g h t e r s p i r a l s . The growth r a t e i s expressed :

where f , t h e f r a c t i o n o f growth s i t e s on t h e i n t e r f a c e , can be approximated :

( b ) Growth a t s t e p s i t e s a t t h e p e r i m e t e r s o f two-dimensional n u c l e i formed on t h e i n t e r f a c e : Here t h e f o r m a t i o n o f s u r f a c e n u c l e i r e p r e s e n t s t h e c r i t i c a l f a c t o r i n growth, a l t h o u g h t h e l a t e r a l growth o f t h e n u c l e i across t h e i n t e r f a c e s h o u l d a l s o be i n c l u d e d i n t h e a n a l y s i s . The g r o w t h r a t e can be expressed :

where A and B a r e c o n s t a n t s , whose v a l u e s depend on t h e d e t a i l e d model used t o des- c r i be growth ( 1 4 ) .

These two models a r e based on two d i f f e r e n t views o f t h e i n t e r f a c e and s t e p s i t e s thereon. One c o n s i d e r s a smooth b u t i m p e r f e c t i n t e r f a c e , t h e o t h e r an i n t e r f a c e which i s b o t h smooth and p e r f e c t . Both n e g l e c t any roughness on t h e i n t e r f a c e i n r e g i o n s o t h e r than t h e i d e n t i f i e d sources o f s t e p s . The l a t t e r r e p r e s e n t s a c o n s i d e r a b l e assumption, w h i c h s h o u l d be most i n e r r o r f o r m a t e r i a l s w i t h ASM = 4R

-

5R c r y s t a l - l i z i n g a t l a r g e u n d e r c o o l i n g s .

More r e a l i s t i c t r e a t m e n t s o f c r y s t a l growth have been p r o v i d e d by computer s i m u l a - t i o n t e c h n i q u e s ( f o r d i s c u s s i o n and r e f e r e n c e s , see Ref. 14). These techniques, em- p l o y e d by w o r k e r s such as Jackson, G i l m e r and Leamy, have y i e l d e d i m p o r t a n t new i n - s i g h t s i n t o t h e c r y s t a l l i z a t i o n process, and have p r o v i d e d s t r o n g s u p p o r t f o r t h e views o f Jackson c o n c e r n i n g t h e n a t u r e o f t h e c r y s t a l - l i q u i d i n t e r f a c e . They have i n d i c a t e d t h a t t h e s i m p l e models s h o u l d b e most a p p r o p r i a t e ( 1 ) f o r m a t e r i a l s w i t h s m a l l e n t r o p i e s o f f u s i o n and hence r o u g h i n t e r f a c e s ; and ( 2 ) f o r m a t e r i a l s w i t h v e r y l a r g e e n t r o p i e s o f f u s i o n (as ASM >lOR-15R) c r y s t a l l i z i n g a t modest undercoo- l i n g s .

C9-180 JOURNAL DE PHYSIQUE

I n c o n s i d e r i n g any o f t h e growth models

-

whether computer models o r t h e expressions o f Eqns (12), (13) o r (15)

-

i t i s necessary t o r e l a t e t h e frequency o f t r a n s p o r t a t t h e i n t e r f a c e , v , t o some measurable q u a n t i t y . I t i s suggested t h a t Eqn. ( 4 ) , t o - gether w i t h Eqn. ( 5 ) o r ( 6 ) , be used f o r t h i s purpose. For simple organic m a t e r i a l s , i t can r e a d i l y be v i s u a l i z e d t h a t t h e r e o r i e n t a t i o n o f t h e molecules r e q u i r e d f o r c r y s t a l 1 iz a t i o n i n v o l v e s motions s i m i l a r t o those i n viscous flow. For complex o x i - des such as the s i l i c a t e s , i t seems reasonable t h a t t r a n s p o r t a t the i n t e r f a c e i n - volves t h e breaking of d i r e c t i o n a l bonds and r e o r g a n i z a t i o n o f the network,proces- ses which again a r e s i m i l a r t o those i n viscous flow. Support f o r t h i s view i s pro- vided by t h e r e s u l t s o f c r y s t a l l i z a t i o n s t u d i e s on Si02, Ge02 and Na20.3Si02. For each o f these m a t e r i a l s , t h e growth r a t e s a r e w e l l described by t h e o r e t i c a l models, provided Eqns (4) and ( 6 ) are used t o represent v. The q u a l i t y o f t h e agreement between theory and experiment, and t h e wide range o f temperature over which t h i s agreement i s observed (hundreds o f Centigrade degrees i n each case), i s almost i n - conceivable w i t h o u t t h e suggested v-q r e l a t i o n being a p p l i c a b l e .

The d i s c u s s i o n t o here has been concerned e x c l u s i v e l y w i t h m a t e r i a l s whose r a t e s of c r y s t a l l i z a t i o n a r e l i m i t e d by i n t e r f a c e attachment k i n e t i c s . For m a t e r i a l s which m e l t t o f l u i d l i q u i d s and m a t e r i a l s which c r y s t a l l i z e w i t h s i z a b l e changes i n compo-

s i t i o n , t h e c r y s t a l growth process i s l i m i t e d by d i f f u s i o n a l processes (heat f l o w o r mass d i f f u s i o n ) over much i f n o t a l l o f t h e range where c r y s t a l l i z a t i o n takes place.

The morphology o f such c r y s t a l l i z a t i o n o f t e n takes t h e form o f dendrites, f r e q u e n t l y p a r a l l e l a r r a y s o f d e n d r i t e s . Faceted c r y s t a l s a r e sometimes observed as w e l l , p a r t i - c u l a r l y a t small undercoolings (21, 22).

The d e n d r i t i c morphologies observed a t s i z a b l e undercoolings represent t h e most com- mon form o f d i f f u s i o n - c o n t r o l l e d growth. Such growth t y p i c a l l y takes place a t r a t e s which a r e independent of time, r e f l e c t i n g the s c a l e o f the d i f f u s i o n f i e l d being i n - dependent o f time (17). T h e o r e t i c a l d e s c r i p t i o n s a r e a v a i l a b l e f o r t h e growth o f i s o l a t e d d e n d r i t e s (18) o r p a r a l l e l arrays o f d e n d r i t e s (19, 20). The growth r a t e i n t h e l a t t e r case i s determined by t h e i n t e r d i f f u s i o n c o e f f i c i e n t i n t h e m e l t (which i s apparently i n v e r s e l y r e l a t e d t o the v i s c o s i t y ) , the s i z e and spacing o f the dendrites, and t h e d i f f e r e n c e s between t h e concentration a t t h e i n t e r f a c e and those i n b u l k l i q u i d and i n t h e c r y s t a l .

The c e n t r a l problems a t the present w r i t i n g i n t h i s area a r e : (1) The determina- t i o n o f an independent r e l a t i o n between t h e r a d i u s o f c u r v a t u r e o f t h e d e n d r i t e t i p s and t h e undercooling; and (2) t h e determination o f whether t h e faceted c r y s t a l s ob- served a t small undercoolings r e f l e c t i n t e r f a c e - c o n t r o l l e d growth o f d i f f u s i o n - c o n t r o l l e d growth. The f a c e t i n g i n t h e l a t t e r case c l e a r l y i n d i c a t e s a n i s o t r o p y i n growth; b u t i t remains t o be e s t a b l i s h e d whether t h i s represents l o c a l a n i s o t r o p y w i t h c r y s t a l s whose o v e r a l l growth r a t e i s determined by d i f f u s i o n a l processes (ana-

logous t o t h e growth o f bismuth hopper c r y s t a l s ) , o r whether i t represents i n t e r f a - c e - c o n t r o l l e d growth a t small undercoolings ( w i t h a t r a n s i t i o n t o d i f f u s i o n - c o n t r o l - l e d growth a t l a r g e r undercoolings). Experiments t o r e s o l v e t h i s issue a r e present underway a t S a i n t Gobain and a t MIT.

3. K i n e t i c Treatments o f Glass Formation

As should be c l e a r from t h e discussion i n S e c t i o n 1 above, t h e c r i t i c a l q u e s t i o n i n discussing glass f o r m a t i o n i s n o t WHETHER a m a t e r i a l i s a glass-former w i t h respect t o c o o l i n g from t h e l i q u i d s t a t e , b u t HOW FAST must t h e l i q u i d be cooled t o a v o i d d e t e c t a b l e c r y s t a l l i z a t i o n . I n answering t h i s question, our a t t e n t i o n w i l l t h e r e f o r e be d i r e c t e d t o t h e k i n e t i c s o f c r y s t a l l i z a t i o n and t o t h e c o o l i n g r a t e s achievable w i t h bodies o f v a r i o u s s i z e s and m a t e r i a l c h a r a c t e r i s t i c s .

I n addressing the question o f glass formation, T u r n b u l l (23) adopted t h e c r i t e r i o n of avoiding a s i n g l e c r y s t a l n u c l e a t i o n event d u r i n g c o o l i n g . This approach has j u s - t i f i c a t i o n f o r f l u i d m e l t s where a s i n g l e n u c l e a t i o n event can r e s u l t i n complete

c r y s t a l l i z a t i o n o f a sample. An a l t e r n a t i v e view was adopted by D i e t z e l and Wickert (241, who c o r r e l a t e d glass-forming a b i l i t y ("glassinesst') w i t h low c r y s t a l growth r a t e s . T h i s approach has some j u s t i f i c a t i o n f o r viscous m e l t s where a s e n s i b l y glassy body can be produced even w i t h copious n u c l e a t i o n .

While each o f these views d i r e c t s a t t e n t i o n t o an i m p o r t a n t k i n e t i c process, n e i t h e r can p r o v i d e a s u f f i c i e n t answer t o t h e q u e s t i o n o f how f a s t must a l i q u i d be cooled t o form a glass. To p r o v i d e the r e q u i s i t e p r e d i c t i v e c a p a b i l i t y , i t i s necessary t o consider both n u c l e a t i o n and c r y s t a l growth. Such an approach was adopted by t h e pre- s e n t author about a decade ago (25), and has been used w i t h considerable success t o describe t h e process o f glass formation.

With t h i s approach, one adopts t h e formal t h e o r y o f t r a n s f o r m a t i o n k i n e t i c s (26-29) t o r e l a t e t h e volume f r a c t i o n c r y s t a l l i z e d , Vc/V, t o t h e n u c l e a t i o n r a t e p e r u n i t volume, I,, and the c r y s t a l growth r a t e , u :

Both I,and u must be considered as f u n c t i o n s o f t i m e through t h e i r dependences on temperature i n continuous c o o l i n g s i t u a t i o n s .

By measuring o r c a l c u l a t i n g the n u c l e a t i o n r a t e and c r y s t a l growth r a t e a t a given temperature, t h e degree o f c r y s t a l l i n i t y can be evaluated as a f u n c t i o n o f time. By r e p e a t i n g t h e c a l c u l a t i o n s f o r a s e r i e s o f temperatures, t h e l o c u s o f times a t va- r i o u s temperatures r e q u i r e d t o form a given f r a c t i o n c r y s t a l l i z e d can be determined.

Such l o c i a r e c a l l e d time-temperature-transformation o r TTT curves. A r e p r e s e n t a t i v e TTT curve i s presented i n F i g . 2 f o r a n o r t h i t e f o r a volume f r a c t i o n c r y s t a l l i z e d o f T h i s f r a c t i o n c r y s t a l l i z e d has been suggested (25) t o represent a j u s t - d e t e c t a - b l e degree o f c r y s t a l l i n i t y

-

i.e., a V c / V which must be avoided i n order t o form a glass.

Fig. 2. Time-temperature t r a n s f o r m a t i o n curve f o r a n o r t h i t e .

vc/v

=

The nose o f each TTT curve denotes t h e l e a s t t i m e r e q u i r e d a t any temperature t o form the p a r t i c u l a r f r a c t i o n c r y s t a l l i z e d . I f one assumes t h a t t h e k i n e t i c s o f crys- t a l l i z a t i o n over the f u l l range o f temperature between the m e l t i n g p o i n t TE, and the nose, TN, a r e as r a p i d as a t t h e nose temperature, onemay o b t a i n a u s e f u l e s t i - mate (overestimate) o f t h e c r i t i c a l c o o l i n g r a t e r e q u i r e d t o form a glass :

where AT,,, = TE-TN; and tN i s the time a t t h e nose o f t h e TTT curve.

C9-182 JOURNAL DE PHYSIQUE

The t h i c k n e s s of m a t e r i a l o b t a i n a b l e a s a g l a s s can correspondingly be approximated :

where DTH i s t h e thermal d i f f u s i v i t y of t h e sample.

More a c c u r a t e e s t i m a t e s of t h e c r i t i c a l cooling r a t e s f o r g l a s s formation can be ob- t a i n e d by c o n s t r u c t i n g continuous cooling o r CT curves. This approach was introduced by Grange and Kiefer (30) and applied t o t h e treatment of g l a s s formation by Onorato and Uhlmann (31). Representative CT curves a r e shown i n Fig. 3 f o r c o n s t a n t - r a t e and logarithmic cooling. AS expected, t h e c r i t i c a l cooling r a t e s estimated from CT cur- ves a r e lower than t h o s e obtained from Eqn. ( 1 7 ) , t y p i c a l l y by a f a c t o r of 5-10.

While t h e CT a n a l y s i s provides more a c c u r a t e e s t i m a t e s of c r i t i c a l COO-

l i n g r a t e s than those obtained with Eqn. ( 1 7 ) , i t s t i l l involves some notable appro- ximations ( s e e d i s c u s s i o n i n Ref. 4 ) . An e x a c t d e s c r i p t i o n o f t h e numbers and s i z e d i s t r i b u t i o n s of c r y s t a l s i n a body s u b j e c t t o an a r b i t r a r y thermal h i s t o r y was pro- vided by Hopper e t a l . ( 3 2 ) . The t r e a t m e n t , whose a p p l i c a t i o n involves numerical techniques, can be used t o d e s c r i b e complex phenomena such a s c r y s t a l l i z a t i o n on r e h e a t i n g a g l a s s (a t o p i c t o be discussed e x p l i c i t l y below). With t h i s approach,

t h e volume f r a c t i o n c r y s t a l l i z e d i s represented :

"C - lo 6

-. 67975

1400- V -

-

_x

-

- -

I l l

where Vc/V ( t . ) i s t h e f r a c t i o n c r y s t a l l i z e d t time t j ; I v i i s t h e steady s t a t e nu- J

c l e a t i o n frequency a t time t i ; and Ri ( t i , t j

3 ^,

t h e r a d i i a t time t . of c r y s t a l s

nu-

J c l e a t e d a t time t i , i s expressed :

Fig. 3. Continuous cooling curves f o r cons- t a n t - r a t e and logarithmic cooling f o r l u n a r composition 67975.

Vc/V = l o d . After Ref. 40.

where R; i s t h e s i z e of t h e c r i t i c a l nucleus a t time t i ; and U k i s t h e c r y s t a l growth r a t e a t time t k .

0 2 4 6 8 10 1 2 1 4

loglot (sec)

While t h i s a n a l y s i s , termed c r y s t a l l i z a t i o n s s a t i s t i c s , can provide an e x a c t descrip- t i o n of c r y s t a l l i z a t i o n during cooling of a l i q u i d , i t r e q u i r e s d e t a i l e d k i n e t i c d a t a on t h e m a t e r i a l . In cases where such d a t a a r e not a v a i l a b l e , a s i m p l i f i e d model can be used t o o b t a i n order-of-magni tude e s t i m a t e s of c r i t i c a l cooling r a t e s . This model i s based on t h e observation t h a t t h e noses of TTT curves g e n e r a l l y occur a t tempera- t u r e s i n t h e range of 0.77 T ~ , a n d on a r e l a t i o n between t h e b a r r i e r t o c r y s t a l nu- c l e a t i o n a t a temperature of 0.8 T E , AG**, and t h e molar entropy of f u s i o n , ASM :

where T* = 0.8 TE. T h i s r e l a t i o n i s a p p l i c a b l e f o r high-ASM m a t e r i a l s ( i n Jackson's sense). A corresponding r e l a t i o n between AG** and ASM h o l d s f o r low-ASM m a t e r i a l s , b u t w i t h a l a r g e r p r o p o r t i o n a l i t y c o e f f i c i e n t .

I n u s i n g t h i s approach, i t i s necessary t o e s t i m a t e t h e v i s c o s i t y a t TN = 0.77 TE.

-

F o r o x i d e m a t e r i a l s , t h i s can be accomplished b y combining h i g h temperature v i s c o s i - t i e s o b t a i n e d f r o m t h e models o f Shaw (33) o r B o t t i n g a and W e i l l (34) t o g e t h e r w i t h measured v a l u e s o f t h e g l a s s t r a n s i t i o n temperatures. Using a polynomial f i t t o t h e combined "data", t h e o v e r a l l v i s c o s i t y - t e m p e r a t u r e r e l a t i o n can be obtained. F o r o t h e r c l a s s e s o f m a t e r i a l s , t h e v i s c o s i t y a t 0.77 TE can be e s t i m a t e d f r o m t h e f r e e volume model

.

With these assumptions as a b a s i s , one can o b t a i n an e s t i m a t e o f t h e c r i t i c a l c o o l i n g r a t e f o r g l a s s f o r m a t i o n ( f o r d e t a i l s o f t h e model, see Ref. 35) :

where t h e n u c l e a t i o n b a r r i e r i s BkT a t AT/TE = 0.2

I t i s seen t h a t p r e d i c t i o n o f c r i t i c a l c o o l i n g r a t e s u s i n g Eqn. ( 2 2 ) r e q u i r e s o n l y minimal i n f o r m a t i o n about t h e m a t e r i a l . I n l i g h t o f t h e c o n s i d e r a b l e approximations used i n o b t a i n i n g Eqn. ( 2 2 ) , i t i s remarkable how u s e f u l i t seems t o be i n p r e d i c t i n g c r i t i c a l c o o l i n g r a t e s (see d i s c u s s i o n below).

The d i s c u s s i o n t o here has c o n s i d e r e d o n l y hemogeneous n u c l e a t i o n , and has t h u s d i - r e c t e d a t t e n t i o n t o t h e minimum c o o l i n g r a t e s r e q u i r e d t o f o r m glasses o f v a r i o u s m a t e r i a l s . To e x p l o r e t h e e f f e c t s o f n u c l e a t i n g h e t e r o g e n e i t i e s on c r i t i c a l c o o l i n g r a t e s , t h e s p h e r i c a l cap model o f t h e heterogeneous nucleus ( 3 6 ) has been adopted.

The e f f e c t s o f v a r i o u s c o n t a c t angles, c o n c e n t r a t i o n s o f n u c l e a t i n g h e t e r o g e n e i t i e s and d i s t r i b u t i o n s o f h e t e r o g e n e i t i e s have been i n v e s t i g a t e d (31, 37). I t was found t h a t h e t e r o g e n e i t i e s w i t h c o n t a c t angles l e s s t h a n about 100° have a n e g l i g i b l e e f - f e c t on t h e c r i t i c a l c o o l i n g r a t e f o r g l a s s f o r m a t i o n , a t l e a s t f o r c o n c e n t r a t i o n s

7 -3

i n t h e range o f 10 cm

,

b u t t h a t f o r h e t e r o g e n e i t i e s w i t h s m a l l e r c o n t a c t angles, t h e c r i t i c a l c o o l i n g r a t e i n c r e a s e s a p p r e c i a b l y w i t h decreasing c o n t a c t angle. T h i s i s i l l u s t r a t e d b y t h e r e s u l t s shown i n F i g . 4, which compares continuous c o o l i n g c u r v e s f o r a n o r t h i t e ^- f o r homogeneous n u c l e a t i o n o n l y w i t h those f o r homogeneous nu- c l e a t i o n

+

10 h e t e r o g e n e i t i e s w i t h c o n t a c t angles between 100' and 40'. 7 The crowding o f t h e CT curves f o r small c o n t a c t angles r e f l e c t s d e p l e t i o n o f t h e s u p p l y o f nuclea- t i ng h e t e r o g e n e i t i e s .

F o r l i q u i d s c o n t a i n i n g d i s t r i b u t i o n s o f h e t e r o g e n e i t i e s , t h e CT curves a r e dominated by even s m a l l c o n c e n t r a t i o n s o f h e t e r o g e n e i t i e s h a v i n g small c o n t a c t angles. As t h e h e t e r o g e n e i t i e s w i t h s m a l l c o n t a c t a n g l e s a r e d e p l e t e d (as t h e y n u c l e a t e c r y s t a l s ) , h e t e r o g e n e i t i e s w i t h l a r g e r c o n t a c t a n g l e become e f f e c t i v e , a l b e i t a t p r o g r e s s i v e l y l a r g e r undercoolings. T h i s i s i l l u s t r a t e d by t h e p l o t o f l o g ( I v n ) V S .

(AT: T:)-' shown i n F i g . 5. Pronounced c u r v a t u r e i n such r e l a t i o n s r e f l e c t s t h e pre- sence o f n u c l e a t i n g h e t e r o g e n e i t i e s , l i k e l y c h a r a c t e r i z e d by a range o f c o n t a c t an- g l e s . The o b s e r v a t i o n o f s t r a i g h t - l i n e l o g ( I v n ) vs. (ATZT")"

.

. r e l a t i o n s , such as t h a t shown i n F i g . 1 above f o r , a n o r t h i t e , suggest t h a t t h e n u c l e a t i o n b e i n g observed i s homogeneous.

C9-184 JOURNAL DE PHYSIQUE

Fig. 4. Continuous c o o l i n g curves f o r a n o r t h i t e f o r homogeneous nucl ea- t i o n

+

b u l k heterogeneous nuclea- t i o n w i t h c o n t a c t angles as i n d i - cated (10 h e t e r o g e n e i t i e s cmm3). 7 A f t e r Ref. 37.

I n a l l o f t h e above discussion o f glass formation, steady-state n u c l e a t i o n has been assumed. I t i s recognized, however, t h a t steady-state c o n d i t i o n s are reached o n l y a f t e r some t r a n s i e n t t i m e (see discussion o f Eqn. (10) above). I n e v a l u a t i n g t h e effects o f t r a n s i e n t s on the c r i t i c a l c o o l i n g r a t e s f o r glass formation, t h e analy- s i s o f c r y s t a l l i z a t i o n s t a t i s t i c s has been m o d i f i e d t o i n c l u d e t r a n s i e n t nucleation,

2 3 -1 F i g . 5. Logarithm ( I $ ) vs (ATrTr)

r e l a t i o n f o r o-terphenyl

.

A f t e r Ref. 37.

as represented by Eqns. (10) and (11). I t was found ( 8 ) t h a t i n t h e case o f homoge- neous nucleation, t h e c r i t i c a l c o o l i n g r a t e s f o r most m a t e r i a l s are s u f f i c i e n t l y low t h a t t h e t r a n s i e n t times are much s m a l l e r than t h e t i m e i n t e r v a l s r e l e v a n t i n assessing t h e k i n e t i c s o f c r y s t a l l i z a t i o n ; and hence steady-state c o n d i t i o n s are e f f e c t i v e l y maintained d u r i n g c o o l i n g . The e f f e c t s o f t r a n s i e n t s on t h e o v e r a l l c r y s t a l l i z a t i o n process should be i m p o r t a n t o n l y f o r c o o l i n g r a t e s which a r e n o t a b l y f a s t e r than those r e q u i r e d t o form glasses.

When n u c l e a t i n g h e t e r o g e n e i t i e s a r e present, f a s t e r c o o l i n g r a t e s are r e q u i r e d t o form glasses ( a t l e a s t f o r h e t e r o g e n e i t i e s w i t h c o n t a c t angles l e s s than 100°). I n such cases, the n u c l e a t i o n r a t e a t l a r g e undercoolings can be s i g n i f i c a n t l y time- dependent; and one might be concerned about t h e e f f e c t s o f t h i s time dependence on the c o o l i n g r a t e s r e q u i r e d t o form glasses. I t t u r n s out, however, t h a t i n t h e pre- sence o f good n u c l e a t i n g c a t a l y s t s , t h e c r y s t a l l i z a t i o n process i s dominated by heterogeneous n u c l e a t i o n t a k i n g p l a c e a t s m a l l e r undercoolings. A t such undercoo- l i n g s , t h e v i s c o s i t i e s are lower, t h e t r a n s i e n t times correspondingly smaller, and t h e steady-state r a t e ( o f heterogeneous n u c l e a t i o n ) i s a p p l i c a b l e . Hence t h e c r i t i - c a l c o o l i n g r a t e s f o r glass formation, even i n t h e presence o f p o t e n t heterogeneous n u c l e i , do n o t seem t o be s t r o n g l y a f f e c t e d by t r a n s i e n t s .

As i n d i c a t e d above, t h e a n a l y s i s of c r y s t a l l i z a t i o n s t a t i s t i c s can be used t o d e s c r i - be c r y s t a l l i z a t i o n , n o t o n l y on c o o l i n g a l i q u i d b u t a l s o on r e h e a t i n g a glass. I n

t h e l a t t e r a p p l i c a t i o n , d e t a i l e d c o n s i d e r a t i o n i s g i v e n t o t h e number and s i z e d i s - t r i b u t i o n s o f s m a l l c r y s t a l 1 i t e s i n t h e body, and t o t h e thermodynamic s t a b i l i t y o f such c r y s t a l l i t e s . Since t h e s i z e o f t h e c r i t i c a l n u c l e u s decreases w i t h i n c r e a s i n g undercool i n g , s m a l l c r y s t a l 1 i t e s which a r e s t a b l e a t l o w temperatures can become u n s t a b l e and m e l t o u t as t h e sample i s heated ( i f t h e y do n o t grow s u f f i c i e n t l y d u r i n g h e a t i n g ) . These e f f e c t s have been c o n s i d e r e d i n t h e a n a l y s i s . I t has been f o u n d t h a t s m a l l c r y s t a l l i t e s g e n e r a l l y grow t o a s t a b l e s i z e on r e h e a t i n g a g l a s s a t reasonable r a t e s ( o n l y a t v e r y f a s t h e a t i n g r a t e s i s t h e r e m e l t i n g o f c r y s t a l l i t e s s i g n i f i c a n t ) . The t r e a t m e n t i s p r e s e n t e d a t l e n g t h i n Ref. 40.

The c a l c u l a t e d v a r i a t i o n o f 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 ( t e m p e r a t u r e o f maximum c r y s - t a l l i z a t i o n r a t e ) w i t h h e a t i n g r a t e agrees w e l l w i t h e x p e r i m e n t a l data. T h i s i s il- l u s t r a t e d by t h e r e s u l t s f o r a n o r t h i t e shown i n F i g . 6. Both t h e a b s o l u t e 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 and i t s dependence on h e a t i n g r a t e a r e w e l l p r e d i c t e d by t h e a n a l y s i s . The r e s u l t s can a l s o be used t o determine t h e magnitude o f t h e b a r r i e r t o c r y s t a l n u c l e a t i o n w i t h i n about 2 3kTX. F o r example, t h e v a l u e e s t i m a t e d f r o m t h e DTA r e s u l t s shown i n F i g . 6

-

80 kT*

-

agrees q u i t e f a v o r a b l y w i t h t h a t o b t a i n e d f r o m t h e d i r e c t measurements shown i n F i g . 1 (82 kT*). The approach o f f e r s c o n s i d e r a - b l e promise, t h e r e f o r e , f o r o b t a i n i n g u s e f u l e s t i m a t e s o f n u c l e a t i o n b a r r i e r s

-

and hence o f c r y s t a l

-

l i q u i d s u r f a c e e n e r g i e s

-

f r o m s i m p l e

DTA

experiments.

F i g . 6. V a r i a t i o n o f 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 w i t h h e a t i n g r a t e f o r a n o r t h i t e . A f t e r Ref. 11.

When t h e a n a l y s i s i s a p p l i e d t o a v a r i e t y of m a t e r i a l s , i t i s f o u n d t h a t t h e 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 depends s t r o n g l y on h e a t i n g r a t e ( a dependence which p r o v i d e s t h e b a s i s f o r e s t i m a t i n g n u c l e a t i o n b a r r i e r s ) and l e s s s t r o n g l y b u t s i g n i - c a n t l y on t h e c o o l i n g r a t e used t o form t h e g l a s s . The dependence o f t h e 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 on h e a t i n g r a t e and c o o l i n g r a t e had been f o u n d e x p e r i m e n t a l l y by Thornburg ( 3 8 ) and Lasocka ( 3 9 ) . F o r volume f r a c t i o n s c r y s t a l l i z e d a t

Tg o f

lo-'

t o

lod1'

o r l e s s , t h e c a l c u l a t e d c r y s t a l 1 iz a t i o n t e m p e r a t u r e becomes independent o f c o o l i n g r a t e ; and t h i s has been suggested t o p r o v i d e a d e f i n i t i o n f o r a g l a s s whose thermal s t a b i l i t y i s independent o f thermal h i s t o r y .

The p r e d i c t i o n s o f t h e k i n e t i c t r e a t m e n t s o f g l a s s f o r m a t i o n a r e i n good a c c o r d w i t h e x p e r i m e n t a l measurements. Comparisons between model p r e d i c t i o n s and e x p e r i - ment have been e f f e c t e d i n two p r i n c i p a l ways. F i r s t , t h e maximum s i z e of body o b t a i n a b l e as a g l a s s c a n be determined and t h e h e a t f l o w problem s o l v e d f o r t h e geometry o f concern. As examples o f t h i s approach, t h r e e s t u d i e s may be c i t e d : ( 1 ) f o r l u n a r c o m p o s i t i o n 60095 (40), t h e c r i t i c a l c o o l i n g r a t e e s t i m a t e d from t h e a n a l y s i s o f c r y s t a l l i z a t i o n s t a t i s t i c s was 2-3 K sec-1, w h i l e t h a t o b t a i n e d f r o m e x p e r i m e n t

+

h e a t f l o w a n a l y s i s was a b o u t 5 K sec-1; ( 2 ) f o r a g o l d - s i l i c o n - g e r m a - nium metal g l a s s (41 ), a c r i t i c a l c o o l i n g r a t e o f a b o u t 3 x 107 K s e c - l was e s t i - mated ( r o u g h l y ) f r o m k i n e t i c a n a l y s i s , w h i l e t h e v a l u e i n d i c a t e d by h e a t flow ana- l y s i s was 0.2-1 x 107 K s e c - l ; and ( 3 ) f o r Zr-Be and Ti-Be g l a s s e s (42), good agreement was o b t a i n e d between p r e d i c t e d and e x p e r i m e n t a l c r i t i c a l c o o l i n g r a t e s , b u t t h e agreement r e q u i r e d assumed ( a l b e i t r e a s o n a b l e ) v a l u e s o f t h e n u c l e a t i o n b a r r i e r .

C9-186 _JOURNAL DE PHYSIQUE

The second approach i n v o l v e s t h e d i r e c t determination o f c r i t i c a l c o o l i n g r a t e s i n programmed c o o l i n g experiments. As one example o f t h i s approach, t h e c r i t i c a l coo- l i n g r a t e s were determined (43, 44) f o r a s e r i e s o f l u n a r compositions. I n several cases, t h e n u c l e a t i o n b a r r i e r s were determined from t h e h e a t i n g r a t e dependence of t h e c r y s t a l 1 iz a t i o n temperature. Comparisons were made between experimental da- t a and t h e p r e d i c t i o n s o f both t h e exact a n a l y s i s o f c r y s t a l l i z a t i o n s t a t i s t i c s and t h e s i m p l i f i e d model. The r e s u l t s i n d i c a t e c l o s e agreement between experimen- t a l data and p r e d i c t i o n s o f t h e exact a n a l y s i s . They a l s o i n d i c a t e good agreement, t y p i c a l l y w i t h i n about an o r d e r o f magnitude, between experimental data and c r i t i - c a l c o o l i n g r a t e s p r e d i c t e d u s i n g t h e s i m p l i f i e d model. Considering t h e s i m p l i c i t y o f t h e model and t h e approximations used i n i t s d e r i v a t i o n , t h i s agreement i s r e - markable.

As a second example, Fang e t a1. (45) determined t h e c r i t i c a l c o o l i n g r a t e s f o r forming glasses of v a r i o u s Na20-Si02 and K 0-Si02 compositions. N u c l e a t i o n b a r r i e r s

2

f o r l i m i t i n g compositions i n t h e Na20-Si02 system were determined as about 60 kT*, and t h i s value was used f o r a l l . T y p i c a l r e s u l t s a r e shown i n Fig. 7. A pronounced minimum i n c r i t i c a l c o o l i n g r a t e i s seen i n t h e v i c i n i t y of t h e e u t e c t i c . T h i s

2 0 -

Fig. 7. V a r i a t i o n o f c r i t i c a l c o o l i n g r a t e w i t h composition i n t h e Na20

-

Si02 system. r = experimental data; b = p r e d i c t i o n s o f s i m p l i - f i e d model; o = p r e d i c t i o n s o f exact a n a l y s i s . A f t e r Ref. 45

70 NozO(wl%)

minimumis p r e d i c t e d b y t h e s i m p l i f i e d m o d e l as w e l l as t h e exact analysis, and r e f l e c t s t h e minimum i n c r y s t a l growth r a t e observed i n t h i s range o f composition.

By a p p l y i n g t h e k i n e t i c a n a l y s i s t o a v a r i e t y o f m a t e r i a l s , i t has been found t h a t case o f glass f o r m a t i o n i s favored by a number o f m a t e r i a l c h a r a c t e r i s t i c s . These i n c l u d e :

1. A h i g h v i s c o s i t y a t t h e nose o f t h e TTT o r CT curve. T h i s i m p l i e s a h i g h v i s c o s i - t y a t t h e m e l t i n g p o i n t and / o r a v i s c o s i t y which increases r a p i d l y w i t h f a l l i n g tem- p e r a t u r e below t h e m e l t i n g p o i n t . I t a l s o suggests t h a t h i g h r a t i o s o f T /T would

q E be f a v o r a b l e f o r g l a s s formation.

2. The absence o f p o t e n t n u c l e a t i n g h e t e r o g e n e i t i e s . T h i s i s favored by compositions which a r e good solvents, such as PbO-containing s i l i c a t e s and metal a l l o y s w i t h l a r - ge concentrations of Fe and N i . I t i s a l s o favored by superheating t h e m e l t w e l l above TE before c o o l i n g .

3. A l a r g e b a r r i e r t o c r y s t a l nucleation. This i m p l i e s a l a r g e c r y s t a l - l i q u i d s u r f a - ce energy, and f o r m a t e r i a l s w i t h i n a given c l a s s i n Jackson's sense, a l a r g e heat o f f u s i o n .

4. The n e c e s s i t y o f extensive s o l u t e r e d i s t r i b u t i o n d u r i n g c r y s t a l 1 iz a t i o n . This d i - r e c t s a t t e n t i o n t o compositions which d i f f e r appreciably from the c r y s t a l l i z i n g phases, and i n p a r t i c u l a r t o regions o f e u t e c t i c s ( e s p e c i a l l y deep e u t e c t i c s ) i n the r e s p e c t i v e phase diagrams. The e f f e c t of compositional o r d e r i n g on me1 t s t a b i l i-

t y i s o f importance here. T h i s c o n d i t i o n a l s o suggests, where p o s s i b l e , t h e formu l a - t i o n of multicomponent compositions, and suggests a p r i n c i p l e o f maximum m e l t con- f u s i o n .

4. Concluding D i s c u s s i o n . K i n e t i c t r e a t m e n t s o f g l a s s f o r m a t i o n have made i t p o s s i - b l e t o p r e d i c t w i t h c o n f i d e n c e t h e c r i t i c a l c o o l i n g r a t e s r e q u i r e d t o form g l a s s e s of v a r i o u s m a t e r i a l s . They have a l s o p r o v i d e d a d e s c r i p t i o n o f c o m p l i c a t e d k i n e t i c processes, such as c r y s t a l l i z a t i o n on r e h e a t i n g a g l a s s , and have p e r m i t t e d t h e e- v a l u a t i o n o f n u c l e a t i o n b a r r i e r s f r o m s i m p l e DTA experiments.

The success o f t h e s e t r e a t m e n t s depends on t h e a b i l i t y t o d e s c r i b e t h e i n d i v i d u a l processes o f c r y s t a l n u c l e a t i o n and growth. I n t h e s e areas, s e v e r a l i s s u e s r e m a i n t o be c l a r i f i e d i n s a t i s f a c t o r y d e t a i l . These i n c l u d e :

1. D e t a i l e d d e t e r m i n a t i o n o f t h e r e 1 a t i o n between i n t e r d i f f u s i o n c o e f f i c i e n t

D,

and v i s c o s i t y ( o r between v and q ) f o r o x i d e m e l t s . T h i s i s e s s e n t i a l f o r q u a n t i - t d t i v e t r e a t m e n t s o f b o t h n u c l e a t i o n and growth. A v a i l a b l e e v i d e n c e suggests t h a t a m o d i f i e d S t o k e s - E i n s t e i n r e l a t i o n can be used ( 7 ) ; h u t more e x p e r i m e n t a l d a t a on

6

a r e needed (and a r e p r e s e n t l y b e i n g o b t a i n e d a t MIT).

2. Experimental d e t e r m i n a t i o n s o f t h e k i n e t i c s o f heterogeneous n u c l e a t i o n and t h e r o l e o f d i s s o l v e d i m p u r i t i e s (such as t r a n s i t i o n m e t a l and r a r e e a r t h i o n s ) i n p r o - m o t i n g c r y s t a l n u c l e a t i o n . A v a i l a b l e evidence suggests t h a t homogeneous n u c l e a t i o n r e p r e s e n t s t h e dominant c o n t r i b u t i o n t o t h e f o r m a t i o n o f c r y s t a l n u c l e i i n t h e r e - g i o n o f l a r g e u n d e r c o o l i n g s which a r e c r i t i c a l i n g l a s s f o r m a t i o n ; b u t c a l c u l a t i o n s based on t h e s p h e r i c a l cap model and c l a s s i c a l n u c l e a t i o n t h e o r y suggest t h a t even s m a l l c o n c e n t r a t i o n s o f h e t e r o g e n e i t i e s w i t h s m a l l c o n t a c t angles can have a l a r g e e f f e c t i n i n c r e a s i n g t h e c r i t i c a l c o o l i n g r a t e s r e q u i r e d t o f o r m g l a s s e s . E l i m i n a - t i o n o f h e t e r o g e n e i t i e s , as by d i v i d i n g a l i q u i d sample i n t o many s m a l l d r o p l e t s ( 4 6 ) so t h a t most a r e f r e e o f h e t e r o g e n e i t i e s , s h o u l d f a v o r g l a s s f o r m a t i o n ; b u t i t remains u n c l e a r why t h e e f f e c t s o f n u c l e a t i n g h e t e r o g e n e i t i e s a r e n o t more o f t e n observed.

3. Experimental and t h e o r e t i c a l s t u d i e s o f t r a n s i e n t n u c l e a t i o n . The p r e s e n t a u t h o r i s unaware o f any d e t a i l e d d a t a i n t h i s area. T h i s

agreement between e x p e r i m e n t a l d a t a and c r i t i c a l c o o l i n g r a t e s c a l c u l a t e d u s i n g s t u d y s t a t e n u c l e a t i o n f r e q u e n c i e s i s encouraging and i n a c c o r d w i t h t h e o r e t i c a l e x p e c t a t i o n s . B u t agreement i s a l s o o b t a i n e d between t h e o r e t i c a l and e x p e r i m e n t a l v a r i a t i o n s o f c r y s t a l 1 i z a t i o n temperature w i t h h e a t i n g r a t e ; y e t p r e l i m i n a r y c a l - c u l a t i o n s i n d i c a t e t h a t t r a n s i e n t s a r e s i g n i f i c a n t d u r i n g t h e r e h e a t i n g o f a g l a s s .

4. D e t e r m i n a t i o n o f t h e r a t e - l i m i t i n g process f o r c r y s t a l growth a t modest under- c o o l i n g s i n systems c r y s t a l l i z i n g w i t h s i z a b l e changes i n c o m p o s i t i o n . I f such g r o w t h i s 1 im i t e d by i n t e r f a c e k i n e t i c s , these systems would p r o v i d e a t t r a c t i v e v e h i c l e s f o r s t u d y i n g t h e t r a n s i t i o n f r o m i n t e r f a ~ e ~ c o n t r o l l e d t o d i f f u s i o n

-

c o n t r o l l e d growth; i f n o t , t h e i r s t u d y s h o u l d p r o v i d e s i g n i f i c a n t i n s i g h t i n t o d i f f u s i o n - c o n t r o l l e d growth m o r p h o l o g i e s w i t h a n i s o t r o p i c growth f e a t u r e s . 5. Experimental and t h e o r e t i c a l s t u d i e s o f c o u p l e d d i f f u s i o n - c o n t r o l l e d growth.

The e s t a b l i s h m e n t o f a r e l a t i o n between u n d e r c o o l i n g and c r y s t a l r a d i u s seems es- s e n t i a l , as do more e x t e n s i v e d e t e r m i n a t i o n s o f growth r a t e s , r a d i i o f and spa- c i n g s between c r y s t a l s as f u n c t i o n s o f undercool i n g . A l s o needed h e r e i s under- s t a n d i n g o f t h e o r i g i n o f s p h e r u l i t i c growth morphologies.

Improved knowledge i n areas such as t h e s e would add c o n s i d e r a b l y t o o u r under- s t a n d i n g o f c r y s t a l l i z a t i o n and g l a s s f o r m a t i o n . I t seems c l e a r , however, t h a t t h e u l t i m a t e t r e a t m e n t s o f g l a s s f o r m a t i o n w i l l depend on e s t a b l i s h i n g r e l a t i o n s b e t -

C9-188 JOURNAL DE PHYSIQUE

ween c r y s t a l l i z a t i o n and t r a n s p o r t on t h e one hand and s t r u c t u r e and i n t e r - a t o m i c i n t e r a c t i o n s on t h e o t h e r . T h i s p r e s e n t s a f o r m i d a b l e c h a l l e n g e , w h i c h t h e p r e s e n t a u t h o r expects t o be t a k e n up d u r i n g t h e ccming decade.

ACKNOWLEDGMENTS

The MIT work d e s c r i b e d i n t h e p r e s e n t paper was s u p p o r t e d b y t h e N a t i o n a l Aeronau- t i c s and Space A d m i n i s t r a t i o n and by t h e N a t i o n a l Science Foundation. T h i s s u p p o r t i s g r a t e f u l l y acknowledged, as i s t h e s u p p o r t o f t h e John Simon Guggenheim Founda- t i o n , who p r o v i d e d t h e a u t h o r w i t h a F e l l o w s h i p f o r t h e 1981-82 y e a r . P a r t i c u l a r thanks a r e due t o i n d i v i d u a l s who c a r r i e d o u t much o f t h e work d e s c r i b e d h e r e i n , and s p e c i a l l y t o Drs. R.W. Hopper, G.W. Scherer, P. I .K. Onorato, L . C. K l e i n , H. Yinnon and 0. Cranmer. T h e i r h e l p and t h e i r f r i e n d s h i p have made t h e work a p l e a s u r e .

References

1. SIMON F.E., Z. Phys.

9

(1927) 806.

2. KAUZMANN W., Chem. Rev.

-

43 (1948) 219.

3. SCHERER G.W. and SCHULTZ P.C., "Techniques o f Forming Glasses" t o appear i n T r e a t i s e on Glass, Vol. 3 (Academic Press, New York,'1983).

4. UHLMANN D.R. and YINNON H., "The F o r m a t i o n o f Glasses", t o appear i n T r e a t i s e on Glass, Vol

.

3 (Academic Press, New York, 1983).

5. THOMPSON C.V. and SPAEPEN F., A c t a Met. - 27, 1855 (1979).

6. HOFFMANN J.D., J. Chem. Phys. 29, 1192 (1958). -

7. UHLMANN D.R., KLEIN L.C. and SCHERER G.W., "On D i f f u s i o n and Viscous Flow i n Potassium S i l i c a t e M e l t s " , s u b m i t t e d f o r p u b l i c a t i o n , J . Non-Cryst. S o l i d s . 8. YINNON H. and UHLMANN D.R., "A K i n e t i c Treatment o f Glass Formation. V I I :

T r a n s i e n t N u c l e a t i o n i n Non-Isothermal C r y s t a l l i z a t i o n d u r i n g Coolingn,accepted f o r p u b l i c a t i o n , J. Non-Cryst. S o l i d s .

9. ROWLANDS E.G. and JAMES P.F., Phys. Chem. Glasses - 20, 1 (1979).

10. NEILSON G.F. and WEINBERG M.C., J . Non-Cryst. S o l i d s

2,

137 (1979).

11. CRANMER D., SALOMAA R., YINNON H. and UHLMANN D.R., J . Non-Cryst. S o l i d s

45,

127 (1981).

12. TURNBULL D., J . Chem. Phys. - 20, 411 (1952).

13. ZELDOVICH J.B., A c t a Phys.-Chim., URSS

18,

1 (1943)

14. UHLMANN D.R., " C r y s t a l Growth i n Glass-Forming Systems, a Ten-Year P e r s p e c t i v e " , accepted f o r pub1 i c a t i o n , Advances i n Ceramics I V .

15. JACKSON K.A., i n Growth and P e r f e c t i o n o f C r y s t a l s ( W i l e y , New York, 1958).

16. JACKSON K.A., i n Progress i n S o l i d S t a t e Chemistry, V o l . 4 (Pergamon Press, Oxford, 1967).

17. UHLMANN D.R., i n Advances i n N u c l e a t i o n and C r y s t a l l i z a t i o n i n Glasses and M e l t s (American Ceramic S o c i e t y , Columbus, 1972).

18. HORVAY G. and CAHN J.W., Acta Met. - 9, 695 (1961).

19. CHRISTENSEN N.H., COOPER A.R. and RAWAL B.S., J . Am. Ceram. Soc.

56,

⁵⁵⁷

( 1 973).

20. SCHERER G.W. and UHLMANN D.R., J . C r y s t a l Growth

30,

304 (1975).

21. GUILLEMET C. and DENONCIN J., " K i n e t i c s o f t h e P r e c i p i t a t i o n o f W o l l a s t o n i t e i n I n d u s t r i a l Soda-Lime-Sil i c a t e Glasses", accepted f o r pub1 i c a t i o n , Advances i n Ceramics I V .

22. KIRKPATRICK R.J., KLEIN L. UHLMANN D.R. and HAYS J .F., J . Geophys. Res.

84,

3671 (1979).

23. TURNBULL D., Contemp. Phys.

10,

473 (1969).

24. DIETZEL A. and WICKERT H., Glastech. Ber. - 29, 1 (1956).

25. UHLMANN D.R., J . Non-Cryst. S o l i d s

7,

337 (1972).

26. JOHNSON W.A. and MEHL K.F., Trans AIME

135,

416 (1939) 27. AVRAMI M., J . Chem. Phys.

7,

1103 (1939).

28. AVRApI M., J . Chem. Phys.

8,

212 (1940).

29. AVRAMI M., J . Chem. Phys.

2,

177 (1941).

30. GRANGE R.A. and KIEFER J.M., Trans. ASM

9,

85 (1941).

31. ONORATO P.I.K. and UHLMANN D.R., J. Non-Cryst. S o l i d s

22,

367 (1976).

32. HOPPER R.W., SCHERER G.W. and UHLMANN, J. Non-Cryst. S o l i d s

15,

45 (1974).

33. SHAW H.R., Am. J . S c i .

272,

870 (1972).

34. BOTTINGA Y. and WEILL D.F., Am. J . S c i . - 272, 438 (1972).

35. ONORATO P.I.K., SCHERER G.W., UHLMANN D.R. and YINNON H., "A K i n e t i c Treatment of Glass Formation. V I I I . S i m p l i f i e d Model", s u b m i t t e d f o r p u b l i c a t i o n , J. Non- C r y s t . Sol i d s .

36. TURNBULL D., i n S o l i d S t a t e Physics, Vol. 3 (Academic Press, New York, 1956).

37. YINNON H. and UHLMANN D.R., J . Non-Cryst. Sol i d s

2,

37 (1981).

38. THORNBURG D.D., M a t l s . Res. B u l l .

2,

148 (1974).

39. LASOCKA M., J . M a t l s . S c i . 11, 1771 (1976).

40. ONORATO P.I.K., UHLMANN D.R. and HOPPER R.W., J . Non-Cryst. S o l i d s

41,

^189,

(1980).

41. DAVIES H.A., Phys. Chem. Glasses

17,

159 (1976).

42. TANNER L.E. and RAY R., A c t a - ~ e t .

7,

1727 (1979).

43. UHLMANN D.R., YINNON H. and FANG C.-Y., i n Proceedings o f t h e XI1 Lunar and P l a n e t a r y Science Conference (Pergamon Press, New York, 1981).

C9-190 JOURNAL DE PHYSIQUE

44. FANG C.-Y., YINNON H . and UHLMANN D . R . , "Cooling Rates f o r Glass Containing Lunar Compositions", t o appear in. Proceedings of t h e XI11 Lunar and Planetary Science Conference.

45. FANG C . - Y . , YINNON H . and UHLMANN D . R . , "A Kinetic Treatment of Glass Formation.

VIII C r i t i c a l Cooling Rates f o r Na20-Si02 and K20-Si02 Glasses", submi t t e d f o r p u b l i c a t i o n , J . Non-Cryst. S o l i d s .

46. TURNBULL D., J . Chem. Phys. 18, 817 (1979).