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ON THE LIQUID GLASS TRANSITION
M. Cyrot
To cite this version:
M. Cyrot. ON THE LIQUID GLASS TRANSITION. Journal de Physique Colloques, 1980, 41 (C8),
pp.C8-107-C8-109. �10.1051/jphyscol:1980828�. �jpa-00220334�
JOURNAL DE PHYSIQUE CoZZoque C8, suppZ&ent au n o 8, Tome 41, a o c t 1980, page
C8-107
ON THE LIQUID GLASS TRANSITION
M . Cyrot
L a b o r a t o i r e L o u i s NdeZ, C. N. R. S . , 166X, 38042 GrenobZe Cedex, France
ABSTRACT : We proposed a new model f o r t h e l i q u i d g l a s s t r a n s i t i o n based on t h e f o l l o w i n g assumptions : i n t h e l i q u i d phase, it e x i s t s f l u c t u a t i o n s o f l o n g l i f e time which l e a d t o inhomogeneities i n t h e l o c a l f r e e energy. A s t h e f r e e energy i s p o s i t i o n dependant, we d e s c r i b e t h e t r a n s i t i o n a s a p e r c o l a t i o n process.
We d i s c u s s t h e c o n d i t i o n s under which such a p r o c e s s i s p o s s i b l e .
The l i q u i d - g l a s s t r a n s i t i o n i s f a r from b e i n g understood. There a r e numbers o f reasons. The experimental s i t u a t i o n ( I ) i s n o t c l e a r due t o com- p l e t e l y d i f f e r e n t methods o f formation o f a g l a s s . T r a d i t i o n a l l y g l a s s e s a r e d e f i n e d a s bodies formed by continuous hardening o f a l i q u i d b u t now t h e y
can b e formed by v a r i o u s discontinuous condensation p r o c e s s e s . The approach t o t h e t r a n s i t i o n can b e s t u d i e d only i n t h e f i r s t case. I n t h e second pro- c e s s , one can only s t u d y t h e p r o p e r t i e s o f t h e l i q u i d phase which p e r m i t s amorphous s o l i d s . Thus it is c l e a r t h a t t h e r e c o u l d b e no phase t r a n s i t i o n a t a l l and t h a t t h e g l a s s t r a n s i t i o n would b e k i n e - t i c i n n a t u r e a g r a d u a l f r e e z i n g out o f e q u i v a l e n t c o n f i g u r a t i o n s . It seems s u r e t h a t t h e c r y s t a l l i n e s t a t e i s always more s t a b l e and g e n e r a l l y t h e r e e x i s t a r e c r y s t a l l i s a t i o n p r o c e s s when h e a t i n g t h e amorphous b o d i e s . The amorphous s t a t e i s t h u s me- t a s t a b l e and t h i s i s a reason o f t h e d i f f i c u l t i e s o f a t h e o r e t i c a l s t u d i e s . A simple s t u d y o f t h e l i q u i d - g l a s s t r a n s i t i o n would r e q u i r e t h a t no crys- t a l l i n e phase e x i s t s . Happily n a t u r e p r o v i d e s us with a system which can shed l i g h t on t h i s p r o b l e m : t h e s p i n g l a s s t r a n s i t i o n . I n t h a t c a s e , t h e ran- dom i n t e r a c t i o n between s p i n s does not permit an o r d e r e d s t a t e . Thus i n d e c r e a s i n g t e m p e r a t u r e s p i n s a r e o b l i g e d t o f r e e z e i n a d i s o r d e r e d manner and we have t h e e q u i v a l e n t of a l i q u i d - g l a s s t r a n s i t i o n without t h e d i f f i c u l t i e s o f a p o s s i b l e o r d e r e d s t a t e . We have c o n s i d e r e d i n a p r e v i o u s paper ( 2 t h i s problem and have shown t h e importance of t h e inhomogeneity i n t h e l o c a l f r e e energy o f t h e sys- tem. I f t h i s inhomogeneity i s l a r g e compared t o t h e f r e e z i n g t e m p e r a t u r e , we can d e f i n e d t h i s tempera- t u r e with t h e h e l p o f a p e r c o l a t i o n p r o c e s s . I f not,we have a g r a d u a l f r e e z i n g which i s p u r e l y k i -
n e t i c . I n t h i s p a p e r , we w i l l show t h a t t h e s e two p r o c e s s e s have t h e i r e q u i v a l e n t i n t h e l i q u i d - g l a s s t r a n s i t i o n . However t h e d i f f i c u l t i e s , t h e r e , stem from t h e f a c t t h a t f l u c t u a t i o n s o r inhomoge- n e i t i e s i n t h e system p e r s i s t o n l y d u r i n g a t y p i c a l time c o n t r a r y t o t h e s p i n g l a s s problem where inho- mogeneities a r e due t o s t a t i s t i c i n t h e random in- t e r a c t i o n s . We t h u s have t o c o n s i d e r a new parame- t e r which i s t h e t i m e of f l u c t u a t i o n s .
I n t h i s p a p e r , we want t o p r e s e n t a model f o r t h e g l a s s t r a n s i t i o n . This model i s based on t h e assump t i o n t h a t i n o r d e r t o be a b l e t o f r e e z e , t h e ma- t e r i a l s must show i n t h e l i q u i d phase l a r g e f l u c - t u a t i o n s which l a s t a l o n g t i m e compared t o t h e t i m e o f quenching t h e m a t e r i a l s . These f l u c t u a t i o n s can b e o f any t y p e s . I n a monoatomic m a t e r i a l s it can be f l u c t u a t i o n s o f d e n s i t y , i n a l l o y s f l u c t u a - t i o n s o f c o n c e n t r a t i o n s e t c . . . It i s only assumed t h a t t h i s inhomogeneity p e r s i s t s under t h e t i m e o f quenching t h e m a t e r i a l s . These inhomogeneities mean t h a t t h e energy o r r a t h e r t h e f r e e energy o f t h e s o l i d phase o b t a i n e d by quenching o r r a p i d de- c r e a s e o f t h e t e m p e r a t u r e i s p o s i t i o n dependant.
For i n s t a n c e i n an a l l o y p a r t o f t h i s a l l o y s cor- respond t o a w e l l d e f i n e d compounds with a l a r g e cohesive energy, o t h e r p a r t does not correspond t o a d e f i n e d compound and h a s a lower f r e e energy. Du- r i n g t h e d e c r e a s e o f t e m p e r a t u r e p a r t o f t h e a l l o y w i l l behave a s a s o l i d c l u s t e r d u r i n g t h e t i m e con- s i d e r e d and t h e thermodynamic o f t h i s p a r t c o r r e s - pond t o t h e thermodynamic o f t h e c l u s t e r . Other p a r t w i l l correspond t o t h e l i q u i d phase and have t h e thermodynamics p r o p e r t i e s o f t h e l i q u i d . More p r e c i s e l y a group o f atoms w i l l behave a s a c l u s t e r embedded i n t h e l i q u i d p h a s e i f i t s l i f e t i m e i s g r e a t e r t h a n t h e t i m e o f o b s e r v a t i o n -tM. An atom
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980828
JOURNAL DE PHYSIQUE
w i l l b e l i n k e d t o a s o l i d - l i k e c l u s t e r if t h e pro- b a b i l i t y o f b r e a k i n g i t s l i n k s with t h e c l u s t e r i s s m a l l during t i m e TM. This p r o b a b i l i t y i s given by an Arrhenuis law. I f E i s t h e f r e e energy of i t s l i n k s with t h e c l u s t e r , we have :
- = - T
'
I e - k T ET ( 1
t h e atom i s l i n k e d w i t h t h e s o l i d - l i k e c l u s t e r i f T < 'I&, if T < .rM t h e atom i s i n t h e l i q u i d phase.
Our assumption o f l a r g e inhomogeneities means t h a t it e x i s t s a range o f temperature where c l u s t e r s of t h i s t y p e can e x i s t d u r i n g t h e time TM. The f l u c - t u a t i o n i n f r e e energy W must b e such t h a t
W > kTG Log
-
L~ (?To
Where T i s t h e g l a s s temperature. The l i f e t i m e o f G
t h e inhomogeneitymust be l o n g enough. This i s our second assumption. For t h e s p i n g l a s s problem, t h i s time i s i n f i n i t e .
When t h e temperature i s decreased, t h e s o l i d - l i k e c l u s t e r s grow. A t temperature TG
,
a l e c l u s t e r becomes i n f i n i t e and t h e v i s c o s i t y i s i n - f i n i t e . T h e r e f o r e , t h e g l a s s t e m p e r a t u r e i s d e f i n e d by a p e r c o l a t i o n p r o c e s s .The F u l c h e r law ( 3 ) f o r v i s c o s i t y i s only r e l a t e d t o t h e growth of an i n f i n i t e c l u s t e r a s we w i l l d i s c u s s i n a next paper. It i s obvious t h a t t h i s F u l c h e r law can be observed only i n s i t u a t i o n where t h e time of f l u c t u a t i o n s i s very l o n g , which means t h a t t h e m a t e r i a l s w i l l n o t c r i s t a l l i z e . But t h i s i s n o t a s u f f i c i e n t c o n d i t i o n we must a l s o have a range o f W l a r g e enough. I T t h i s range i s n o t l a r g e enough, we w i l l have a d i f f e r e n t t y p e o f t r a n s i t i o n . We w i l l n o t be a b l e t o s e p a r a t e l i q u i d region from s o l i d - l i k e region. There w i l l b e a qua- s i homogeneous n u c l e a t i o n o f s o l i d - l i k e r e g i o n s . No i n f i n i t e c l u s t e r w i l l grow b u t t h e r e w i l l b e a con- t i n u o u s f r e e z i n g . This i s more s i m i l a r t o a micro- c r y s t a l l i n e m a t e r i a l s .
This model o f p e r c o l a t i o n i s very d i f f i - c u l t t o handle. I n o r d e r t o extend t h e d e s c r i p t i o n , we s e t up t h e f o l l o w i n g semi-phenological model.
We devide t h e m a t e r i a l s o f N atoms i n t o blocks o f n atoms. We have p = ; b l o c k s . N These b l o c k can b e e i t h e r i n t h e l i q u i d phase w i t h an energy which w i l l be t a k e n a s t h e zero o f energy. The amount of phase space w i l l be t a k e n a s t h o s e o f h a r d spheres i n t h e volume v o f t h e blocks
( v
-
n 6 ) nb l o c k can b e a l s o i n t h e amorphous s t a t e with a f r e e energy E p e r atom which depends on t h e posi- t i o n o f t h e b l o c k s and which i s measured from t h e l i q u i d s t a t e . We c a l l P ( E ) t h e p r o b a b i l i t y t h a t a block h a s a f r e e e n e r g y € p e r atom. The p a r t i t i o n
f u n c t i o n i s i f we have p t l i q u i d - l i k e blocks and ( 1 - t ) p s o l i d - l i k e b l o c k s . (1-t)p
v - n 6 n p t
-8
.Z nEjL
e jj.1Q =
d t ( 3 )o
(n! )ptSuch an e x p r e s s i o n f o r t h e p a r t i t i o n f u n c t i o n i s n o t v a l i d f o r an homogeneous system because t h e p t l i q u i d - l i k e b l o c k s can b e choosen a t random.
Let us c a l l -E t h e energy minimum f o r P ( E ) we de- f i n e E ( t ) as
E ( t
I - t =
P ( E ) dE -E0
1
Thus we have Q = d t exp $ ( t )
When N o r p i s going t o i n f i n i t y , we can approxi- mate $ by t h e value o f t h e i n t e g r a n t a t t h e maxi- mum o f $ ( t ) .
I f f o r i n s t a n c e , we assume t h a t P ( E ) i s c o n s t a n t from -E t o 0 , we have :
and t h e maximum v a l u e o f $ ( t ) i s a t t a i n e d f o r
t = ( 1
+
log-) ( 9 )€ 0
As one d e c r e a s e s t h e t e m p e r a t u r e , t h e number o f b l o c k s which becomes s o l i d - l i k e i n c r e a s e s . When t h e s e blocks form an i n f i n i t e c l u s t e r , we have t h e l i q u i d g l a s s t r a n s i t i o n . T G i s r e l a t e d t o a perco- l a t i o n p r o c e s s . The v a l u e o f t a t t h e p e r c o l a t i o n
1 1
t h r e s h o l d i s between
-
t ot.
A tTG,
t h e v i s - 3c o s i t y i s going t o i n f i n i t y b u t t h e r e i s no s i n - g u l a r i t y a s f a r a s t h e thermodynamic q u a n t i t i e s a r e concerned.
Our thermodynamic d e s c r i p t i o n assumes t h a t a s o l i d - l i k e b l o c k remains s o l i d - l i k e d u r i n g a time- g r e a t e r t h a n T* i. e.
nE > kT Log
=-
TM Where6
i s f o u r times t h e h a r d s p h e r e volume. TheseAccording t o t h e r e s u l t ( 9 ) we must have i n o r d e r t o b e s e l f - c o n s i s t e n t
T~ V-n6
n > Log
-
/ ( I + Log -)T o
However n cannot be t o o l a r g e because t h e range of P ( E ) i s a d e c r e a s i n g f u n c t i o n o f n and we must keep t h i s range much g r e a t e r t h a n TG.
Cohen and Grest ( 5 ) have a l s o d e f i n e d TG through a p e r c o l a t i o n t h r e s h o l d . The p e r c o l a t i o n p r o c e s s stems from a d e f i n i t i o n by t h e volume o f l i q u i d - l i k e and s o l i d - l i k e c e l l s . This i s an e x t e n s i o n o f t h e f r e e volume t h e o r y ( 6 ) and t h e . t r a n s i t i o n i s d r i v e n by t h e communal entropy. However s e r i o u s o b j e c t i o n s have been r a i s e d ( 7 ) t o t h i s p i c t u r e . The most s e v e r e i s t h e p e r f e c t l y l i n e a r b e h a v i o r o f t h e t h e r m a l expansion a s t h e f r e e volume be- comes q u i t e s m a l l . We p o i n t o u t t h a t o u r t h e o r y does not l i e on volume b u t on f r e e energy f l u c t u a - t i o n s and s o l i d - l i k e c l u s t e r s a r e n o t d e f i n e d through volume. Secondly, we t h i n k t h a t t h e time must e x p l i c i t l y e n t e r a t h e o r y of t h e l i q u i d g l a s s t r a n s i t i o n and o u r p e r c o l a t i o n p r o c e s s depends on it through e q u a t i o n 1 . Up t o now it seems very d i f f i c u l t t o s t u d y t h i s p e r c o l a t i o n p r o c e s s except perhaps i n t h e s p i n g l a s s t r a n s i t i o n which can b e looked a t a s a model.
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,
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S. K i r k p a t r i c k , Rev. Mod. Phys.
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574 (1973) 5 M.H. Cohen and G.S. G r e s t , Phys. Rev. B. 20,1077 (1979)
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