State of Substance Near the Melting Point

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T i t l e :

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C O U N C I L O F CANADA

T e e h n i e a l T r a n s l a t i o n TT-314 The s t a t e of s u b s t a n c e n e a r t h e m e l t i n g p o i n t , ( 0 s o s t o y a n f i v e s h c h e s t v a v b l f z i t o e h k l p l a v l e n i y a

1

By: G ,

M,

Bartenev Reference: Zhur, E k s p e r , T e o r e t , F i z , ,

-

20: 218-223, 1950. T r a n s l a t e d by: E s t h e r Rabkin

THE STATE OF SUBSTANCE NEAR ..- THE MELTING _{- .---} POINT R e l a t i o n s h i p s ape ~ b t ~ i n e d which d e s c r i b e

the. c r y s t a l l i n e - l i q u i d s t a t e and t h e ano- malous phenomena n e a r t h e m e l t i n g p o i n t s . A comparison of t h e o r y w i t h experinlent i s

g i v e n

..

1. The problem r e g a r d i n g t h e s t a t e o f s u b s t a n c e n e a r t h e m e l t i n g p o i n t has been i n v e s t i g a t e d v e r y l i t t l e , Accord- i n g t o t h e g e n e ~ a a l c o n c e p t s of t h e phase t h e o r y , t h e i n t e r - s e c t i o n of t h e thermodynamic s u r f a c e s

*

( p , T ) o f t h e l i q u i d and s o l i d p h a s e s does n o t i n t r o d u c e any s p e c i a l f e a t u r e s i n t o t h e s t a t e o f s u b s t a n c e n e a r t h e m e l t i n g p o i n t s . Thus, t h e s p e c i f i c h e a t cp, c a l c u l a t e d by t h e thermodynamic f o r m u l a c~ = -T (

b

$/

T ' ) ~ , h a s a c c o r d i n g t o t h e s e c o n c e p t s a smooth temperat,ilre c o u r s e up t o t h e m e l t i n g p o i n t ; a t t h e a c t u a l m e l t i n g p o i n t cp g o e s t o i n f i n i t y . However, i n e x p e r f m e n t s n e a r t h e m e l t f n g p o i n t s o f p u r e s u b s t a n c e s a n o m a l i e s a r e o b s e r v e d i n t h e s p e c i f i c h e a t , i n t h e t h e r m a l e x p a n s i o n c o e f f i c i e n t and i n o t h e r p h y s f e a l v a l u e s . For t h e e x p l a n a t i o n o f t h e s e phenomena, Brody (1) proposed t h a t near" t h e m e l t i n g p o i n t s , owing t o t h e f a c t t h a t t h e c h e m i c a l p o t e n t i a l s o f b o t h p h a s e s become v e r y c l o s e t o - g e t h e r , f l u c t u a t i o n s a r e p o s s i b l e w i t h a t r a n s i t i o n o f t h e s u b s t a n c e from t h e s o l i d i n t o t h e l f q u i d p h a s e . These f l u s - t u a t i o n s c o u l d be c a l l e d phase f l u c t u a t i o n s i n d i ~ e c t a g r e e - ment w i t h t h e t e r m "phase t r a n s i t i o n s " ,

Brody d i d n o t g i v e a s a t i s f a c t o r y c a l c u l a t i o n o f t h e p h a s e f l u c t u a t i o n s n e a r t h e m e l t i n g p o i n t , and t h e c a l c u l a t i o n o f ~ r e n k e l ' ~ ) i n t h i s d i r e c t i o n i s based on a method g i v i n g r e s u l t s which, u n f o r t u n a t e l y , c a n n o t be v e r i f i ed by d i r e c t ex- p e r i m e n t . T h e r e f o r e , i n t h e work of F r e n k e l no comparison o f t h e o r y w i t h experiment i s g i v e n . The e x p e r i m e n t a l f a c t s s u b s t a n t i a t e t h e i d e a t h a t n e a r t h e m e l t i n g p o i n t a s u b s t a n c e i s found i n a p a r t i c u l a r c r y s t a l l i n e - l i q u i d s t a t e . There e x i s t two methods o f t h e o r e t i c a l c o n s i d e r a t i o n o f t h e problem r e g a r d i n g t h e s t , a t e o f a s u b s t a n c e n e a r t h e m e l t i n g p o i n t , One of t h e s e , t h e m o l e c u l a r - k i n e t i c method, s t e m s from

s t a t i s t i c a l thermodynamics and c a n be reduced t o a c o m p u t a t i o n w i t h t h e h e l p of a s t a t i s t i c a l i n t e g r a l o f t h e thermodynamic p o t e n t i a l

4

( p , T ) o f t h e system. T h i s s o l u t i o n e n c o u n t e r s con-

s i d e r a b l e d i f f i c u l t f e s ,

The o t h e r , t h e semi- thermodynamic method o f s o l u t i o n , stems from t h e p h a s e t h e o r y and p h a s e f l u c t u a t i o n s , I n t h i s p a p e r we have s e l e c t e d t h e second method of a p p r o a c h t o t h e s o l u t i o n of t h e problem o f t h e s t a t e o f s u b s t a n c e n e a r t h e m e l t i n g p o i n t , 2 , L e t be t h e a v e r a g e s t a t i s t i c a l q u a n t i t y of t h e l i q u i d p h a s e n e a r t h e m e l t i n g p o i n t f o r a n e q u i l i b r i u m s t a t e , Then, from t h e a s s u m p t i o n s r e g a r d i n g t h e a d d i t i v i t y of

ii

and t h e a d d i t f v f t y of t h e thermodynamic p o t e n t i a l

d?

( p , T ) of t h e s y s t e m one c a n f i n d t h e dependence o f

ii

on t e m p e r a t u r e and p r e s s u r e . I n f a c t , i f

P

( m ) i s t h e d i s t r i b u t i o n f u n c t i o n of t h e p h a s e f l u c t u a t f o n s , t h e n t h e f i r s t a s s u m p t i o n can be e x p r e s s e d i n t h e f o l l o w i n g manner:

where M f s t h e mass of t h e system, m i s t h e i n s t a n t a n e o u s q u a n t i t y

o f t h e l i q u i d phase, The d i s t r f b u t f o n f u n c t i o n f' i s a f u n c t i o n not o n l y of m, b u t a l s o of t h e parameter M, Vie w f l l prove t h i s , Transforming (1) by an i n t e g r a t i o n by a r t s and t a k i n g i n t o ac-

E

count t h e c o n d i t i o n of n o r m a l i z a t i o n

\

p d m = 1, we o b t a i n t h e J f o l l o w i n g f u n c t i o n a l e q u a t i o n s : 0

M

1 ~ ( 0 )

-

p

(rn) d m = c o n s t . and

p

( m )

= a g / b m

A s o l u t i o n of t h e f u n c t i o n a l e q u a t i o n i s t h e f u n c t i o n :

p=g(m/~).

From here ( r n ) =

l , q g

( ; I = t v c

i

1

F u r t h e r , it should be t a k e n i n t o account t h a t one and t h e same q u a n t i t y m of l i q u i d phase may be l o c a l i z e d i n t h e system by a number of methods, t h e sane a s i n gaseous s t a t i s t i c s each m a c ~ o s t a t e can be r e a l i z e d by a m u l t i p l i c i t y of m i c r o s t a t e s ,

Henceforth, we w f l l c o n s i d e r o n l y systems which a r e found i n a n e q u i l f b ~ f u m s t a t e , For a system placed i n a t h e r m o s t a t under t h e c o n d i t i o n t h a t p = c o n s t , and found i n t h e e q u i l i b r i u m , t h e p r o b a b i l i t y of t h e o r i g i n a t i o n by one of t h e methods of t h e l i q u i d phase i n t h e q u a n t i t y m i s p r o p o r t i o n a l t o t h e Boltzmann f a c t o r .

where

A $

i s t h e d e v i a t i o n of t h e thermodynamic p o t e n t i a l of t h e system from t h e o r i g i n a l p u r e l y c r y s t a l l i n e s t a t e , The v a l u e d rf, may be expressed i n t h e f o l l o w i n g form:

where4 2,

d

a r e t h e chemical o r t h e s p e c i f i c p o t e n t i a l s of t h e l i q u i d and t h e s o l i d pimses,

6

i s t h e s u r f a c e t e n s i o n on k t h e boundary o f t h e d i v i s i o n of t h e p h a s e s ; i s t h e number of t h e s e c t i o n s ( i n i t i a t o r s ) of t h e l i q u i d phase c o n s i s t i n g k of j atoms o r molecules, S i s t h e v a l u e of t h e s u r f a c e of j t h e d i v i s i o n of t h e phase corresponding t o t h e s e c t f o n a of t h e l i q u i d phase, k i s t h e index corresponding t o one of t h e p o s s i b l e s t a t i s t i c a l d i s t r i b u t i o n s i n t h e system of t h e l i q u i d phase i n t h e q.~ a n t i t y rn,

A t t h e m e l t i n g p o i n t t h e d i f f e r e n c e of t h e chemical poten- t i a l s become z e r o , As we g e t f u r t h e r away from t h e p o f n t of m e l t i n g , t h e d i f f e r e n c e of t h e chemical p o t e n t f a l s i n c r e a s e s w i t h o u t l i m i t ( a t t h e b e g i n n i n g a c c o r d i n g t o a l i n e a r r e l a t f o n -

s h i p ) , As r a g a r d s t h e s u r f a c e t e n s i o n , t h e n i n g e n e r a l i t changes v e r y l i t t l e w i t h a change i n temperature and f t does n o t go over t o z e r o a t t h e m e l t i n g p o i n t , I n a d d i t i o n , t h e s u r f a c e t e n s i o n on t h e boundary s o l i d - m e l t i s a f a i r l y s m a l l v a l u e q . From h e r e i t f o l l o w s t h a t s u f f i c i e n t l y f a r from t h e m e l t i n g p o i n t a t m ) 0, t h e second term i n e x p r e s s i o n ( 4 ) may

be n e g l e c t e d . A t m = 0 , t h i s term becomes 0 ,

his

c o n d i t i o n f o l l o w s from t h e f a c t t h a t t h e d i f f e r e n c e i n t h e d e n s i t i e s of b o t h phases i s s m a l l , and t h a t b o t h phases c o n s i s t of p a r t i c l e s of one k i n d .

Henceforth, we w i l l s o l v e t h e problem f o r t h e

temperature r e g i o n vhere t h i s assumption h o l d s , I n s t e a d of ( 3 ) and ( 4 ) we w i l l w r i t e t h e approximate e x p r e s s i o n f o r t h e p r o b a b i l i t y q(m) :

d m )

-e

-

~ ( P ~ - P ~ ) / ~ T ₍₅

₁

I n t h i s form t h e p r o b a b i l i t y q(m) of t h e l o e a l i z a t i o n of t h e f f x e d q u a n t i t y m of t h e l i q u i d phase i s t h e same f o r any of the methods mentioned e a r l i e r . Something s i m i l a r e x i s t s i n gaseous s t a t i s t i c s when, by n e g l e c t i n g t h e In- t e r a c t i o n of t h e p a r t i c l e s , t h e p r o b a b i l i t y of any group i s assumed a p r i o r i t o be t h e same, However, i n our c a s e a c a l c u l a t i o n of t h e p r o b a b i l i t y of t h e s t a t e cannot be c a r r f ed o u t by a n e l e m e n t a r y method,

I n f a c t , l e t 4 ( m ) d r n be t h e p r o b a b i l i t y t h a t i n g e n e r a l a m e t a s t a b l e phase can be d e t e c t e d i n a q u a n t i t y

from

m t o m

*

dm. T h i s w i l l be t h e l a r g e r t h e g r e a t e r t h e number of ways by which t h i s s t a t e can be r e a l i z e d . I f we could con-

s i d e r t h a t a l l t h e elementary p r o b a b i l i t i e s q(m) a r e inde-

pendent, t h e n t h e above-mentioned p r o b a b i l i t y , a p p a r e n t l y , would be p r o p o r t i o n a l t o t h e p r o b a b i l i t y q(m) and t o t h e number of arrangements, However, we have no r i g h t t o make

such a n assumption, s i n c e we do n o t d e a l ' w i t h a g a s , Hence, i n o r d e r t o o b t a i n a c o r r e c t e x p r e s s i o n f o r P (m), f t i s n e c e s s a r y t o f i n d a more compJex dependence on q(m), which i n a g e n e r a l form we w i l l e x p r e s s a s f o l l o w s :

4

(mb = F

The f u n c t i o n F, i n p a r t i c u l a r , should be s u c h t h a t f ( m ) would s a t i s f y c o n d i t i o n ( 2 ) , I f we t a k e i n t o c o n s i d e r a t f o n

e x p r e s s i o n ( 5 ) f o r q ( m ) , t h l s w i l l l e a d t o a s o l u t i o n o f t h e form: p (m) = f ( z ) , vlhere f i s an a r b i t r a r y f u n c t l o n o f t h e

I

i n d e p e n d e n t v a r i a b l e z = exp

(

-

@ 2

-e?1

p

)

Here k

T

i s a n a r b i t r a r y c o n s t a n t i n d e p e n d e n t o f

rn

and I;?* S i n c e , a c c o r d i n g t o o u r a s s u m p t i o n , w e d e a l w f t h t h e t e m p e r a t u r e r e g i o n where t h e p r o b a b i l i t y of phase f l u c t u a t i o n s i s s m a l l , hence z w i l l b e a s m a l l v a l u e . I n a d d i t i o n , a s we g e t away f u r t h e r from t h e m e l t i n g p o i n t ,

z

a p p r o a c h e s z e r o , Hence, i f we expand t h e f u ~ l c t l o n f i n t o a s e r i e s w i t h ~ e s p e c t t o z and i f we l i m i t o u r s e l v e s t o t h e l i n e a r t e r m of t h e s e r i e s a e w i l l o b t a i n , by k e e p i n g i n mfnd t h a t f ( 0 ) = 0 ; a d i s t r i b u - t i o n f u n c t i o n o f t h e f o l l o w i n g form: S u b s t i t u t i n g t h l s e x p r e s s i o n i n t o formula ( 1 ) we f i n d t h e r e l a t i o n s h i p f o r m i d e n o t i n g x

=,u(O2 -

Q = I / ' ~ T : m =

M ( ;

+ _{1 - e A ) (71} 3 , Gachkovskii and S t r e l k o v i n v e s t i g a t e d t h e t h e r m a l expansf on of a z i n c m o n o c r y s t a l n e a r t h e m s l t f n g p o i n t , They have found an anomalous i n c r e a s e i n t h e c o e f f i c i e n t s of l i n e a r e x p a n s i o n o Using r e l a t i o n s h i p ( 7 ) i t i s p o s s i b l e t o c a l c u l a t e t h e anomalous p a r t of t h e v o l u m e t r i c t h e r m a l c o e f f i c i e n t of e x p a n s i o n by t h e f o r m u l a :

-

he

v a l u e o f p m a y n a t u r a l l y depend on p and T , b u t a s a comparison w i t h experiment shows t h i s dependence i s v e r y weak, and i n t h e r e -

v2

-

v 1

1

C( anomalous =

--

-

v1

where V2 and V 1 a r e t h e s p e c i f i c volumes o f t h e l i q u i d and s o l i d p h a s e s , and t h e n t o compare i t w i t h e x p e r i m e n t , It i s of i n t e r e s t t o e x p l a i n f i r s t of a l l f o r what t e m p e r a t u r e r e - g i o n our s e l e c t i o n of t h e d i s t r i b u t i o n f u n c t i o n ( 6 ) i s jus- t i f i e d , I n F i g . 1 t h e r e s u l t s a r e g i v e n ( t h e p o i n t s and c u r v e 1) f o r t h e c o e f f i c i e n t of t h e volume e x p a n s i o n & n e a r t h e m e l t - i n g t e m p e r a t u r e of z i n c c a l c u l a t e d from t he e x p e r i m e n t a l d a t a ( 4 ) o f S t r e l k o v and Gachkovskii The d o t t e d e x t r a p o l a t e d s t r a i g h t l i n e 3 g i v e s

o(,

-

t h e nortmal p o r t i on of t h e e x p a n s i o n c o e f f i c i e n t

.

The theoretical c u r v e may be c a l c u l a t e d from t h e e x p r e s s i o n

O(

= o(

*

_{o( a n ,}

* e r e o( an i s determined by f o r m u l a ( 8 ) . The t h e o r e t i c a l

c u r v e c a l c u l a t e d a t / = 1 0 0 1 0 g i s given i n t h e same f i g i r e a s number 2 , I t shows a s a t i s f a c t o r y a g r e e m e n t w i t h t h e ex-

p e r i m e n t a l dependence up t o t e m p e r a t u r e s of 2-3O b e f o r e t h e m e l t i n g p o i n t , c o r r e s p o n d i n g t o 4 1 9 . ~ ~ ~ . A c a l c u l a t i o n by formula ( 7 ) a t t h e above v a l u e o f / g f v e s a t 3' b e f o r e t h e m e l t i n g p o i n t t h e q u a n t i t y of l i q u i d p h a s e of a p p r o x i m a t e l y o n l y 1$, I n ~ i g . 2 t h e r e s u l t s a r e o i v e n ( 5 b f t h e i n v e s t i g a t i o n of t h e s p e c i f i c h e a t of t i n n e a r t h e m e l t i n g p o i n t , c o r r e s p o n d i n g t o 2 3 1 , ~ ~ ~ . The s m a l l maximum a t 166O, a p p a r e n t l y , c a n be ex- p l a i n e d by a t r a n s f o r m a t i o n o c c u r r i n g i n t h e s o l i d phase, and we do n o t have t o t a k e i t i n t o a c c o u n t ,

The t o t a l s p e c i f i c h e a t i s e q u a l t o t h e sum of t h e normal p o r t i o n of t h e s p e c i f i c h e a t , which a t h i g h t e m p e r a t u r e s has a l i n e a r c o u r s e , and t h e anomalous p o r t i o n of t h e s p e c i f i c h e a t , which can be c a l c u l a t e d by t h e f o l l o v ~ i n g formula:

C~ (anomalous) = (+)p ( 9

where i s t h e s p e c i f i c h e a t of t h e s o l i d a t c o n s t a n t p r e s s u r e ( e q u a l t o 14.0 ~ a l / ~ )

.

Taking i n t o account t h e normal p o r t i o n of t h e s p e c i f i c h e a t , whl.ch i n F i g . 2 i s shown by a d o t t e d e x t r a p o l s t e d s t r a i g h t l i n e , we w i l l o b t a i n t h e c a l c u l a t e d r e l a t i o n s h i p i n t h e form of cupve 2 ,

This r e l a t i o n s h i p corresponds t o t h e value/(*= 3 0 1 0 - l ~

8

4. Vie w i l l e x p l a i n t h e p h y s i c a l meaning of t h e c o n s t a n t e n t e r i n g i n t o t h e d i s t r i b u t i o n f u n c t i o n ( 6 ) , For t h i s purpose we w i l l s e p a r a t e t h e sub-system, t h e mass of which i s e q u a l t o p

,

and we w i l l c o n s i d e r t h e meaning of t h e d i s t r i b u t i o n f u n c t i o n a s a p p l i e d t o t h e sub-system, t a k i n g i n t o account t h a t now t h e d i s - t r i b u t l o n f u n c t i o n r e l a t e s t o t h e phase f l u c t u a t i o n s i n which p a r t i c i p a t e only t h e molecules of t h e sub-system, I f we assume t h a t M

=/u,

we can say t h a t t h e d i s t r i b u t i m f u n c t i o n ( 6 ) goes o v e r i n t o t h e elementary p r o b a b i l i t y of t r a n s i t i o n q(m), e o r r e s - ponding t o t h e p r o b a b i l i t y of t r a n s i t i o n i n t o a l i q u i d s t a t e by a s i n g l e method, T h i s means t h a t t h e sub-system e i t h e r complete- l y goes over i n t o a l i q u i d s t a t e o r does n o t go over a t a l l , Any p a r t i a l t r a n s i t i o n o f t h e sub-system i n t o a l i q u i d s t a t e

This v a l u e i s l e s s t h a n f o r t h e c a s e of z i n c , a p p a r e n t l y , owing t o t h e e f f e c t of s m a l l i m p u r i t i e s p r e s e n t i n t h e t i n i n v e s t i g a t e d .

m y be r e a l i z e d by more t h a n one method and, hence, should be excluded from t h e a n a l y s i s , On t h e c o n t r a r y , t h e t r a n s i t i o n of t h e sub-system/(* a s a w h o l e i n t o a l i q u i d s t a t e may be r e - p r e s e n t e d as a t r a n s i t i o n by one method. T h e r e f o r e , no por-

t i o n of t h e sub-system,hcan go over i n t o a l i q u i d s t a t e I n - dependently of i t s o t h e r p o r t i o n s ,

Thus, t h e s u b - s y s t e m / c c r e p r e s e n t s a c e r t a i n c r i t i c a l molecular complex i n d i c a t i n g , a p p a r e n t l y , a t r a n s i t i o n w i t h a d e c r e a s e i n t h e number of p a r t i c l e s from t h e macro- t o t h e

mi crosystems.

A s f a r a s we know, an approximate e v a l u a t i o n of t h i s ( 2 1

molecular complex was f f rst given by F r e n k e l This cor- responded t o a n o r d e r of hundreds of atoms. According t o t h e e x p e r i m e n t a l d a t a g i v e n i n F i g . 2 , from t h e anomalous s l o p e of t h e s p e c i f i c h e a t of t i n n e a r t h e m e l t i n g p o i n t it i s p o s s i b l e t o e v a l u a t e approximately t h i s complex by u s i n g t h e F r e n k e l formula Thi s v a l u e i s of t h e o r d e r of 0,5010 3 atoms of t i n , while a c c o r d i n g t o our d a t a t h e value o f , w

3

should correspond t o 1.5.10 atoms of t i n . However, t h i s

3 v a l u e f o r z i n c corresponds t o 9.10 atoms. 5. As one of t h e r e s u l t s we w i l l c o n s i d e r t h e problem r e g a r d i n g t h e c e n t e r s of m e l t i n g and c r y s t a l l i z a t i o n . By t h e c e n t e r s of m e l t i n g o r , i n g e n e r a l , by t h e c e n t e r s of phase t r a n s i t i o n s , we w i l l t a k e i t t o mean t h e p o i n t s ( o r s m a l l r e g i o n s ) where f l u c t u a t i n g phase t r a n s i t i o n s o r i g i n a t e , and from h e r e t h i s t r a n s i t i o n proceeds f u r t h e r , Any p o i n t of a

From the considerations of the physical meaning of ,.tc

,'

follows the assurance that if a given point is a center of melting, then the points found in its vicinity, enclosed in a small volume equal to the volume of the critical molecular

complex,^^,

cannot be at the same time independent centers of melting, Hence, in a crystal of mass

M

the maximum number of independent centers of melting existing at the same time cannot be larger than the number n =

M

/

P

,

where n is the number of all the molecular complexes entering into the crystal,

We will consider the probability of melting of the total crystal as a whole, For this purpose we will substitute

rn

=

M

into the distribution function (6). VJe obtain:

?(M)

= c(e' M(C2- P1)/k~) l b (10)

This expression, owing to the large value of n, is much larger than the probability q ( h l ) ~ e - " ( P 2

- fi)/kT

<

1, corresponding to the probability of phase transitions by

a

single method, *om this we can conclude that the melting of the total crystal may be accomplished by a multiplicity of methods, We will con-

sider the two possible extreme methods of phase transitions during the melting of the total crystal: 1) a transttion with the formation of

n

centers of melting, and 2) a transition with the formation of one center of melting.

The first method of transition is clear from a physical point of view, The second method of phase transitions may be

of t h e p o s s f b l e c e n t e r s of m e l t i n g u o f l i q u i d phase i s p r o d u c e d .

Then two c a s e s a r e p o s s i b l e : e i t h e r t h e c e n t e r of m e l t i n g will vanfsh, o r i t w i l l develop owing t o t h e t r a n s f e r o f t h e i n d f v f - d u a l atoms o r molecules of t h e c r y s t a l through t h e d f v i s i o n of t h e phases, u n t i l t h e t o t a l c r y s t a l i s d i s s o l v e d . I n g e n e r a l , t h e volume of t h e l i q u i d phase produced may e i t h e r d e c r e a s e o r i n c r e a s e f n a f l u c t u a t i n g manner a s a r e s u l t of t h e c h a r a c t e r i s - t f c mechanism of " e v a p o r a t i o n and condensationt1 of atom3 from t h e s u r f a c e of t h e d i v i s i o n of p h a s e s , Thus, when a c r y s t a l approaches t h e m e l t i n g p o i n t and goes o v e r i t , i t p o s s e s s e s n u c l e i of t h e l i q u i d phase, some of which exceed t h e c r i t i c a l dfmensions,

I f we c o n s i d e r t h e phase f l u c t u a t i o n s i n t h e l i q u i d phase, t h a t i s , i f we approach from t h e s i d e of t h e l i q u i d phase t h e temperature of c r y s t a l l i z a t f o n , t h e n a l l our c o n s i d e r a t i o n s re.- main i n f o r c e , But i n t h e d i s t r i b u t i o n f u n c t i o n ( 6 ) t h e chemi- c a l p o t e n t i a l s T 2 andql should be t r a n s p o s e d , so t h a t t h e metas-. t a b l e phase i s now t h e s o l i d phase, Then, i n t h i s c a s e , we d e a l wf t h c e n t e r s of c r y s t a l l i z a t f on.

Although up u n t i l now w e considered o n l y e q u i l i b r i u m pro- c e s s e s , but t h e i d e a of the c r y s t a l l i z a t i o n c e n t e r s remains in

f o r c e a l s o d u r i n g t h e t r a n s i t i o n t h r o u g h t h e p o i n t of c r y s t a l - l i z a t i o n , i f t h e p r o c e s s d i f f e r s from an e q u i l f b r f u m p r o c e s s

( s u p e r c o o l i n g ) .

O f t h e numerous p o s s i b l e niethods by which t h e l i q u i d phase may become c r y s t a l l i z e d t h e r e a r e two extreme methods:

a s a minimum a t one c e n t e r of c r y s t a l l i z a t i o n , and a s a maximum a t n c e n t e r s o f c r y s t a l l i z a t i o n e T h i s w i l l g i v e a m o n o c r y s t a l and a n i d e a l p o l y c r y s t a l , r e s p e c t i v e l y A p p a r e n t l y , I t i s pos- s i b l e t o i n c r e a s e t h e p r o b a b i l i t y e i t h e r o f t h e f f r s t o r t h e second c a s e by means o f t h e e x p e r i m e n t a l c o n d i t i o n s , f o r example, by one o r o t h e r methods of c o o l i n g .

Thus, t h e i d e a l p o l y c r y s t a l w i l l g i v e t h e lower boundary o f p o l y d i s p e r s f o n . The dimensions of t h e i n d i v i d u a l s m a l l e r y s - t a l s o f s u c h a p o l y c r y s t a l s h o u l d c o r r e s p o n d t o t h e dimensions o f t h e c r i t i c a l m o l e c u l a r eomplex. C o n c l u s i o n s

-

1, On t h e b a s i s o f t h e g e n e r a l c o n s i d e r a t f o n s a r e l a t f o n - s h i p f s o b t a i n e d d e s c r i b i n g t h e c r y s t a l l i n e - l i q u i d s t a t e o f p u r e s u b s t a n c e s n e a r t h e m e l t i n g p o i n t s , C a l c u l a t i o n shows t h a t i n a z i n c m o n o c r y s t a l 3' b e f o r e t h e m e l t i n g p o i n t , t h e q u a n t i t y o f l f quid p h a s e i s a p p r o x i m a t e l y 1%.

2 , The anomalous p o r t i o n o f t h e s p e c i f i c h e a t and t h e anomalous p o r t f on of t h e c o e f f i c f a n t o f vo1umetrj.c e x p a n s i o n n e a r t h e p o i n t o f m e l t i n g i s c a l c u l a t e d from t h e above r e l a - t i o n s h i p . A comparison of t h e e x p e r i m e n t a l d a t a w i t h t h a t

c a l c u l a t e d f o r z i n c and t i n g i v e s s a t i s f a c t o r y a g r e e m e n t ,

3, The c o n s t a n t p L , which e n t e r s i n t o t h e t h e o r e t f c a l formulae and c a n be d e t e r m i n e d from e x p e r i m e n t , i s a c c o r d i n g t o t h e p h y s i c a l meaning t h a t c r i t i c a l m o l e c u l a r complex which i n d i c a t e s a t r a n s i t i o n w i t h a d e c r e a s e i n t h e number o f p a r -

o f t h e p h a s e t o t h e m i c r o p r o p e r t i e s o f t h e i n d i v i d u a l m o l e c u l e s , The dfmensf ons o f t h e c r f t i c a l m o l e c u l a r complex a r e 9,000

atoms f o r z i n c and 1 , 5 0 0 atoms f o r t i n .

4, The maximum number of i n d e p e n d e n t c e n t e r s o f m e l t i n g o r c r y a t a l l i z a t f o n which c a n be produced s i m u l t a n e o u v l y in a s y s t e m i.5 e q u a l t o t h e number of' t h e c r i t i c a l m o l e c u l a r corn-=

p l e x e s c o n s t i t u t i n g t h e s y s t e m , The i d e a l p o l y c r y s t a l c o r r e s - ponds t o t h e e a s e when i n t h e l i q u i d phase t h i s maxfmum quan- t i t y of c r y s t a l l i z a t i o n c e n t e r s c a n be produced s i m u l t a n e o u s l y , Hence, t h e v a l u e , u i n d i c a t e s t h e lower l i m i t of t h e p o l g d i a p e r s f on

/

o f p o l y c r y s t a l l f n e m a t e r i a l .

I n s t i t u t e of F i n e Chemical Technology Received 2 9 t h September, 1949

L i t e r a t u r e .. .

1,

E,

Brody, Phys, Z s , , 2 3 , 1 9 7 , 1 9 2 2 ; J o u r n , Chem, P h y s , , 7 , 972, 1 9 3 9 ,

-

2 , Yao I . E'renkel, Z h u r o E k s p e r , T e o r e t F i z . , ,

-

9, 952, 1 9 3 9 , 3 ,

V t

Gachkovskii and

P o

S t r e l k o v , Z h u r , E k s p e r , T e o r e t F f z , , 7, 532, 1937. 4 , G o B a r t e n e v , Zhur, F f z , Khim,,

--

23, 1075, 1949.

Fig.

I.

The volumetric coefficient of expansion of zinc near the melting point.

Fig. 2. The specific heat of tin near the melting point.