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A new approach to the prediction of glass formation
Mat. Res. Bull. Vol. 3, pp. 265-280, 1968.
P e r g a m o n P r e s s , Inc.
P rin ted
in the United States.
A NEW APPROACH TO THE PREDICTION OF GLASS FORMATION*
P. T. Sarjeant and Rustum Roy Materials Research Laboratory The Pennsylvania State University University Park, Pennsylvania
(Received J a n u a r y 19, 1968; Communicated by R. Roy)
ABSTRACT The rate of cooling has usually been ignored in t r e a t m e n t s of glassforming capability. The rate r e q u i r e d to form n o n - c r y s t a l l i n e solid** phases is shown h e r e i n to depend d i r e c t l y on the melting t e m p e r a t u r e and a v e r a g e coefficient of self diffusion, and i n v e r s e l y upon melt viscosity, relaxation time and linear d i m e n s i o n s of the ionic s pe c i e s c o m p r i s i n g the melt. Calculated dependence of the formation of NCS phases as a function of these p a r a m e t e r s is shown graphically. A d i m e n s i o n l e s s e x p r e s s i o n is d e r i v e d (named the " g l a s s number") for the prediction of glass fo rmatio n at any p r e s c r i b e d cooling rate for any melt of known viscosity.
Introduction Although other a p p r o a c h e s to the problem of predicting glass formation f r o m melts utilize principally the concepts of coordination (1), bond strength or type (2, 3) and "free volume" (4), the effect of the cooling rate has not gene r a l l y been taken into consideration, at least quantitatively.
Since NCS phases
a r e always metastable, their f or m a t io n from m e l t s and retention is p r i m a r i l y *This paper was p r e s e n t e d at the International Conference on the C h a r a c t e r ization of Materials, R o c h e s t e r , N . Y . , USA, November 8-10, 1967. **The abbreviated form of the e x p r e s s i o n " n o n - c r y s t a l l i n e solid" (NCS) used here r e f e r s specifically to an a r r a n g e m e n t w h e r e i n only the first, second or perhaps third n e a r e s t neighbours to a given atom or ion a r e s i m i l a r . Long range o r d e r is absent. We have previously used the t e r m short range o r d e r only (SROO) for the same purpose. 265
266
GLASS FORMATION
Vol. 3, No. 3
a question of kinetics and thus the cooling rate must be of p r i m a r y importance. This paper p r e s e n t s an a n a l y s i s of the importance of this p a r a m e t e r to the theory of gl a s s formation. In the following sections, f i r s t an a n a l y s i s of heat flow during the cooling of m e l t s under different conditions will be considered.
Calculated cooling
r a t e s will then be c o m p a r e d with typical quench r a t e s and observed phases obtained for different m a t e r i a l s using v a r i o u s laboratory techniques.
The au-
thors (5-7) have r e c e n t l y applied splat-cooling and flame spray techniques to a number of ionic compounds in o r d e r to widen the s p e c t r u m of e x p e r i m e n t a l cooling rate data available. One r e c o g n i z e s as a n e c e s s a r y but not sufficient condition for c r y s t a l lization, the f or m a t i on of an o r d e r e d , c r y s t a l l i n e , nucleus of c r i t i c a l radius. Naturally then, the total diffusion over a t i m e - t e m p e r a t u r e integrated interval will be d e t e r m i n a t i v e of whether or not c r y s t a l l i z a t i o n o ccu rs .
In other words,
one must deal with those v a r i a b l e s principally affecting the t r a n s p o r t of m a t e rial to build on " c r i t i c a l " sized nuclei.
T h e r e f o r e , the p a r a m e t e r s affecting
the coefficient of self-diffusion will be d i s c u s s e d , with p a r t i c u l a r e m p h a s i s on melt viscosity. The concept of the " c r i t i c a l quenching r a t e " ("Q" in ° C / s e c , defined as the cooling rate below which detectable c r y s t a l l i n e phases a r e obtained from the melt) will be introduced.
As a f irst approximation, (Q) is a s s u m e d to vary
d i r e c t l y as the melting t e m p e r a t u r e and i n v e r s e l y as the relaxation time at the melting point (Tm).
Functional relationships between relaxation time and vis-
cosity (or coefficient of self-diffusion) will be substituted in the e x p r e s s i o n for (Q). The coefficient of variability is evaluated using e x p e r i m e n t a l data for the c r y s t a l l i z a t i o n rate of NaC1 under rapid quenching conditions.
Theoretical
values predicting (Q) for other m a t e r i a l s will be calculated using known p a r a m eters.
These will then be checked against e x p e r i m e n t a l r e s u l t s and the calcu-
lated cooling r a t e s obtainable by splat-cooling and other e x p e r i m e n t a l methods used in this work.
A d i m e n s i o n l e s s quantity (called h e r e i n the "glass number")
will be de ri ved which is useful for the prediction of g las s formation from melts when a p r e d e t e r m i n e d cooling rate is inserted in the expression.
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GLASS FORMATION
267
C a l c u l a t i o n of Cooling R a t e s The equation for the l o w e r i n g of t e m p e r a t u r e with t i m e for a finite body in contact with a heat sink ( s u b s t r a t e ) at l o w e r t e m p e r a t u r e m a y be w r i t t e n Tt - T
sT (1) = e x p ( t K m / p C p d 2) Tt = o s w h e r e T t is the t e m p e r a t u r e at t i m e t, T s is the s u b s t r a t e t e m p e r a t u r e , K m -
is the t h e r m a l conductivity at the m e l t i n g point, C
is the s p e c i f i c heat, P is P the d e n s i t y , and d is the t h i c k n e s s (or r a d i u s if the cooling body is s p h e r i c a l ) (8).
T h i s r e l a t i o n holds for ideal cooling, i . e . w h e r e skin e f f e c t s a r e small.
When s u r f a c e e f f e c t s a r e l a r g e , and c o n t r o l l i n g , this equation has the f o r m for Newtonian cooling and (hd) is s u b s t i t u t e d for (Km) , w h e r e (h) is the heat t r a n s f e r coefficient.
Ruhl (9) has u s e d the c r i t e r i o n that if ( h d / K m ) d0. 015,
Newtonian cooling p r e v a i l s , but if ( h d / K m ) ~30, cooling is ideal. It is r e c o g n i z e d that in the c a l c u l a t i o n of cooling r a t e s for thin flakes, s o m e cooling mode i n t e r m e d i a t e b e t w e e n t h e s e two e x t r e m e s is a p p r o p r i a t e . This will be g o v e r n e d p r i m a r i l y by (d), the film t h i c k n e s s .
For thicker films,
(Km) is d e t e r m i n a t i v e , while at the cooled i n t e r f a c e and in v e r y thin f i l m s (h) is controlling.
Although t h e r e is no d i r e c t r e l a t i o n b e t w e e n (Km) and (h), v a l -
u e s of (h) for the m e l t d u r i n g s p l a t - c o o l i n g will c e r t a i n l y be g r e a t e r than (Km) b e c a u s e of the rapid motion of the m e l t with r e s p e c t to the s u b s t r a t e .
As a
f i r s t a p p r o x i m a t i o n , t h e r e f o r e , the following t r e a t m e n t will a s s u m e the s i t u a tion for Newtonian (i. e. s u r f a c e ) cooling at the high cooling r a t e s .
Further-
m o r e , since v a l u e s for (h) f o r c e r a m i c m e l t s cannot be found in the l i t e r a t u r e , they will be a s s u m e d to be a p p r o x i m a t e l y one o r d e r of m a g n i t u d e g r e a t e r than (Km)*.
The c a l c u l a t e d values for the cooling r a t e obtained may t h e r e f o r e be
slightly high,
but should lie within an o r d e r of magnitude of the a c t u a l value.
F o r e x a m p l e , making the a p p r o x i m a t i o n noted above and c a l c u l a t i n g the quantity ( h d / K m ) for a t y p i c a l s i l i c a m e l t cooled as a 20p film" *'Values for (h) of 2 . 7 to 6.8 c a l / c m 2 sec °C for a l u m i n u m splatted on n i c k e l s u b s t r a t e s have b e e n obtained by P r e d e c k i (10). High s p e e d t h e r m a l m e a s u r e m e n t s w e r e m a d e by cathode r a y tube t r a c e t e c h n i q u e s . T h e s e values a r e app r o x i m a t e l y an o r d e r of m a g n i t u d e g r e a t e r than (Km) which is 1.01 c a l / c m sec °C for a l u m i n u m .
268
GLASS FORMATION K •
m
Vol. 3, No. 3
= 0. 004 c a l / c m sec °C [Kingery, (11)] 0. 04 c a l / c m 2 s e c °C
h
= 20 m i c r o n s o r 20 x 10 -4 c m
d hd K
m
(0. 04)(20 x 10 -4) (0. 004
0.02.
R u h l ' s c r i t e r i o n i n d i c a t e s that p r i m a r i l y Newtonian cooling is involved in r a p i d q u e n c h i n g of m e l t s when the p a r t i c l e t h i c k n e s s o r d i a m e t e r is l e s s than a p p r o x i m a t e l y 20 m i c r o n s . Taking l o g a r i t h m s of Eqn. (1) and d i f f e r e n t i a t i n g with r e s p e c t to t, the following e x p r e s s i o n f o r the i n s t a n t a n e o u s cooling r a t e (Q) is obtained: fl'P
h = - (T t - T s ) ~ Cpd.
(2)
This relation can now be used to calculate the approximate cooling rate at the melting point for any melt (T t = T m) when the appropriate parameters are inserted. The latent heat of fusion (AHf) does not enter the calculation since crystallization only intervenes when the cooling rate is not great enough to prevent diffusional ordering from taking place. This is readily apparent from experimental observations. Theoretically, when film thickness is on the order of the ionic dimensions (-~ I0•), the fastest cooling rate will prevail for any melt. The cooling rate in this case would be 4.0 x 109 °C/sec for NaCI (or other ionic compounds) and approximately twice this for lead and other metals (see Table 1). The coefficient of heat transfer (conductivity) accounts for the difference in ultimate cooling rates. For van der Waals compounds, the cooling rate will probably be less than that for NaCl under similar conditions• Cooling rates obtainable experimentally by c o m m o n laboratory techniques m a y be compared as set forth below: I. Annealing Furnace (1000°C to 25°C in 24 hrs.)
---10 -2 °C/sec
2• Pellet (20 m g m ) in Pt (dropped into Hg)
-~102 - 103 °C/sec.
3. Standard Strip Furnace
...103 _ 104 °C/sec.
4. Splat Cooling (also Verneuil or Plasma Technique)
...i05
_
107 °C/sec.
Vol. 3, No. 3
GLASS FORMATION
269
TABLE 1 C a l c u l a t i o n of T h e o r e t i c a l V a l u e s for Cooling R a t e s at V a r i o u s Splat T h i c k n e s s e s Using Equation (4)
SiO2
MgAI204
NaCI
Pb
H20
(°K)
2010
2378
1073
600
2.73
(°K)
300
300
300
100
100
h
0. 04 e
0. 2 e
0. 2 a
0. 8 b
0. 01 c
p c (at Tm) d C P Cooling Rate (°C/sec)
2.1
3.3
2.0
0.2
0.2
0.2
T T
m S
11.2
1.0
0.04
1.0
d=lp
1 . 6 x 106
6.2 x 106
3.8x
106
8.9x
106
1. T x 104
d = 20~
0.8x
10 5
3 . 1 x 10 5
1 . 9 x 10 5
4.4x
10 5
8
1.6x 109
6.2 x 109
3.8x 109
8 . 9 x 109
d
=
lOb
x 10 2
1.7x 107
a v a l u e e s t i m a t e d at ten t i m e s t h e r m a l conductivity given by K i n g e r y et al (11), a s shown by P r e d e c k i et al (10). b p r e d e c k i et a l (10). CInternational C r i t i c a l T a b l e s , M c G r a w - H i l l Book Company, I n c . , New Y o r k d(1928). Handbook of C h e m i s t r y and P h y s i c s , 44th ed. C h e m i c a l R u b b e r P u b l i s h i n g Company, Cleveland, Ohio (1963). e K i n g e r y (12).
It m u s t be r e c a l l e d that the cooling r a t e s obtained u s i n g Eqn. (2) a r e only a p p r o x i m a t e .
Not only a r e few p r e c i s e high t e m p e r a t u r e p r o p e r t y data
obtainable in the l i t e r a t u r e , but t e c h n i q u e p a r a m e t e r s influence the r e s u l t s also.
F o r e x a m p l e , the v i s c o s i t y of the m e l t m a y be l o w e r e d c o n s i d e r a b l y by
heating s e v e r a l h u n d r e d d e g r e e s above the liquidus.
This in t u r n p e r m i t s
g r e a t e r flow and h e n c e t h i n n e r f i l m s on s u b s t r a t e contact, with s u p e r i o r c o n tact f o r f a s t e r heat r e l e a s e .
The end r e s u l t is that the a p p e a r a n c e of NCS
p h a s e s would then s e e m to be a function of the m e l t t e m p e r a t u r e , when it is r e a l l y s t i l l a function of heat t r a n s f e r , or cooling r a t e .
In spite of its s h o r t -
c o m i n g s , Eqn. (2) is u s e f u l in f u r n i s h i n g the o r d e r of m a g n i t u d e for the quench rate.
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GLASS FORMATION
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Diffusion P r o c e s s e s in Melts Changes in structure required for nucleation p r o c e s s e s in melts cause unique supply and transport phenomena to occur on solidification. Diffusion may be required only over s m a l l distances, but diffusion theory is least well developed for small scale effects and high local gradients of concentration and strain. Consideration of Fick's law of diffusion and the application of E y r i n g ' s theory (12) of absolute rates leads to the following form of e x p r e s s i o n for the diffusion coefficient: (cm 2) Dx ~ : ~2f
(3)
where ~, is the distance between successive equilibrium positions ( j u m p d i s tance) and f is the rate constant ( j u m p s / s e c ) . When shearing s t r e s s is applied to a liquid, the height of the potential b a r r i e r to flow is altered.
By considering this difference, Eyring et al ob-
tained the e x p r e s s i o n for viscosity, kT I?= k-~f or
kT f= k-~"
(4)
If (f) of Eqn. (4) is now substituted in (3), an expression similar in form to the Stokes-Einstein relation results, viz.
Das shown by Eyring.
kT
(5)
The rate constant (f) may be considered to be s i m i l a r to
the "relaxation frequency" of Maxwell's theory.
It e x e r t s a dominant effect on
the viscosity (and thus the diffusion coefficient). The e x p r e s s i o n given in Eqn. (4) above, is closely s i m i l a r to that proposed by Debye (13) for the relaxation time (~") of polar molecules in a liquid placed in an alternating e l e c t r i c field: r = 4Try?aS or kT where (a) is of m o l e c u l a r dimensions.
kT 1 (6) 4~va T At the so-called "anomalous d i s p e r s i o n f_
frequency" (f), a d e c r e a s e is observed in the d i e l e c t r i c constant at long wave: lengths (radio and audio range).
This is most easily m e a s u r e d in g l a s s e s or
other highly viscous s y s t e m s and in certain solids just below the solidus.
How.
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GLASS F O R M A T I O N
271
e v e r , the a p p l i c a t i o n of Eqn. (6) r e s u l t s in v a l u e s of (a 3) w h i c h a r e too s m a l l at t e m p e r a t u r e s n e a r T
m
(14).
F u r t h e r , the f o r m of Eqn. (6) is not u n l i k e that due to E i n s t e i n (15) for rotational Brownian motion.
He d e r i v e d the f o r m u l a 2RT
30 2V~ in which w 0 is the r a t e of r o t a t i o n t h r o u g h an angle ~.
(7)
T h u s , it is e v i d e n t that in s p i t e of d i s s i m i l a r a p p r o a c h e s to the p r o b l e m , the e q u a t i o n s of E y r i n g , Debye and E i n s t e i n for the p r e d i c t i o n of the r e l a x a t i o n f r e q u e n c y in l i q u i d s a l l include the s a m e p a r a m e t e r s in like d e p e n d e n c y ( t h e r m a l e n e r g y , v i s c o s i t y and p a r t i c l e v o l u m e ) . E s t i m a t i o n of the C r i t i c a l Cooling Rate for NaC1 A m o n g the m a n y i n o r g a n i c c o m p o u n d s t e s t e d , NaC1 w a s c h o s e n a s a t y p i c a l ionic p h a s e f o r e s t i m a t i o n of the c r y s t a l l i z a t i o n v e l o c i t y u n d e r r a p i d quenching conditions. strip microfurnace. inch).
A thin c o a t i n g of t h i s m a t e r i a l w a s a p p l i e d to the hot T h e s t r i p u s e d w a s i r i d i u m ( 1 / 3 2 inch x 1 inch x 0. 002
By s h u t t i n g t h e c u r r e n t off with the m e l t j u s t a few d e g r e e s above the
l i q u i d u s , a m a x i m u m c o o l i n g r a t e e s t i m a t e d at b e t w e e n 104-105 d e g r e e s p e r second was obtained.
T h e m o v e m e n t of the c r y s t a l l i z a t i o n f r o n t w a s m e a s u r e d
and a v e l o c i t y of a p p r o x i m a t e l y 10-15 c m / s e c
found u n d e r t h e s e c o n d i t i o n s ; it
w a s a l s o found to be s o m e w h a t s l o w e r at s l o w e r q u e n c h i n g r a t e s .
T h i s is a
r e a s o n a b l e value c o m p a r a b l e to o t h e r s i m i l a r data. V a l u e s f o r the g r o w t h v e l o c i t y of c r y s t a l l i n e p h a s e s have b e e n m e a s u r e d by e a r l i e r w o r k e r s , n o t a b l y G r a u e r and H a m i l t o n (16) and T a m m a n (17), a s d i s c u s s e d by Hillig and T u r n b u l l (18). by Van Hook (19).
A c o m p i l a t i o n of t h e s e d a t a w a s m a d e
To m a k e the m e a s u r e m e n t ,
the l i q u i d s w e r e c a r e f u l l y s u p e r -
c o o l e d and the s p o n t a n e o u s c r y s t a l l i z a t i o n v e l o c i t y was r e p o r t e d a s a f u n c t i o n of this s u p e r c o o l i n g .
T h e v a l u e s f o r c e r t a i n van d e r W a a l s c o m p o u n d s , e . g .
m - d i n i t r o b e n z e n e , p - d i c h l o r o b e n z e n e and c h l o r o f o r m , having low v i s c o s i t y a r e 10 to 20 c m / s e c
at 20 to 40 d e g r e e s of i n i t i a l s u p e r c o o l i n g .
In spite of t h i s
d i f f e r e n t a p p r o a c h , and v e r y d i f f e r e n t p h a s e s , the m a x i m u m v e l o c i t y of c r y s t a l l i z a t i o n f o r low v i s c o s i t y l i q u i d s c a n be s e e n to lie at a value l e s s than 100
272
cm/sec.
GLASS FORMATION
Vol. 3, No. 3
The v i s c o s i t y of c h l o r o f o r m is 0. 009 poise at T m ( - 1 3 ° C ) and for
NaC1, 0. 013 poise (20) at T m. In addition, P r e d e c k i (10) c a l c u l a t e d that the c r y s t a l l i z a t i o n v e l o c i t y for m o l t e n a l u m i n u m and s i l v e r in the splat cooling p r o c e s s a r e a p p r o x i m a t e l y 10 c m and 50 c m p e r second, r e s p e c t i v e l y .
The v i s c o s i t y of m o l t e n a l u m i n u m
at the m e l t i n g point is 0. 027 poise [Mitra and S h a r m a , (21)]. F r o m t h e s e data on the c r y s t a l l i z a t i o n v e l o c i t y of NaC1, an e s t i m a t e of the c r i t i c a l quenching r a t e (defined a s the cooling r a t e for m e l t s below which solid c r y s t a l l i n e p h a s e s a r e obtained) m a y be c a l c u l a t e d .
It will be a s s u m e d
that, for NaC1, the spontaneous c r y s t a l n u c l e a t i o n and d e n d r i t i c g r o w t h r a t e b e c o m e v e r y low at a p p r o x i m a t e l y 10 d e g r e e s u p e r c o o l i n g .
If the l o w e r l i m i t
of d e t e c t a b l e c r y s t a l l i n e o r d e r i n g o c c u r s when the c r y s t a l l i t e s i z e is ---10~, then this c r i t i c a l cooling r a t e is the r a t e which will not allow the g r o w t h of a 10~ n u c l e u s within a 10 ° i n t e r v a l and this is found to be (12.5 c m / s e c x 1 0 ° C / 1 0 -7 cm) or 1.2 x 109 ° C / s e c ,
approximately.
C o r r e l a t i o n of C r i t i c a l Cooling Rate and R e l a x a t i o n T i m e It is difficult to e s t a b l i s h a g e n e r a l r e l a t i o n b e t w e e n the cooling r a t e and r e l a x a t i o n t i m e for all types of m a t e r i a l s .
However, as a first approximation,
c l e a r l y the c r i t i c a l cooling r a t e (Q)* m a y be s i m p l y a s s u m e d to depend d i r e c t l y on the m e l t i n g t e m p e r a t u r e (Tm) and the r e l a x a t i o n ( R e s t s t r a h l e n ) f r e q u e n c y if) o r j u m p r a t e of the s t r u c t u r a l u n i t s c o m p r i s i n g the m e l t . Then: T m Q = Z. - ZfW T m w h e r e Z is s o m e constant. F o r NaC1, the value for (Q) has been shown to be a p p r o x i m a t e l y 1.2 x 109 ° C / s e c , v a l u e s for T
m
hence solving for (Z) by substitution of a c t u a l
and T for NaC1 we obtain, Z~
2.0x
10 -6 .
(8)
If now it is a s s u m e d that the s i m p l e s t of the g e n e r a l r e l a t i o n s connecting r e l a x a tion t i m e with v i s c o s i t y , t e m p e r a t u r e , ionic volume, etc. will hold, b a s e d on Eqns. (4), ( 6 ) a n d (7), T=~
(9) m
*'Defined as that cooling rate, belowwhich detectable crystalline phases are obtained on quenching the melt.
Vol. 3, No. 3
GLASS FORMATION
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.-4
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oo
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d •
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0J
~+
eo
~
o ~ O o
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,,,-d
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co
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0
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t:z0
Z 1,-4
r..) r~
--
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•
:~
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o
. -
¢~ (~
[-,oo
'z:s..~-,.-~
~ .= .,..,
r.x.1 o . , ~
h
274
GLASS FORMATION
Vol. 3, No. 3
the c r i t i c a l cooling rate (Q) b e c o m e s Q = 2 . 0 x 10 - 6 . where (V/N) is the same as (a3).
(Wm)2 • R / V ~
(I0)
Although this is an e x t r e m e simplification
of the interaction of the many p a r a m e t e r s involved, if the values for SiO2 are inserted (see Table 2), a (Q) value of approximately 4 ° C / s e c is obtained, which at least is of the right o r d e r of magnitude. If this calculation is applied to the simple low viscosity oxide (insulators) and metallic melts, values obtained for the critical quenching rates a r e (6.6 x 109 ° C / s e c ) for MgA1204 and (l. 0 x 108 ° C / s e c ) for lead.
This is consistent
with the inability to obtain g l a s s e s from these m a t e r i a l s at cooling rates up to l07 ° C / s e c . The values calculated for MgSiO3, Mg2SiO4 and BaTiO 3 are also in accord with the fact that an NCS phase is obtained for MgSiO 3 but not for the others by quenching using common laboratory techniques.
On the other hand,
using splat techniques (107 ° C / s e c ) both Mg2SiO4 and BaTiO 3 a r e made into NCS. Reference to the heat t r a n s f e r calculations in Table 1 shows that the maximum cooling rate for NaCl is approximately (3.8 x l09 ° C / s e c ) for very thin films formed in splat cooling.
This indicates that it is unlikely that s i m -
ple structured purely ionic insulators (or metals) can ever be "quenched" to a true glassy phase.
An e x t r e m e l y high degree of d i s o r d e r may a r i s e from the
thinness of the film, or very small ' c r y s t a l l i t e ' size. Results of the calculation of the dependence of the c r i t i c a l cooling rate on melt viscosity at varying T
using Eqn. (10) using a constant value (30 cc) m for the g r a m ionic (molecular) volume are shown plotted in Fig. la. This d i a g r a m is only meant to convey a qualitative picture of the relation.
The ef-
fect of ionic (molecular) volume at constant t e m p e r a t u r e (Tm) is illustrated in Fig. lb.
The numerical values calculated for these and subsequent figures
are available e l s e w h e r e (22). Similarly, by substitution in Eqn. (8) for the relation connecting the relaxation time with the coefficient of self-diffusion in Eqn. (3), the critical cooling rate becomes
T Q = 2 . 0 x 10 - 6 .
• D m a2 x
(11)
Vol. 3, No. 3
GLASS FORMATION
%*`% ' % %%*`
6.0
6.0
%*`*`*`
GLASSYPHASES(SRO0)
%~.%%*` *`*`%',
\\\
4.0
~
"G
oooo.
\\'k--
3.0
R
\ \ \
Associoted Liquids Bonds
)
\\\
-I.0 Unassociated Liquids (Covalent, Van der Woals, or M e t a l l i c B o n d s
- 2.0
-5.C
Ionic
o3, ,oo
/ooo./, ? , ')',V ,,\/ oo.o
2.0
I0,000~/ 3 /
"6 ~- 1.0
~
\ / 3a=
,000~?
~ o
\\\ \ \ \ kkk
(Covalent-IonicBonds,
Hydrogen
)',,\,,"\\
5.0
500 o K.
CRYSTALLINE
c. 1.0
',\\\
4.0
,oooo.
~
2.0
~%GLASSYPHASES(SRO0)
5.0
5.0 Viscous Melts:
275
~k
I0~3
! \ k\ \
CRYSTALLINE
-I.0
\ \ \ \ \ \ kkk
%%~ , ~
PHASES
T m
-2.0
%
= I000 °K.)
\\\ I
-2
I
I
0
I
I
2
I
I
I
4
I
6
i
I
B
~ "C
-2
I0
i
i
0
2ogQ(°C.Isec.)
i
i
i
i
i
i
2 4 6 2og Q (°C.lsec.)
i
1
B
i
I0
FIG. la
FIG. lb
Plot of the Logarithm of Melt V i s c o s i t y vs. Logarithm of C r i t i c a l Cooling i~ate at Various Melting Temperatures.
Plot of the Logarithm of Melt V i s c o s i t y vs. Logarithm of Critical Cooling Rate at Different Melt Moiety V o l u m e s .
The r e s u l t s of calculations of (Q) as a function of (Dx) and at varying T
are shown in Fig. 2a, using a constant value of 15~ 2 for the c r o s s m s e c t i o n a l area of the diffusing s p e c i e s . The effect of volume of the diffusing s p e c i e s at constant t e m p e r a t u r e (Tm) is shown graphically in Fig. 2b. P r e d i c t i o n of P h a s e A s s e m b l a g e on Quenching Melts If the e x p r e s s i o n for the calculation of the c r i t i c a l cooling rate, Eqn. (11) is r e a r r a n g e d to the form: q.a
3 .,~
: 1
(12)
2 . 0 x 10 -6 k(Tm )2 and l o g a r i t h m s taken, we obtain for the c r i t i c a l condition between g l a s s f o r m a tion and c r y s t a l l i z a t i o n the s a t i s f a c t i o n of the relation:
276
GLASS FORMATION
Vol. 3, No. 3
Molten Metals ona salts van der Wools Compounds HzO (Low Viscosity Liquids
-6~
CRYSTALLINE PHASES
CRYSTALLINE PHASES
-4.C -5.C
~. -7
//~ o2:50~'2
-6.0 ¸
5 -8
500 °K.
Associated Liquids
E -9
*%
-7.0~
/.~;/---~ o2--500~,~ ////
E -8.0
o
p-
o~ - 1 0
GLASS PHASES(SRO0)
Viscous Melts
-I
_151
o
-9.0
~ o
-I0.0
etc.
-12.0 f -2
i 0
I 2
I I 4 6 ,tog Q { °C./sec.] FIG.
Plot
/ / /
_GLASSY P..~ASES{SRO0}
-I 1.0
SiOz, B203 -If
oZ= 5~ 2
/ /
(Tm = I O 0 0 ° K ' ) / /
u) e4.
/,/___
i 8
i
-2
i
1
0
i
o,; 5ooo /
2
i
i
i
4
i
i
6
i
8
i
i
I0
J~og Q (°C./sec.)
FIG. 2b
2a
of the Logarithm
/-/-/--
of the
Coefficient of Self D i f f u s i o n vs. L o g a r i t h m of C r i t i c a l Cooling Rate at V a r i o u s Melting T e m p e r a t u r e s .
P l o t of the L o g a r i t h m of the Coefficient of Self D i f f u s i o n vs. L o g a r i t h m of C r i t i c a l Cooling Rate at Different Melt Moiety C r o s s sectional Areas.
3 log [Q" a • ~] + 5.70 = 0. (13) kT2 m Now if (R), the selected cooling rate available in any particular experimental arrangement is substituted for (Q), and the other melt parameters inserted, and the equation yields a positive number, it indicates that the critical cooling rate for the melt would be exceeded and NCS phases will probably result experimentally. crystallize.
Conversely negative numbers will indicate that the melt will For example, substituting for a melt of SiO2: 3 log [R" a " 7] + 5.70 = N ("quenchnumber") kT 2 q m
R = 10°C/sec;
(14)
n ~ 107 poise; ( e s t . ) a 3 ~ 1 0 0 0 ~ 3 = 10 - 2 1 c m 3 ; T m ~ 2 0 0 0 ° K
Vol. 3, No. 3
GLASS FORMATION
log[
10" 10 - 2 1 " 107
(4 x 106)(1.38 x 10 -16)
277
] + 5 . 7 0 = +2.16
which is in a c c o r d with e x p e r i m e n t a l f o r m a t i o n of g l a s s at this cooling rate. Analogously, substituting for a typical spinel m e l t (e. g. MgA1204): R
106 ° C / s e c ; ~ ~ 2 x 10 -2 poise; a 3
20~k3 ( e s t . ) ~ 2 x 10 -23 cm; T
m
2378°K; we obtain, 106" 2 x 10 - 2 3 " 2 x 10-21 x
x i0
, +
The negative sign indicates that one will obtain c r y s t a l l i n e m a t e r i a l , again in a c c o r d with e x p e r i m e n t . A s i m i l a r r e l a t i o n can be obtained by r e a r r a n g i n g Eqn. (11) which contains the coefficient of s e l f - d i f f u s i o n for the m e l t at Tm, viz. Q" a 2 log i T D ] + 5.70 = (Nq) (16) m m and substituting (R) for (Q) as before. H e r e again, if (Nq) is positive, (R) is g r e a t e r than (Q) and NCS p h a s e s will probably r e s u l t . The magnitude of (Nq) may t h e r e f o r e be i n t e r p r e t e d as a m e a s u r e of the p r o x i m i t y of the p a r t i c u l a r cooling rate to the c r i t i c a l g l a s s f o r m i n g value. If (Nq) is close to zero, the p r e s c r i b e d cooling rate will be close to c r i t i c a l and mixed NCS and c r y s t a l l i n e p h a s e s may be p r e s e n t e x p e r i m e n t a l l y . can be u n d e r s t o o d by r e f e r r i n g to Tables 1 and 2. a r e qualitative. value of (a3).
This
o f course, t h e s e r e l a t i o n s
The l a r g e s t variable e n t e r s b e c a u s e of the e s t i m a t i o n of the However, it may be r e c a l l e d that for s t r i c t l y ionic m e l t s , (a 3)
is close to the a v e r a g e ionic volume.
Large values for (a 3) would apply for
" p o l y m e r i z e d " m e l t s with v i s c o s i t i e s higher than one poise at the melting point. W h e r e a s the p r e s e n t a p p r o a c h may a d e q u a t e l y d e s c r i b e the k i n e t i c s of o r d e r i n g during rapid quenching in the vicinity of Tm, it is r e c o g n i z e d that d i f f u s i o n - c o n t r o l l e d p r o c e s s e s p e r s i s t during subsolidus cooling right down to (and below) r o o m t e m p e r a t u r e .
M o r e o v e r t h e r e is, in principle, no limit to
the size of the s a m p l e which can be t r e a t e d by these techniques.* To account *Indeed in the d i s c u s s i o n following the p r e s e n t a t i o n of the paper, Dr. E. Deeg r e f e r r e d to his unpublished w o r k along t h e s e lines in the case of large t a n k - s i z e glass melts.
278
GLASS FORMATION
Vol. 3, No. 3
a d e q u a t e l y for t h e s e p r o c e s s e s would r e q u i r e a c o m p l e x t i m e - t e m p e r a t u r e i t e r a t i o n p r o c e d u r e to be p e r f o r m e d , s i n c e the cooling r a t e c h a n g e s exponentially with t i m e .
Although d e t a i l e d m a t h e m a t i c a l t e c h n i q u e s a r e a v a i l a b l e to
p r o v i d e e x a c t solutions to this p r o b l e m , e x p e r i m e n t a l data on solid diffusion show that this c o n t r i b u t e s r e l a t i v e l y little to ion m i g r a t i o n d u r i n g the subsolidus cooling p e r i o d p r o v i d e d that the s u b s t r a t e t e m p e r a t u r e and T m differ by 1000°C or m o r e .
N e g l e c t of this p r o c e s s in the p r e s e n t c a l c u l a t i o n m a y
t h e r e f o r e lead to values of the c r i t i c a l cooling r a t e which a r e slightly low, but c e r t a i n l y by l e s s than one o r d e r of m a g n i t u d e . In o t h e r p a p e r s the a u t h o r s (5-7) have p r o v i d e d a c o n s i d e r a b l e volume of new data on the " q u e n c h a b i l i t y " to a g l a s s of a wide r a n g e of ionic oxide melts.
The " g l a s s n u m b e r " , Nq, d e v e l o p e d above, s e r v e s quite a d e q u a t e l y in
such c a s e s to s e p a r a t e those m e l t s which can be r e t a i n e d a s NCS f r o m those which c r y s t a l l i z e . Summary This p a p e r has i n t r o d u c e d the r a t e of quenching a s a m a j o r p a r a m e t e r in the p r e d i c t i o n of " g l a s s f o r m a t i o n " .
An e m p i r i c a l s e m i - q u a n t i t a t i v e
a p p r o a c h has led to the d e v e l o p m e n t of a s o - c a l l e d g l a s s n u m b e r , Nq, which can be r e l a t e d to only two e x p e r i m e n t a l p a r a m e t e r s (Tm, the m e l t i n g point and 77 the v i s c o s i t y at the m e l t i n g point), and an e s t i m a t e of the volume of the diffusing s p e c i e s , via the r e l a t i o n 3 a ~7] + 5.70. kT 2 m P o s i t i v e v a l u e s of N obtained at any g i v e n quenching r a t e R, indicate that a Nq = log j R .
g l a s s or NCS will f o r m . Acknowledgement The w o r k was s u p p o r t e d by the A d v a n c e d R e s e a r c h P r o j e c t s A g e n c y u n d e r C o n t r a c t No. DA-49-083 OSA-3140. Refe r e nce s 1. W. H. Z a c h a r i a s e n , J. Am. Chem. Soc. 54, 3841 (1932).
2. W. A. Weyl and E. C. M a r b o e , The Constitution of Glass, Vol. l, Wiley, N. Y. (1962); K. Han Sun, J. Am. Soc. 30, 277 (1947).
Vol. 3, No. 3
GLASS FORMATION
279
3. A. Smekal, J. Soc. G1. Techn. 35, 411 (1951). 4. D. Turnbull and M. H. Cohen, J. Chem. Phys. 2.9_9, 1049 (1958). 5. P. T. Sarjeant and Rustum Roy, J. Am. Ceram. Soc. 50, 500 (1967). 6. P. T. Sarjeant and Rustum Roy, J. Appl. Phys. 3_88, 4540 (1967). 7. P. T. Sarjeant and Rustum Roy, J. Am. Ceram. Soc. (to be published). 8. H. J. V. Tyrell, Diffusion and Heat Flow in Liquids, Butterworths, London (1961). 9. R. C. Ruhl, CoolingRates in Splat Quenching, Office of Naval Research, Contract Nonr-1841 (38), Technical Report No. 12-DSR 7618, available from the Dept. of Metallurgy, M. I. T., Cambridge, Mass. (1966). I0. P. Predecki, A. W. Mullendore and N. J. Grant, Trans. Met. Soc. AIME 233, 1581 (1965). ii. S. Glasstone, K. Laidler and H. Eyring, The Theory of Rate Processes, McGraw-Hill, New York (1941). 12. W. D. Kingery, J. Am. Ceram. Soc. 42, 617 (1959). 13. P. Debye, Polar Molecules, pp. 77-108, Dover Publication, New York (1929). 14. W. Jackson, Proc. Roy. Soc. 153A, 158 (1935). 15. A. Einstein, Ann. Phys. 17, 549 (1905). 16. O. H. Grauer and E. H. Hamilton, J. Res. NBS, 495 (1950). 17. G. Tamman and E. Janckel, Z. anorg. Chem. 193, 76 (1930). 18. W. B. Hillig and D. Turnbull, J. Chem. Phys. 24, 914 (1956). 19. A. Van Hook, Crystallization Theory and Practice, ACS Monograph No. 152, Rheinhold Publishing Corp., New York (1961). 20. Handbook of Chemistry and Physics, 44th ed., Chemical Rubber Publishing Co., Cleveland, Ohio (1962). 21. S. S. Mitra and L. P. Sharma, J. Chem. Phys. 38, 1287 (1963). 22. P. T. Sarjeant, Experimental and T h e o r e t i c a l Approaches to the Glassy State, Ph.D. d i s s e r t a t i o n in Solid State Science, The Pennsylvania State University, University Park, Pennsylvania (1967).
P e r g a m o n P r e s s , Inc.
P rin ted
in the United States.
A NEW APPROACH TO THE PREDICTION OF GLASS FORMATION*
P. T. Sarjeant and Rustum Roy Materials Research Laboratory The Pennsylvania State University University Park, Pennsylvania
(Received J a n u a r y 19, 1968; Communicated by R. Roy)
ABSTRACT The rate of cooling has usually been ignored in t r e a t m e n t s of glassforming capability. The rate r e q u i r e d to form n o n - c r y s t a l l i n e solid** phases is shown h e r e i n to depend d i r e c t l y on the melting t e m p e r a t u r e and a v e r a g e coefficient of self diffusion, and i n v e r s e l y upon melt viscosity, relaxation time and linear d i m e n s i o n s of the ionic s pe c i e s c o m p r i s i n g the melt. Calculated dependence of the formation of NCS phases as a function of these p a r a m e t e r s is shown graphically. A d i m e n s i o n l e s s e x p r e s s i o n is d e r i v e d (named the " g l a s s number") for the prediction of glass fo rmatio n at any p r e s c r i b e d cooling rate for any melt of known viscosity.
Introduction Although other a p p r o a c h e s to the problem of predicting glass formation f r o m melts utilize principally the concepts of coordination (1), bond strength or type (2, 3) and "free volume" (4), the effect of the cooling rate has not gene r a l l y been taken into consideration, at least quantitatively.
Since NCS phases
a r e always metastable, their f or m a t io n from m e l t s and retention is p r i m a r i l y *This paper was p r e s e n t e d at the International Conference on the C h a r a c t e r ization of Materials, R o c h e s t e r , N . Y . , USA, November 8-10, 1967. **The abbreviated form of the e x p r e s s i o n " n o n - c r y s t a l l i n e solid" (NCS) used here r e f e r s specifically to an a r r a n g e m e n t w h e r e i n only the first, second or perhaps third n e a r e s t neighbours to a given atom or ion a r e s i m i l a r . Long range o r d e r is absent. We have previously used the t e r m short range o r d e r only (SROO) for the same purpose. 265
266
GLASS FORMATION
Vol. 3, No. 3
a question of kinetics and thus the cooling rate must be of p r i m a r y importance. This paper p r e s e n t s an a n a l y s i s of the importance of this p a r a m e t e r to the theory of gl a s s formation. In the following sections, f i r s t an a n a l y s i s of heat flow during the cooling of m e l t s under different conditions will be considered.
Calculated cooling
r a t e s will then be c o m p a r e d with typical quench r a t e s and observed phases obtained for different m a t e r i a l s using v a r i o u s laboratory techniques.
The au-
thors (5-7) have r e c e n t l y applied splat-cooling and flame spray techniques to a number of ionic compounds in o r d e r to widen the s p e c t r u m of e x p e r i m e n t a l cooling rate data available. One r e c o g n i z e s as a n e c e s s a r y but not sufficient condition for c r y s t a l lization, the f or m a t i on of an o r d e r e d , c r y s t a l l i n e , nucleus of c r i t i c a l radius. Naturally then, the total diffusion over a t i m e - t e m p e r a t u r e integrated interval will be d e t e r m i n a t i v e of whether or not c r y s t a l l i z a t i o n o ccu rs .
In other words,
one must deal with those v a r i a b l e s principally affecting the t r a n s p o r t of m a t e rial to build on " c r i t i c a l " sized nuclei.
T h e r e f o r e , the p a r a m e t e r s affecting
the coefficient of self-diffusion will be d i s c u s s e d , with p a r t i c u l a r e m p h a s i s on melt viscosity. The concept of the " c r i t i c a l quenching r a t e " ("Q" in ° C / s e c , defined as the cooling rate below which detectable c r y s t a l l i n e phases a r e obtained from the melt) will be introduced.
As a f irst approximation, (Q) is a s s u m e d to vary
d i r e c t l y as the melting t e m p e r a t u r e and i n v e r s e l y as the relaxation time at the melting point (Tm).
Functional relationships between relaxation time and vis-
cosity (or coefficient of self-diffusion) will be substituted in the e x p r e s s i o n for (Q). The coefficient of variability is evaluated using e x p e r i m e n t a l data for the c r y s t a l l i z a t i o n rate of NaC1 under rapid quenching conditions.
Theoretical
values predicting (Q) for other m a t e r i a l s will be calculated using known p a r a m eters.
These will then be checked against e x p e r i m e n t a l r e s u l t s and the calcu-
lated cooling r a t e s obtainable by splat-cooling and other e x p e r i m e n t a l methods used in this work.
A d i m e n s i o n l e s s quantity (called h e r e i n the "glass number")
will be de ri ved which is useful for the prediction of g las s formation from melts when a p r e d e t e r m i n e d cooling rate is inserted in the expression.
Vol. 3, No. 3
GLASS FORMATION
267
C a l c u l a t i o n of Cooling R a t e s The equation for the l o w e r i n g of t e m p e r a t u r e with t i m e for a finite body in contact with a heat sink ( s u b s t r a t e ) at l o w e r t e m p e r a t u r e m a y be w r i t t e n Tt - T
sT (1) = e x p ( t K m / p C p d 2) Tt = o s w h e r e T t is the t e m p e r a t u r e at t i m e t, T s is the s u b s t r a t e t e m p e r a t u r e , K m -
is the t h e r m a l conductivity at the m e l t i n g point, C
is the s p e c i f i c heat, P is P the d e n s i t y , and d is the t h i c k n e s s (or r a d i u s if the cooling body is s p h e r i c a l ) (8).
T h i s r e l a t i o n holds for ideal cooling, i . e . w h e r e skin e f f e c t s a r e small.
When s u r f a c e e f f e c t s a r e l a r g e , and c o n t r o l l i n g , this equation has the f o r m for Newtonian cooling and (hd) is s u b s t i t u t e d for (Km) , w h e r e (h) is the heat t r a n s f e r coefficient.
Ruhl (9) has u s e d the c r i t e r i o n that if ( h d / K m ) d0. 015,
Newtonian cooling p r e v a i l s , but if ( h d / K m ) ~30, cooling is ideal. It is r e c o g n i z e d that in the c a l c u l a t i o n of cooling r a t e s for thin flakes, s o m e cooling mode i n t e r m e d i a t e b e t w e e n t h e s e two e x t r e m e s is a p p r o p r i a t e . This will be g o v e r n e d p r i m a r i l y by (d), the film t h i c k n e s s .
For thicker films,
(Km) is d e t e r m i n a t i v e , while at the cooled i n t e r f a c e and in v e r y thin f i l m s (h) is controlling.
Although t h e r e is no d i r e c t r e l a t i o n b e t w e e n (Km) and (h), v a l -
u e s of (h) for the m e l t d u r i n g s p l a t - c o o l i n g will c e r t a i n l y be g r e a t e r than (Km) b e c a u s e of the rapid motion of the m e l t with r e s p e c t to the s u b s t r a t e .
As a
f i r s t a p p r o x i m a t i o n , t h e r e f o r e , the following t r e a t m e n t will a s s u m e the s i t u a tion for Newtonian (i. e. s u r f a c e ) cooling at the high cooling r a t e s .
Further-
m o r e , since v a l u e s for (h) f o r c e r a m i c m e l t s cannot be found in the l i t e r a t u r e , they will be a s s u m e d to be a p p r o x i m a t e l y one o r d e r of m a g n i t u d e g r e a t e r than (Km)*.
The c a l c u l a t e d values for the cooling r a t e obtained may t h e r e f o r e be
slightly high,
but should lie within an o r d e r of magnitude of the a c t u a l value.
F o r e x a m p l e , making the a p p r o x i m a t i o n noted above and c a l c u l a t i n g the quantity ( h d / K m ) for a t y p i c a l s i l i c a m e l t cooled as a 20p film" *'Values for (h) of 2 . 7 to 6.8 c a l / c m 2 sec °C for a l u m i n u m splatted on n i c k e l s u b s t r a t e s have b e e n obtained by P r e d e c k i (10). High s p e e d t h e r m a l m e a s u r e m e n t s w e r e m a d e by cathode r a y tube t r a c e t e c h n i q u e s . T h e s e values a r e app r o x i m a t e l y an o r d e r of m a g n i t u d e g r e a t e r than (Km) which is 1.01 c a l / c m sec °C for a l u m i n u m .
268
GLASS FORMATION K •
m
Vol. 3, No. 3
= 0. 004 c a l / c m sec °C [Kingery, (11)] 0. 04 c a l / c m 2 s e c °C
h
= 20 m i c r o n s o r 20 x 10 -4 c m
d hd K
m
(0. 04)(20 x 10 -4) (0. 004
0.02.
R u h l ' s c r i t e r i o n i n d i c a t e s that p r i m a r i l y Newtonian cooling is involved in r a p i d q u e n c h i n g of m e l t s when the p a r t i c l e t h i c k n e s s o r d i a m e t e r is l e s s than a p p r o x i m a t e l y 20 m i c r o n s . Taking l o g a r i t h m s of Eqn. (1) and d i f f e r e n t i a t i n g with r e s p e c t to t, the following e x p r e s s i o n f o r the i n s t a n t a n e o u s cooling r a t e (Q) is obtained: fl'P
h = - (T t - T s ) ~ Cpd.
(2)
This relation can now be used to calculate the approximate cooling rate at the melting point for any melt (T t = T m) when the appropriate parameters are inserted. The latent heat of fusion (AHf) does not enter the calculation since crystallization only intervenes when the cooling rate is not great enough to prevent diffusional ordering from taking place. This is readily apparent from experimental observations. Theoretically, when film thickness is on the order of the ionic dimensions (-~ I0•), the fastest cooling rate will prevail for any melt. The cooling rate in this case would be 4.0 x 109 °C/sec for NaCI (or other ionic compounds) and approximately twice this for lead and other metals (see Table 1). The coefficient of heat transfer (conductivity) accounts for the difference in ultimate cooling rates. For van der Waals compounds, the cooling rate will probably be less than that for NaCl under similar conditions• Cooling rates obtainable experimentally by c o m m o n laboratory techniques m a y be compared as set forth below: I. Annealing Furnace (1000°C to 25°C in 24 hrs.)
---10 -2 °C/sec
2• Pellet (20 m g m ) in Pt (dropped into Hg)
-~102 - 103 °C/sec.
3. Standard Strip Furnace
...103 _ 104 °C/sec.
4. Splat Cooling (also Verneuil or Plasma Technique)
...i05
_
107 °C/sec.
Vol. 3, No. 3
GLASS FORMATION
269
TABLE 1 C a l c u l a t i o n of T h e o r e t i c a l V a l u e s for Cooling R a t e s at V a r i o u s Splat T h i c k n e s s e s Using Equation (4)
SiO2
MgAI204
NaCI
Pb
H20
(°K)
2010
2378
1073
600
2.73
(°K)
300
300
300
100
100
h
0. 04 e
0. 2 e
0. 2 a
0. 8 b
0. 01 c
p c (at Tm) d C P Cooling Rate (°C/sec)
2.1
3.3
2.0
0.2
0.2
0.2
T T
m S
11.2
1.0
0.04
1.0
d=lp
1 . 6 x 106
6.2 x 106
3.8x
106
8.9x
106
1. T x 104
d = 20~
0.8x
10 5
3 . 1 x 10 5
1 . 9 x 10 5
4.4x
10 5
8
1.6x 109
6.2 x 109
3.8x 109
8 . 9 x 109
d
=
lOb
x 10 2
1.7x 107
a v a l u e e s t i m a t e d at ten t i m e s t h e r m a l conductivity given by K i n g e r y et al (11), a s shown by P r e d e c k i et al (10). b p r e d e c k i et a l (10). CInternational C r i t i c a l T a b l e s , M c G r a w - H i l l Book Company, I n c . , New Y o r k d(1928). Handbook of C h e m i s t r y and P h y s i c s , 44th ed. C h e m i c a l R u b b e r P u b l i s h i n g Company, Cleveland, Ohio (1963). e K i n g e r y (12).
It m u s t be r e c a l l e d that the cooling r a t e s obtained u s i n g Eqn. (2) a r e only a p p r o x i m a t e .
Not only a r e few p r e c i s e high t e m p e r a t u r e p r o p e r t y data
obtainable in the l i t e r a t u r e , but t e c h n i q u e p a r a m e t e r s influence the r e s u l t s also.
F o r e x a m p l e , the v i s c o s i t y of the m e l t m a y be l o w e r e d c o n s i d e r a b l y by
heating s e v e r a l h u n d r e d d e g r e e s above the liquidus.
This in t u r n p e r m i t s
g r e a t e r flow and h e n c e t h i n n e r f i l m s on s u b s t r a t e contact, with s u p e r i o r c o n tact f o r f a s t e r heat r e l e a s e .
The end r e s u l t is that the a p p e a r a n c e of NCS
p h a s e s would then s e e m to be a function of the m e l t t e m p e r a t u r e , when it is r e a l l y s t i l l a function of heat t r a n s f e r , or cooling r a t e .
In spite of its s h o r t -
c o m i n g s , Eqn. (2) is u s e f u l in f u r n i s h i n g the o r d e r of m a g n i t u d e for the quench rate.
270
GLASS FORMATION
Vol. 3, No. 3
Diffusion P r o c e s s e s in Melts Changes in structure required for nucleation p r o c e s s e s in melts cause unique supply and transport phenomena to occur on solidification. Diffusion may be required only over s m a l l distances, but diffusion theory is least well developed for small scale effects and high local gradients of concentration and strain. Consideration of Fick's law of diffusion and the application of E y r i n g ' s theory (12) of absolute rates leads to the following form of e x p r e s s i o n for the diffusion coefficient: (cm 2) Dx ~ : ~2f
(3)
where ~, is the distance between successive equilibrium positions ( j u m p d i s tance) and f is the rate constant ( j u m p s / s e c ) . When shearing s t r e s s is applied to a liquid, the height of the potential b a r r i e r to flow is altered.
By considering this difference, Eyring et al ob-
tained the e x p r e s s i o n for viscosity, kT I?= k-~f or
kT f= k-~"
(4)
If (f) of Eqn. (4) is now substituted in (3), an expression similar in form to the Stokes-Einstein relation results, viz.
Das shown by Eyring.
kT
(5)
The rate constant (f) may be considered to be s i m i l a r to
the "relaxation frequency" of Maxwell's theory.
It e x e r t s a dominant effect on
the viscosity (and thus the diffusion coefficient). The e x p r e s s i o n given in Eqn. (4) above, is closely s i m i l a r to that proposed by Debye (13) for the relaxation time (~") of polar molecules in a liquid placed in an alternating e l e c t r i c field: r = 4Try?aS or kT where (a) is of m o l e c u l a r dimensions.
kT 1 (6) 4~va T At the so-called "anomalous d i s p e r s i o n f_
frequency" (f), a d e c r e a s e is observed in the d i e l e c t r i c constant at long wave: lengths (radio and audio range).
This is most easily m e a s u r e d in g l a s s e s or
other highly viscous s y s t e m s and in certain solids just below the solidus.
How.
Vol. 3, No. 3
GLASS F O R M A T I O N
271
e v e r , the a p p l i c a t i o n of Eqn. (6) r e s u l t s in v a l u e s of (a 3) w h i c h a r e too s m a l l at t e m p e r a t u r e s n e a r T
m
(14).
F u r t h e r , the f o r m of Eqn. (6) is not u n l i k e that due to E i n s t e i n (15) for rotational Brownian motion.
He d e r i v e d the f o r m u l a 2RT
30 2V~ in which w 0 is the r a t e of r o t a t i o n t h r o u g h an angle ~.
(7)
T h u s , it is e v i d e n t that in s p i t e of d i s s i m i l a r a p p r o a c h e s to the p r o b l e m , the e q u a t i o n s of E y r i n g , Debye and E i n s t e i n for the p r e d i c t i o n of the r e l a x a t i o n f r e q u e n c y in l i q u i d s a l l include the s a m e p a r a m e t e r s in like d e p e n d e n c y ( t h e r m a l e n e r g y , v i s c o s i t y and p a r t i c l e v o l u m e ) . E s t i m a t i o n of the C r i t i c a l Cooling Rate for NaC1 A m o n g the m a n y i n o r g a n i c c o m p o u n d s t e s t e d , NaC1 w a s c h o s e n a s a t y p i c a l ionic p h a s e f o r e s t i m a t i o n of the c r y s t a l l i z a t i o n v e l o c i t y u n d e r r a p i d quenching conditions. strip microfurnace. inch).
A thin c o a t i n g of t h i s m a t e r i a l w a s a p p l i e d to the hot T h e s t r i p u s e d w a s i r i d i u m ( 1 / 3 2 inch x 1 inch x 0. 002
By s h u t t i n g t h e c u r r e n t off with the m e l t j u s t a few d e g r e e s above the
l i q u i d u s , a m a x i m u m c o o l i n g r a t e e s t i m a t e d at b e t w e e n 104-105 d e g r e e s p e r second was obtained.
T h e m o v e m e n t of the c r y s t a l l i z a t i o n f r o n t w a s m e a s u r e d
and a v e l o c i t y of a p p r o x i m a t e l y 10-15 c m / s e c
found u n d e r t h e s e c o n d i t i o n s ; it
w a s a l s o found to be s o m e w h a t s l o w e r at s l o w e r q u e n c h i n g r a t e s .
T h i s is a
r e a s o n a b l e value c o m p a r a b l e to o t h e r s i m i l a r data. V a l u e s f o r the g r o w t h v e l o c i t y of c r y s t a l l i n e p h a s e s have b e e n m e a s u r e d by e a r l i e r w o r k e r s , n o t a b l y G r a u e r and H a m i l t o n (16) and T a m m a n (17), a s d i s c u s s e d by Hillig and T u r n b u l l (18). by Van Hook (19).
A c o m p i l a t i o n of t h e s e d a t a w a s m a d e
To m a k e the m e a s u r e m e n t ,
the l i q u i d s w e r e c a r e f u l l y s u p e r -
c o o l e d and the s p o n t a n e o u s c r y s t a l l i z a t i o n v e l o c i t y was r e p o r t e d a s a f u n c t i o n of this s u p e r c o o l i n g .
T h e v a l u e s f o r c e r t a i n van d e r W a a l s c o m p o u n d s , e . g .
m - d i n i t r o b e n z e n e , p - d i c h l o r o b e n z e n e and c h l o r o f o r m , having low v i s c o s i t y a r e 10 to 20 c m / s e c
at 20 to 40 d e g r e e s of i n i t i a l s u p e r c o o l i n g .
In spite of t h i s
d i f f e r e n t a p p r o a c h , and v e r y d i f f e r e n t p h a s e s , the m a x i m u m v e l o c i t y of c r y s t a l l i z a t i o n f o r low v i s c o s i t y l i q u i d s c a n be s e e n to lie at a value l e s s than 100
272
cm/sec.
GLASS FORMATION
Vol. 3, No. 3
The v i s c o s i t y of c h l o r o f o r m is 0. 009 poise at T m ( - 1 3 ° C ) and for
NaC1, 0. 013 poise (20) at T m. In addition, P r e d e c k i (10) c a l c u l a t e d that the c r y s t a l l i z a t i o n v e l o c i t y for m o l t e n a l u m i n u m and s i l v e r in the splat cooling p r o c e s s a r e a p p r o x i m a t e l y 10 c m and 50 c m p e r second, r e s p e c t i v e l y .
The v i s c o s i t y of m o l t e n a l u m i n u m
at the m e l t i n g point is 0. 027 poise [Mitra and S h a r m a , (21)]. F r o m t h e s e data on the c r y s t a l l i z a t i o n v e l o c i t y of NaC1, an e s t i m a t e of the c r i t i c a l quenching r a t e (defined a s the cooling r a t e for m e l t s below which solid c r y s t a l l i n e p h a s e s a r e obtained) m a y be c a l c u l a t e d .
It will be a s s u m e d
that, for NaC1, the spontaneous c r y s t a l n u c l e a t i o n and d e n d r i t i c g r o w t h r a t e b e c o m e v e r y low at a p p r o x i m a t e l y 10 d e g r e e s u p e r c o o l i n g .
If the l o w e r l i m i t
of d e t e c t a b l e c r y s t a l l i n e o r d e r i n g o c c u r s when the c r y s t a l l i t e s i z e is ---10~, then this c r i t i c a l cooling r a t e is the r a t e which will not allow the g r o w t h of a 10~ n u c l e u s within a 10 ° i n t e r v a l and this is found to be (12.5 c m / s e c x 1 0 ° C / 1 0 -7 cm) or 1.2 x 109 ° C / s e c ,
approximately.
C o r r e l a t i o n of C r i t i c a l Cooling Rate and R e l a x a t i o n T i m e It is difficult to e s t a b l i s h a g e n e r a l r e l a t i o n b e t w e e n the cooling r a t e and r e l a x a t i o n t i m e for all types of m a t e r i a l s .
However, as a first approximation,
c l e a r l y the c r i t i c a l cooling r a t e (Q)* m a y be s i m p l y a s s u m e d to depend d i r e c t l y on the m e l t i n g t e m p e r a t u r e (Tm) and the r e l a x a t i o n ( R e s t s t r a h l e n ) f r e q u e n c y if) o r j u m p r a t e of the s t r u c t u r a l u n i t s c o m p r i s i n g the m e l t . Then: T m Q = Z. - ZfW T m w h e r e Z is s o m e constant. F o r NaC1, the value for (Q) has been shown to be a p p r o x i m a t e l y 1.2 x 109 ° C / s e c , v a l u e s for T
m
hence solving for (Z) by substitution of a c t u a l
and T for NaC1 we obtain, Z~
2.0x
10 -6 .
(8)
If now it is a s s u m e d that the s i m p l e s t of the g e n e r a l r e l a t i o n s connecting r e l a x a tion t i m e with v i s c o s i t y , t e m p e r a t u r e , ionic volume, etc. will hold, b a s e d on Eqns. (4), ( 6 ) a n d (7), T=~
(9) m
*'Defined as that cooling rate, belowwhich detectable crystalline phases are obtained on quenching the melt.
Vol. 3, No. 3
GLASS FORMATION
,--I
O
~D
273
CO
v
o o~
co
©
,,,'-4
g3 O
d
o
¢~,
•
v
(D o
o
o
0o O
co It-..
r/l
o ~ ,.o ,,--4
O
r~
v
o
0o ° ,,,,~
~1
0~..~
.-4
o
oo
eo
¢d
d •
CO
0J
~+
eo
~
o ~ O o
cD
,,,-d
I
.,_; O
co
o
o
o
o
~
Z;
Q
0
rut3
o,...i
t:z0
Z 1,-4
r..) r~
--
0J r~
•
:~
o
o
. -
¢~ (~
[-,oo
'z:s..~-,.-~
~ .= .,..,
r.x.1 o . , ~
h
274
GLASS FORMATION
Vol. 3, No. 3
the c r i t i c a l cooling rate (Q) b e c o m e s Q = 2 . 0 x 10 - 6 . where (V/N) is the same as (a3).
(Wm)2 • R / V ~
(I0)
Although this is an e x t r e m e simplification
of the interaction of the many p a r a m e t e r s involved, if the values for SiO2 are inserted (see Table 2), a (Q) value of approximately 4 ° C / s e c is obtained, which at least is of the right o r d e r of magnitude. If this calculation is applied to the simple low viscosity oxide (insulators) and metallic melts, values obtained for the critical quenching rates a r e (6.6 x 109 ° C / s e c ) for MgA1204 and (l. 0 x 108 ° C / s e c ) for lead.
This is consistent
with the inability to obtain g l a s s e s from these m a t e r i a l s at cooling rates up to l07 ° C / s e c . The values calculated for MgSiO3, Mg2SiO4 and BaTiO 3 are also in accord with the fact that an NCS phase is obtained for MgSiO 3 but not for the others by quenching using common laboratory techniques.
On the other hand,
using splat techniques (107 ° C / s e c ) both Mg2SiO4 and BaTiO 3 a r e made into NCS. Reference to the heat t r a n s f e r calculations in Table 1 shows that the maximum cooling rate for NaCl is approximately (3.8 x l09 ° C / s e c ) for very thin films formed in splat cooling.
This indicates that it is unlikely that s i m -
ple structured purely ionic insulators (or metals) can ever be "quenched" to a true glassy phase.
An e x t r e m e l y high degree of d i s o r d e r may a r i s e from the
thinness of the film, or very small ' c r y s t a l l i t e ' size. Results of the calculation of the dependence of the c r i t i c a l cooling rate on melt viscosity at varying T
using Eqn. (10) using a constant value (30 cc) m for the g r a m ionic (molecular) volume are shown plotted in Fig. la. This d i a g r a m is only meant to convey a qualitative picture of the relation.
The ef-
fect of ionic (molecular) volume at constant t e m p e r a t u r e (Tm) is illustrated in Fig. lb.
The numerical values calculated for these and subsequent figures
are available e l s e w h e r e (22). Similarly, by substitution in Eqn. (8) for the relation connecting the relaxation time with the coefficient of self-diffusion in Eqn. (3), the critical cooling rate becomes
T Q = 2 . 0 x 10 - 6 .
• D m a2 x
(11)
Vol. 3, No. 3
GLASS FORMATION
%*`% ' % %%*`
6.0
6.0
%*`*`*`
GLASSYPHASES(SRO0)
%~.%%*` *`*`%',
\\\
4.0
~
"G
oooo.
\\'k--
3.0
R
\ \ \
Associoted Liquids Bonds
)
\\\
-I.0 Unassociated Liquids (Covalent, Van der Woals, or M e t a l l i c B o n d s
- 2.0
-5.C
Ionic
o3, ,oo
/ooo./, ? , ')',V ,,\/ oo.o
2.0
I0,000~/ 3 /
"6 ~- 1.0
~
\ / 3a=
,000~?
~ o
\\\ \ \ \ kkk
(Covalent-IonicBonds,
Hydrogen
)',,\,,"\\
5.0
500 o K.
CRYSTALLINE
c. 1.0
',\\\
4.0
,oooo.
~
2.0
~%GLASSYPHASES(SRO0)
5.0
5.0 Viscous Melts:
275
~k
I0~3
! \ k\ \
CRYSTALLINE
-I.0
\ \ \ \ \ \ kkk
%%~ , ~
PHASES
T m
-2.0
%
= I000 °K.)
\\\ I
-2
I
I
0
I
I
2
I
I
I
4
I
6
i
I
B
~ "C
-2
I0
i
i
0
2ogQ(°C.Isec.)
i
i
i
i
i
i
2 4 6 2og Q (°C.lsec.)
i
1
B
i
I0
FIG. la
FIG. lb
Plot of the Logarithm of Melt V i s c o s i t y vs. Logarithm of C r i t i c a l Cooling i~ate at Various Melting Temperatures.
Plot of the Logarithm of Melt V i s c o s i t y vs. Logarithm of Critical Cooling Rate at Different Melt Moiety V o l u m e s .
The r e s u l t s of calculations of (Q) as a function of (Dx) and at varying T
are shown in Fig. 2a, using a constant value of 15~ 2 for the c r o s s m s e c t i o n a l area of the diffusing s p e c i e s . The effect of volume of the diffusing s p e c i e s at constant t e m p e r a t u r e (Tm) is shown graphically in Fig. 2b. P r e d i c t i o n of P h a s e A s s e m b l a g e on Quenching Melts If the e x p r e s s i o n for the calculation of the c r i t i c a l cooling rate, Eqn. (11) is r e a r r a n g e d to the form: q.a
3 .,~
: 1
(12)
2 . 0 x 10 -6 k(Tm )2 and l o g a r i t h m s taken, we obtain for the c r i t i c a l condition between g l a s s f o r m a tion and c r y s t a l l i z a t i o n the s a t i s f a c t i o n of the relation:
276
GLASS FORMATION
Vol. 3, No. 3
Molten Metals ona salts van der Wools Compounds HzO (Low Viscosity Liquids
-6~
CRYSTALLINE PHASES
CRYSTALLINE PHASES
-4.C -5.C
~. -7
//~ o2:50~'2
-6.0 ¸
5 -8
500 °K.
Associated Liquids
E -9
*%
-7.0~
/.~;/---~ o2--500~,~ ////
E -8.0
o
p-
o~ - 1 0
GLASS PHASES(SRO0)
Viscous Melts
-I
_151
o
-9.0
~ o
-I0.0
etc.
-12.0 f -2
i 0
I 2
I I 4 6 ,tog Q { °C./sec.] FIG.
Plot
/ / /
_GLASSY P..~ASES{SRO0}
-I 1.0
SiOz, B203 -If
oZ= 5~ 2
/ /
(Tm = I O 0 0 ° K ' ) / /
u) e4.
/,/___
i 8
i
-2
i
1
0
i
o,; 5ooo /
2
i
i
i
4
i
i
6
i
8
i
i
I0
J~og Q (°C./sec.)
FIG. 2b
2a
of the Logarithm
/-/-/--
of the
Coefficient of Self D i f f u s i o n vs. L o g a r i t h m of C r i t i c a l Cooling Rate at V a r i o u s Melting T e m p e r a t u r e s .
P l o t of the L o g a r i t h m of the Coefficient of Self D i f f u s i o n vs. L o g a r i t h m of C r i t i c a l Cooling Rate at Different Melt Moiety C r o s s sectional Areas.
3 log [Q" a • ~] + 5.70 = 0. (13) kT2 m Now if (R), the selected cooling rate available in any particular experimental arrangement is substituted for (Q), and the other melt parameters inserted, and the equation yields a positive number, it indicates that the critical cooling rate for the melt would be exceeded and NCS phases will probably result experimentally. crystallize.
Conversely negative numbers will indicate that the melt will For example, substituting for a melt of SiO2: 3 log [R" a " 7] + 5.70 = N ("quenchnumber") kT 2 q m
R = 10°C/sec;
(14)
n ~ 107 poise; ( e s t . ) a 3 ~ 1 0 0 0 ~ 3 = 10 - 2 1 c m 3 ; T m ~ 2 0 0 0 ° K
Vol. 3, No. 3
GLASS FORMATION
log[
10" 10 - 2 1 " 107
(4 x 106)(1.38 x 10 -16)
277
] + 5 . 7 0 = +2.16
which is in a c c o r d with e x p e r i m e n t a l f o r m a t i o n of g l a s s at this cooling rate. Analogously, substituting for a typical spinel m e l t (e. g. MgA1204): R
106 ° C / s e c ; ~ ~ 2 x 10 -2 poise; a 3
20~k3 ( e s t . ) ~ 2 x 10 -23 cm; T
m
2378°K; we obtain, 106" 2 x 10 - 2 3 " 2 x 10-21 x
x i0
, +
The negative sign indicates that one will obtain c r y s t a l l i n e m a t e r i a l , again in a c c o r d with e x p e r i m e n t . A s i m i l a r r e l a t i o n can be obtained by r e a r r a n g i n g Eqn. (11) which contains the coefficient of s e l f - d i f f u s i o n for the m e l t at Tm, viz. Q" a 2 log i T D ] + 5.70 = (Nq) (16) m m and substituting (R) for (Q) as before. H e r e again, if (Nq) is positive, (R) is g r e a t e r than (Q) and NCS p h a s e s will probably r e s u l t . The magnitude of (Nq) may t h e r e f o r e be i n t e r p r e t e d as a m e a s u r e of the p r o x i m i t y of the p a r t i c u l a r cooling rate to the c r i t i c a l g l a s s f o r m i n g value. If (Nq) is close to zero, the p r e s c r i b e d cooling rate will be close to c r i t i c a l and mixed NCS and c r y s t a l l i n e p h a s e s may be p r e s e n t e x p e r i m e n t a l l y . can be u n d e r s t o o d by r e f e r r i n g to Tables 1 and 2. a r e qualitative. value of (a3).
This
o f course, t h e s e r e l a t i o n s
The l a r g e s t variable e n t e r s b e c a u s e of the e s t i m a t i o n of the However, it may be r e c a l l e d that for s t r i c t l y ionic m e l t s , (a 3)
is close to the a v e r a g e ionic volume.
Large values for (a 3) would apply for
" p o l y m e r i z e d " m e l t s with v i s c o s i t i e s higher than one poise at the melting point. W h e r e a s the p r e s e n t a p p r o a c h may a d e q u a t e l y d e s c r i b e the k i n e t i c s of o r d e r i n g during rapid quenching in the vicinity of Tm, it is r e c o g n i z e d that d i f f u s i o n - c o n t r o l l e d p r o c e s s e s p e r s i s t during subsolidus cooling right down to (and below) r o o m t e m p e r a t u r e .
M o r e o v e r t h e r e is, in principle, no limit to
the size of the s a m p l e which can be t r e a t e d by these techniques.* To account *Indeed in the d i s c u s s i o n following the p r e s e n t a t i o n of the paper, Dr. E. Deeg r e f e r r e d to his unpublished w o r k along t h e s e lines in the case of large t a n k - s i z e glass melts.
278
GLASS FORMATION
Vol. 3, No. 3
a d e q u a t e l y for t h e s e p r o c e s s e s would r e q u i r e a c o m p l e x t i m e - t e m p e r a t u r e i t e r a t i o n p r o c e d u r e to be p e r f o r m e d , s i n c e the cooling r a t e c h a n g e s exponentially with t i m e .
Although d e t a i l e d m a t h e m a t i c a l t e c h n i q u e s a r e a v a i l a b l e to
p r o v i d e e x a c t solutions to this p r o b l e m , e x p e r i m e n t a l data on solid diffusion show that this c o n t r i b u t e s r e l a t i v e l y little to ion m i g r a t i o n d u r i n g the subsolidus cooling p e r i o d p r o v i d e d that the s u b s t r a t e t e m p e r a t u r e and T m differ by 1000°C or m o r e .
N e g l e c t of this p r o c e s s in the p r e s e n t c a l c u l a t i o n m a y
t h e r e f o r e lead to values of the c r i t i c a l cooling r a t e which a r e slightly low, but c e r t a i n l y by l e s s than one o r d e r of m a g n i t u d e . In o t h e r p a p e r s the a u t h o r s (5-7) have p r o v i d e d a c o n s i d e r a b l e volume of new data on the " q u e n c h a b i l i t y " to a g l a s s of a wide r a n g e of ionic oxide melts.
The " g l a s s n u m b e r " , Nq, d e v e l o p e d above, s e r v e s quite a d e q u a t e l y in
such c a s e s to s e p a r a t e those m e l t s which can be r e t a i n e d a s NCS f r o m those which c r y s t a l l i z e . Summary This p a p e r has i n t r o d u c e d the r a t e of quenching a s a m a j o r p a r a m e t e r in the p r e d i c t i o n of " g l a s s f o r m a t i o n " .
An e m p i r i c a l s e m i - q u a n t i t a t i v e
a p p r o a c h has led to the d e v e l o p m e n t of a s o - c a l l e d g l a s s n u m b e r , Nq, which can be r e l a t e d to only two e x p e r i m e n t a l p a r a m e t e r s (Tm, the m e l t i n g point and 77 the v i s c o s i t y at the m e l t i n g point), and an e s t i m a t e of the volume of the diffusing s p e c i e s , via the r e l a t i o n 3 a ~7] + 5.70. kT 2 m P o s i t i v e v a l u e s of N obtained at any g i v e n quenching r a t e R, indicate that a Nq = log j R .
g l a s s or NCS will f o r m . Acknowledgement The w o r k was s u p p o r t e d by the A d v a n c e d R e s e a r c h P r o j e c t s A g e n c y u n d e r C o n t r a c t No. DA-49-083 OSA-3140. Refe r e nce s 1. W. H. Z a c h a r i a s e n , J. Am. Chem. Soc. 54, 3841 (1932).
2. W. A. Weyl and E. C. M a r b o e , The Constitution of Glass, Vol. l, Wiley, N. Y. (1962); K. Han Sun, J. Am. Soc. 30, 277 (1947).
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3. A. Smekal, J. Soc. G1. Techn. 35, 411 (1951). 4. D. Turnbull and M. H. Cohen, J. Chem. Phys. 2.9_9, 1049 (1958). 5. P. T. Sarjeant and Rustum Roy, J. Am. Ceram. Soc. 50, 500 (1967). 6. P. T. Sarjeant and Rustum Roy, J. Appl. Phys. 3_88, 4540 (1967). 7. P. T. Sarjeant and Rustum Roy, J. Am. Ceram. Soc. (to be published). 8. H. J. V. Tyrell, Diffusion and Heat Flow in Liquids, Butterworths, London (1961). 9. R. C. Ruhl, CoolingRates in Splat Quenching, Office of Naval Research, Contract Nonr-1841 (38), Technical Report No. 12-DSR 7618, available from the Dept. of Metallurgy, M. I. T., Cambridge, Mass. (1966). I0. P. Predecki, A. W. Mullendore and N. J. Grant, Trans. Met. Soc. AIME 233, 1581 (1965). ii. S. Glasstone, K. Laidler and H. Eyring, The Theory of Rate Processes, McGraw-Hill, New York (1941). 12. W. D. Kingery, J. Am. Ceram. Soc. 42, 617 (1959). 13. P. Debye, Polar Molecules, pp. 77-108, Dover Publication, New York (1929). 14. W. Jackson, Proc. Roy. Soc. 153A, 158 (1935). 15. A. Einstein, Ann. Phys. 17, 549 (1905). 16. O. H. Grauer and E. H. Hamilton, J. Res. NBS, 495 (1950). 17. G. Tamman and E. Janckel, Z. anorg. Chem. 193, 76 (1930). 18. W. B. Hillig and D. Turnbull, J. Chem. Phys. 24, 914 (1956). 19. A. Van Hook, Crystallization Theory and Practice, ACS Monograph No. 152, Rheinhold Publishing Corp., New York (1961). 20. Handbook of Chemistry and Physics, 44th ed., Chemical Rubber Publishing Co., Cleveland, Ohio (1962). 21. S. S. Mitra and L. P. Sharma, J. Chem. Phys. 38, 1287 (1963). 22. P. T. Sarjeant, Experimental and T h e o r e t i c a l Approaches to the Glassy State, Ph.D. d i s s e r t a t i o n in Solid State Science, The Pennsylvania State University, University Park, Pennsylvania (1967).