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The effect of thermal diffusion on constitutional supercooling during solidification
Scripta METALLURGICA
Vol. I, pp. 93-96, 1967
Pergamon P r e s s , Inc.
Printed in the United States.
THE
EFFECT OF THERMAL SUPERCOOLING
DIFFUSION ON CONSTITUTIONAL DURING SOLIDIFICATION*
J. D. V e r h o e v e n I n s t i t u t e f o r A t o m i c R e s e a r c h and D e p a r t m e n t of M e t a l l u r g y , I o w a S t a t e U n i v e r s i t y , A m e s , I o w a 50010
(Received August 3, 1967) By m a k i n g a f l u x b a l a n c e on t h e s o l u t e at the i n t e r f a c e of a s o l i d i f y i n g a l l o y , t h e e q u a t i o n f o r the o n s e t of c o n s t i t u t i o n a l s u p e r c o o l i n g m a y be d e r i v e d , G ~
mRX D
(1)
w h e r e X ° i s t h e f r a c t i o n a l c o m p o s i t i o n of s o l u t e in t h e o r i g i n a l a l l o y and t h e o t h e r s y m b o l s have their usual meaning.
T h i s e q u a t i o n w a s f i r s t d e r i v e d by T i l l e r et a l . ,
(!) and it a p p l i e s
o n l y when the e f f e c t i v e d i s t r i b u t i o n c o e f f i c i e n t , ke, i s one (the bulk l i q u i d c o m p o s i t i o n = s o l i d c o m p o s i t i o n = Xo).
M a n y d i f f e r e n t e x p e r i m e n t a l i n v e s t i g a t i o n s on d i l u t e a l l o y s h a v e shown
that the e q u a t i o n p r e d i c t s th e o n s e t of i n t e r f a c e i n s t a b i l i t y w i t h i n e x p e r i m e n t a l e r r o r . In d e r i v i n g eq (I) it i s a s s u m e d that o r d i n a r y d i f f u s i o n i s the o n l y m e c h a n i s m of m a s s t r a n s p o r t f o r t h e s o l u t e in t h e l i q u i d at the i n t e r f a c e . the i n t e r f a c e ,
S i n c e a t e m p e r a t u r e g r a d i e n t e x i s t s at
m a s s t r a n s p o r t w i l l a l s o o c c u r due to t h e r m a l d i f f u s i o n ( S o r e t e f f e c t ) .
The
e q u a t i o n f r e q u e n t l y u s e d to d e s c r i b e u n i d i r e c t i o n a l m a s s t r a n s p o r t in a b i n a r y s y s t e m r e s u l t ing f r o m b o t h a c o n c e n t r a t i o n g r a d i e n t ( o r d i n a r y d i f f u s i o n ) and a t e m p e r a t u r e m a l diffusion) is,
gradient (ther-
(Z) dX J -- - p D ~ Z
w h e r e X is f r a c t i o n a l composition,
. D' P X(1 - X )
p is density,
dT ~
(2)
D i s t h e o r d i n a r y d i f f u s i o n c o e f f i c i e n t , D' i s
a t h e r m a l d i f f u s i o n c o e f f i c i e n t , T i s t e m p e r a t u r e and Z i s d i s t a n c e .
Th e f l u x b a l a n c e at t h e
i n t e r f a c e of a u n i d i r e c t i o n a l s o l i d i f y i n g a l l o y i n c l u d i n g t h e r m a l d i f f u s i o n g i v e s the f o l l o w i n g
equation, dX dT RC~, - R C s = - p~,D~, (~-~)~ - D'%p~.X%(I-X~) (~--~),
(B)
w h e r e a l l t e r m s a r e r e s t r i c t e d to the i n t e r f a c e p o s i t i o n , R i s t h e r a t e of i n t e r f a c e m o t i o n , C i s a v o l u m e c o n c e n t r a t i o n , and the s u b s c r i p t s ~ and s r e f e r to the l i q u i d and s o l i d s i d e s of the i n t e r f a c e ,
respectively.
As in th e d e r i v a t i o n of eq (I), d i f f u s i o n in t h e s o l i d i s n e g l e c t e d .
T h e o n s e t of c o n s t i t u t i o n a l s u p e r c o o l i n g i s d e s c r i b e d by t h e f o U o w i n g e q u a t i o n a p p l i e d at th e i n t e r f a c e ,
93
94
THERMAL DIFFUSION DURING SOLIDIFICATION
f d T "~
(dX /~
d T F "~
( d X "i
temperature
of t h e l i q u i d .
\-a-fJ~ ~ where
T F is the freezing
and assuming
interface
equilibrium
,."bE'/~
(4)
mRX~ D (1 - k ) G& < I - m X & ( l - X ~ ) S ST is the Soret coefficient
k e is one,
the composition
L e t t i n g G~ -- ( d T / d Z ) ~ ,
m = -(dTF/dX)~,
and a constant density for solid and liquid,
f o l l o w i n g e q u a t i o n f o r t h e o n s e t of c o n s t i t u t i o n a l
where
Vol. 1, No. 2
supercooling
by combining
one finds the
e q (3) a n d (4),
T
(5)
defined as'the
ratio,
D'/D.
In freezing
on the liquid side of the interface
experiments
where
is Xo/k after the initial transient
and one obtains, mRX
o
(l-k)
D
k
G& < 1 - m X 6 ( l Comparing
- X6)S T
(6)
e q s (6) a n d (1), i t i s s e e n t h a t t h e f r a c t i o n a l
the Soret effect is given by the thermal has been measured Winter
in three
and Drickamer
molten binary
systems.
of t h e s e a u t h o r s
agree
would expect the thermal the Sn-Pb
Kawakami
(3, 4).
fusion introduces interface
stability
a relatively
system
diffusion factor
an error
experiments
is the eutectic
mal diffusion introduces w o u l d b e 27%.
Consider,
Hence,
of 9%.
on concentrated
system.
Hence,
the error
For
neglecting
thermal
dif-
Since most
and since the Sn-Pb
system
likely that thermal
alloys however,
has
diffusion
it is possible
supercooling
t h e w o r k of M o l l a r d
require-
and Flemmings
In this work the liquid composition of t h e v a l u e of X • o
d i f f u s i o n f a c t o r i s 0. 0 9 2 .
an error
solute
i n t h e s e t y p e of s t u d i e s .
irrespective
the thermal
on i n t e r -
The maximum
case in these investigations.
for example,
in the Sn-Pb
composition
Sn r i c h s i d e o f t h i s s y s t e m the error
conducted
(5) a n d
investigations
in the literature.
diffusion will be significant when using the constitutional stability.
1 that one
Using this value for X° and the above Sorer
of o n l y 1% i n t h e w o r s t
o n t h e g r o w t h of c o m p o s i t e s
from the
Fig.
of P b i n Sn a t t h e m e l t i n g p o i n t
six different
h i g h v a l u e of m a n d S T a n d l o w v a l u e of k ° i t s e e m s
ment for interface
The data
i n d i l u t e s o l u t i o n s f o r a g i v e n S T.
i s f o u n d to b e 0 . 0 1 Z .
error
systems
of S T r a n g i n g
determined
It can be seen from
studies are made at lower concentrations,
In solidification
interface
At least
have been reported
is usually negligible within experimental
that thermal
I.
t h e i r d a t a to d i l u t e s o l u t i o n s
s t u d i e s w a s 0. 1 a t . %*Pb.
the thermal
and give values
taXi and X6maybe
as shown on Fig.
i s a r o u n d 4 x 10 -3 ° K - I .
c o n t e n t i n a n y of t h e s e
(3, 4) u s i n g e q (Z).
which has been studied by both Winter and Drickamer
Extrapolating
face stability in the Sn-Pb coefficient,
The terms
by neglecting
The Soret coefficient
to t h e S o r e t c o e f f i c i e n t .
system
d i f f u s i o n e f f e c t to b e s m a l l e s t
system
the Soret coefficient
by Kawakarni
This ratio is easily converted
0. 18 x 10 - 3 to 6 . 3 x 10 -3 ° K - I .
introduced
diffusion ratio in the same three
fairly well except in the Sn-Zn
v a l u e of X ° a n d t h e p h a s e d i a g r a m
Consider
a thermal
error
m X 6 ( 1 -X4~)S T .
alloy systems
(5) h a v e m e a s u r e d
plus 4 additional from
diffusion factor,
experiments
becomes
(6)
at the
For their work on the
Hence,
neglecting
the ther-
o n t h e P b r i c h s i d e of t h i s s y s t e m
significant
in concentrated
alloys,
Vol.
1, N o . 2
THERMAL
DIFFUSION
DURING SOLIDIFICATION
95
FIG. I
reX/
Xo
XZ
Determination of the t e r m s reX& and X L f r o m the phase diagrams and t h e o r i g i n a l a l l o y c o m p o s i t i o n X o.
provided molten
the data of Kawakami metals
are accurate.
i n t h e w o r k of M o l l a r d to s t a b i l i z e
state conditions, striction
arises
condition, diffusivity. experiment
the thermal
dilute solutions, requires
on the order
A representative
(5) o n S o r e t c o e f f i c i e n t s
in
d i f f u s i o n to b e s i g n i f i c a n t
gradients
as high as 480°C/cm
interface.
from the boundary
in effect,
T 1, a t a d i s t a n c e
(6) s i n c e t h e y e m p l o y e d
Y u e (7) h a s d e r i v e d
cooling which also includes
and Drickamer
One might intuitively expect thermal
and Flemmings
the solid-liquid
In a recent paper
(3, 4) a n d W i n t e r
a n e q u a t i o n f o r t h e o n s e t of c o n s t i t u t i o n a l
diffusion effect.
ke = l,
His equation is restricted
and no mixing in the liquid. to a p p r o a c h
a constant
of a t l e a s t 2 K / R f r o m t h e i n t e r f a c e , v a l u e of K f o r l i q u i d m e t a l s
with a liquid specimen
One additional
condition used for the heat flow equation.
the liquid temperature
l e n g t h of 20 c m a t r a v e l
is around
superto s t e a d y
where
measurable
value,
K is the thermal
0. 1 c m 2 / s e c .
r a t e of a t l e a s t
re-
This boundary
For an
10 . 2 c m / s e c
96
THERMAL DIFFUSION DURING SOLIDIFICATION
would b e r e q u i r e d .
Vol. I, No. 2
Hence, Y u e ' s e q u a t i o n i s r e s t r i c t e d to r e l a t i v e l y high t r a v e l r a t e s .
Eq. (5) of this w o r k i s s u b j e c t to n o n e o f t h e s e r e s t r i c t i o n s . S o r e t c o e f f i c i e n t s a r e e x t r e m e l y d i f f i c u l t to m e a s u r e without i n t r o d u c i n g e r r o r s due to Liquid c o n v e c t i o n w i t h i n the m e a s u r i n g c e l l .
In v i e w of r e c e n t w o r k by Cole and W i n e g a r d (8)
and U t e c h and E a r l y (9) t h e r e c a n be Little doubt that K a w a k a m i (3, 4) had c o n s i d e r a b l e c o n v e c t i o n in h i s e x p e r i m e n t s .
S i n c e c o n v e c t i o n e r r o r s would t e n d to r e d u c e S T b e l o w the t r u e
v a l u e i t i s p o s s i b l e that the v a l u e s of S T in m o l t e n a l l o y s a r e h i g h e r t h a n t h o s e found by K a w a k a m i and D r i c k a m e r .
Yue (7) h a s r e c e n t l y r e p o r t e d a t h e r m a l d i f f u s i o n c o e f f i c i e n t of
1 . 6 5 x 10 -5 c m 2 / s e c = ° C for F e i n A1.
F o r the u s u a l v a l u e s of l i q u i d d i f f u s i o n c o e f f i c i e n t s ,
t h i s c o r r e s p o n d s to a S o r e t c o e f f i c i e n t a t l e a s t two o r d e r s of m a g n i t u d e h i g h e r t h a n t h o s e r e p o r t e d by K a w a k a m i o r D r i c k a m e r .
Hence, it m a y w e l l be that the S o r e t effect i s v e r y
i m p o r t a n t in d e t e r m i n i n g c o n s t i t u t i o n a l s u p e r c o o l i n g in c o n c e n t r a t e d a l l o y s .
R e f e r e n c es *
W o r k w a s p e r f o r m e d i n the A m e s L a b o r a t o r y of the U. S. A t o m i c E n e r g y C o m m i s s i o n . C o n t r i b u t i o n No. 2122.
(1)
W.A.
T i l l e r , K. A. J a c k s o n , J. W. R u t t e r , and B. C h a l m e r s , Acta Met. I, 428
(1953). (2)
S . R . D e G r o o t , T h e r m o d y n a m i c s of I r r e v e r s i b l e P r o c e s s e s , P u b . C o . , A m s t e r d a m (1952).
(3)
S. K a w a k a m i , Nippon K i n z o k u Gakkai=Si 17, 59, 620 (1953).
(4)
S. K a w a k a m i , Ibid 19, 322 (1955).
(5)
F.R.
W i n t e r and H. G. D r i c k a m e r , 5. P h y s . C h e m . 59, 1229 0 9 5 5 ) .
(6)
F.R.
IV[ollard and Iv[. C. F l e m i n g s , to be p u b l i s h e d i n T r a n s . Met. Soc. AIME.
(7)
A.S.
Yue, C r y s t a l G r o w t b p .
(8)
G. $. Cole a n d W. C. W i n e g a r d , J. I n s t . Ivietals 93, 153 (1965).
(9)
H.P.
197, P e r g a m o n P r e s s ,
1~ 114, N o r t h - H o U a n d
New York (1967).
U t e c h a n d J. G. E a r l y , A c t a Met. 15, 1238 (1967).
Vol. I, pp. 93-96, 1967
Pergamon P r e s s , Inc.
Printed in the United States.
THE
EFFECT OF THERMAL SUPERCOOLING
DIFFUSION ON CONSTITUTIONAL DURING SOLIDIFICATION*
J. D. V e r h o e v e n I n s t i t u t e f o r A t o m i c R e s e a r c h and D e p a r t m e n t of M e t a l l u r g y , I o w a S t a t e U n i v e r s i t y , A m e s , I o w a 50010
(Received August 3, 1967) By m a k i n g a f l u x b a l a n c e on t h e s o l u t e at the i n t e r f a c e of a s o l i d i f y i n g a l l o y , t h e e q u a t i o n f o r the o n s e t of c o n s t i t u t i o n a l s u p e r c o o l i n g m a y be d e r i v e d , G ~
mRX D
(1)
w h e r e X ° i s t h e f r a c t i o n a l c o m p o s i t i o n of s o l u t e in t h e o r i g i n a l a l l o y and t h e o t h e r s y m b o l s have their usual meaning.
T h i s e q u a t i o n w a s f i r s t d e r i v e d by T i l l e r et a l . ,
(!) and it a p p l i e s
o n l y when the e f f e c t i v e d i s t r i b u t i o n c o e f f i c i e n t , ke, i s one (the bulk l i q u i d c o m p o s i t i o n = s o l i d c o m p o s i t i o n = Xo).
M a n y d i f f e r e n t e x p e r i m e n t a l i n v e s t i g a t i o n s on d i l u t e a l l o y s h a v e shown
that the e q u a t i o n p r e d i c t s th e o n s e t of i n t e r f a c e i n s t a b i l i t y w i t h i n e x p e r i m e n t a l e r r o r . In d e r i v i n g eq (I) it i s a s s u m e d that o r d i n a r y d i f f u s i o n i s the o n l y m e c h a n i s m of m a s s t r a n s p o r t f o r t h e s o l u t e in t h e l i q u i d at the i n t e r f a c e . the i n t e r f a c e ,
S i n c e a t e m p e r a t u r e g r a d i e n t e x i s t s at
m a s s t r a n s p o r t w i l l a l s o o c c u r due to t h e r m a l d i f f u s i o n ( S o r e t e f f e c t ) .
The
e q u a t i o n f r e q u e n t l y u s e d to d e s c r i b e u n i d i r e c t i o n a l m a s s t r a n s p o r t in a b i n a r y s y s t e m r e s u l t ing f r o m b o t h a c o n c e n t r a t i o n g r a d i e n t ( o r d i n a r y d i f f u s i o n ) and a t e m p e r a t u r e m a l diffusion) is,
gradient (ther-
(Z) dX J -- - p D ~ Z
w h e r e X is f r a c t i o n a l composition,
. D' P X(1 - X )
p is density,
dT ~
(2)
D i s t h e o r d i n a r y d i f f u s i o n c o e f f i c i e n t , D' i s
a t h e r m a l d i f f u s i o n c o e f f i c i e n t , T i s t e m p e r a t u r e and Z i s d i s t a n c e .
Th e f l u x b a l a n c e at t h e
i n t e r f a c e of a u n i d i r e c t i o n a l s o l i d i f y i n g a l l o y i n c l u d i n g t h e r m a l d i f f u s i o n g i v e s the f o l l o w i n g
equation, dX dT RC~, - R C s = - p~,D~, (~-~)~ - D'%p~.X%(I-X~) (~--~),
(B)
w h e r e a l l t e r m s a r e r e s t r i c t e d to the i n t e r f a c e p o s i t i o n , R i s t h e r a t e of i n t e r f a c e m o t i o n , C i s a v o l u m e c o n c e n t r a t i o n , and the s u b s c r i p t s ~ and s r e f e r to the l i q u i d and s o l i d s i d e s of the i n t e r f a c e ,
respectively.
As in th e d e r i v a t i o n of eq (I), d i f f u s i o n in t h e s o l i d i s n e g l e c t e d .
T h e o n s e t of c o n s t i t u t i o n a l s u p e r c o o l i n g i s d e s c r i b e d by t h e f o U o w i n g e q u a t i o n a p p l i e d at th e i n t e r f a c e ,
93
94
THERMAL DIFFUSION DURING SOLIDIFICATION
f d T "~
(dX /~
d T F "~
( d X "i
temperature
of t h e l i q u i d .
\-a-fJ~ ~ where
T F is the freezing
and assuming
interface
equilibrium
,."bE'/~
(4)
mRX~ D (1 - k ) G& < I - m X & ( l - X ~ ) S ST is the Soret coefficient
k e is one,
the composition
L e t t i n g G~ -- ( d T / d Z ) ~ ,
m = -(dTF/dX)~,
and a constant density for solid and liquid,
f o l l o w i n g e q u a t i o n f o r t h e o n s e t of c o n s t i t u t i o n a l
where
Vol. 1, No. 2
supercooling
by combining
one finds the
e q (3) a n d (4),
T
(5)
defined as'the
ratio,
D'/D.
In freezing
on the liquid side of the interface
experiments
where
is Xo/k after the initial transient
and one obtains, mRX
o
(l-k)
D
k
G& < 1 - m X 6 ( l Comparing
- X6)S T
(6)
e q s (6) a n d (1), i t i s s e e n t h a t t h e f r a c t i o n a l
the Soret effect is given by the thermal has been measured Winter
in three
and Drickamer
molten binary
systems.
of t h e s e a u t h o r s
agree
would expect the thermal the Sn-Pb
Kawakami
(3, 4).
fusion introduces interface
stability
a relatively
system
diffusion factor
an error
experiments
is the eutectic
mal diffusion introduces w o u l d b e 27%.
Consider,
Hence,
of 9%.
on concentrated
system.
Hence,
the error
For
neglecting
thermal
dif-
Since most
and since the Sn-Pb
system
likely that thermal
alloys however,
has
diffusion
it is possible
supercooling
t h e w o r k of M o l l a r d
require-
and Flemmings
In this work the liquid composition of t h e v a l u e of X • o
d i f f u s i o n f a c t o r i s 0. 0 9 2 .
an error
solute
i n t h e s e t y p e of s t u d i e s .
irrespective
the thermal
on i n t e r -
The maximum
case in these investigations.
for example,
in the Sn-Pb
composition
Sn r i c h s i d e o f t h i s s y s t e m the error
conducted
(5) a n d
investigations
in the literature.
diffusion will be significant when using the constitutional stability.
1 that one
Using this value for X° and the above Sorer
of o n l y 1% i n t h e w o r s t
o n t h e g r o w t h of c o m p o s i t e s
from the
Fig.
of P b i n Sn a t t h e m e l t i n g p o i n t
six different
h i g h v a l u e of m a n d S T a n d l o w v a l u e of k ° i t s e e m s
ment for interface
The data
i n d i l u t e s o l u t i o n s f o r a g i v e n S T.
i s f o u n d to b e 0 . 0 1 Z .
error
systems
of S T r a n g i n g
determined
It can be seen from
studies are made at lower concentrations,
In solidification
interface
At least
have been reported
is usually negligible within experimental
that thermal
I.
t h e i r d a t a to d i l u t e s o l u t i o n s
s t u d i e s w a s 0. 1 a t . %*Pb.
the thermal
and give values
taXi and X6maybe
as shown on Fig.
i s a r o u n d 4 x 10 -3 ° K - I .
c o n t e n t i n a n y of t h e s e
(3, 4) u s i n g e q (Z).
which has been studied by both Winter and Drickamer
Extrapolating
face stability in the Sn-Pb coefficient,
The terms
by neglecting
The Soret coefficient
to t h e S o r e t c o e f f i c i e n t .
system
d i f f u s i o n e f f e c t to b e s m a l l e s t
system
the Soret coefficient
by Kawakarni
This ratio is easily converted
0. 18 x 10 - 3 to 6 . 3 x 10 -3 ° K - I .
introduced
diffusion ratio in the same three
fairly well except in the Sn-Zn
v a l u e of X ° a n d t h e p h a s e d i a g r a m
Consider
a thermal
error
m X 6 ( 1 -X4~)S T .
alloy systems
(5) h a v e m e a s u r e d
plus 4 additional from
diffusion factor,
experiments
becomes
(6)
at the
For their work on the
Hence,
neglecting
the ther-
o n t h e P b r i c h s i d e of t h i s s y s t e m
significant
in concentrated
alloys,
Vol.
1, N o . 2
THERMAL
DIFFUSION
DURING SOLIDIFICATION
95
FIG. I
reX/
Xo
XZ
Determination of the t e r m s reX& and X L f r o m the phase diagrams and t h e o r i g i n a l a l l o y c o m p o s i t i o n X o.
provided molten
the data of Kawakami metals
are accurate.
i n t h e w o r k of M o l l a r d to s t a b i l i z e
state conditions, striction
arises
condition, diffusivity. experiment
the thermal
dilute solutions, requires
on the order
A representative
(5) o n S o r e t c o e f f i c i e n t s
in
d i f f u s i o n to b e s i g n i f i c a n t
gradients
as high as 480°C/cm
interface.
from the boundary
in effect,
T 1, a t a d i s t a n c e
(6) s i n c e t h e y e m p l o y e d
Y u e (7) h a s d e r i v e d
cooling which also includes
and Drickamer
One might intuitively expect thermal
and Flemmings
the solid-liquid
In a recent paper
(3, 4) a n d W i n t e r
a n e q u a t i o n f o r t h e o n s e t of c o n s t i t u t i o n a l
diffusion effect.
ke = l,
His equation is restricted
and no mixing in the liquid. to a p p r o a c h
a constant
of a t l e a s t 2 K / R f r o m t h e i n t e r f a c e , v a l u e of K f o r l i q u i d m e t a l s
with a liquid specimen
One additional
condition used for the heat flow equation.
the liquid temperature
l e n g t h of 20 c m a t r a v e l
is around
superto s t e a d y
where
measurable
value,
K is the thermal
0. 1 c m 2 / s e c .
r a t e of a t l e a s t
re-
This boundary
For an
10 . 2 c m / s e c
96
THERMAL DIFFUSION DURING SOLIDIFICATION
would b e r e q u i r e d .
Vol. I, No. 2
Hence, Y u e ' s e q u a t i o n i s r e s t r i c t e d to r e l a t i v e l y high t r a v e l r a t e s .
Eq. (5) of this w o r k i s s u b j e c t to n o n e o f t h e s e r e s t r i c t i o n s . S o r e t c o e f f i c i e n t s a r e e x t r e m e l y d i f f i c u l t to m e a s u r e without i n t r o d u c i n g e r r o r s due to Liquid c o n v e c t i o n w i t h i n the m e a s u r i n g c e l l .
In v i e w of r e c e n t w o r k by Cole and W i n e g a r d (8)
and U t e c h and E a r l y (9) t h e r e c a n be Little doubt that K a w a k a m i (3, 4) had c o n s i d e r a b l e c o n v e c t i o n in h i s e x p e r i m e n t s .
S i n c e c o n v e c t i o n e r r o r s would t e n d to r e d u c e S T b e l o w the t r u e
v a l u e i t i s p o s s i b l e that the v a l u e s of S T in m o l t e n a l l o y s a r e h i g h e r t h a n t h o s e found by K a w a k a m i and D r i c k a m e r .
Yue (7) h a s r e c e n t l y r e p o r t e d a t h e r m a l d i f f u s i o n c o e f f i c i e n t of
1 . 6 5 x 10 -5 c m 2 / s e c = ° C for F e i n A1.
F o r the u s u a l v a l u e s of l i q u i d d i f f u s i o n c o e f f i c i e n t s ,
t h i s c o r r e s p o n d s to a S o r e t c o e f f i c i e n t a t l e a s t two o r d e r s of m a g n i t u d e h i g h e r t h a n t h o s e r e p o r t e d by K a w a k a m i o r D r i c k a m e r .
Hence, it m a y w e l l be that the S o r e t effect i s v e r y
i m p o r t a n t in d e t e r m i n i n g c o n s t i t u t i o n a l s u p e r c o o l i n g in c o n c e n t r a t e d a l l o y s .
R e f e r e n c es *
W o r k w a s p e r f o r m e d i n the A m e s L a b o r a t o r y of the U. S. A t o m i c E n e r g y C o m m i s s i o n . C o n t r i b u t i o n No. 2122.
(1)
W.A.
T i l l e r , K. A. J a c k s o n , J. W. R u t t e r , and B. C h a l m e r s , Acta Met. I, 428
(1953). (2)
S . R . D e G r o o t , T h e r m o d y n a m i c s of I r r e v e r s i b l e P r o c e s s e s , P u b . C o . , A m s t e r d a m (1952).
(3)
S. K a w a k a m i , Nippon K i n z o k u Gakkai=Si 17, 59, 620 (1953).
(4)
S. K a w a k a m i , Ibid 19, 322 (1955).
(5)
F.R.
W i n t e r and H. G. D r i c k a m e r , 5. P h y s . C h e m . 59, 1229 0 9 5 5 ) .
(6)
F.R.
IV[ollard and Iv[. C. F l e m i n g s , to be p u b l i s h e d i n T r a n s . Met. Soc. AIME.
(7)
A.S.
Yue, C r y s t a l G r o w t b p .
(8)
G. $. Cole a n d W. C. W i n e g a r d , J. I n s t . Ivietals 93, 153 (1965).
(9)
H.P.
197, P e r g a m o n P r e s s ,
1~ 114, N o r t h - H o U a n d
New York (1967).
U t e c h a n d J. G. E a r l y , A c t a Met. 15, 1238 (1967).