The solubility of oxides in molten alloys
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m/n
x~:
7 d In "ro.~* + m (dl__d~ff_~/* '
k7;7)
[4]
.\
W h e r e ( d I n Y o / d X M ) * = £M a n d ( d In YM/dXM)* m a y be n e g l e c t e d , x ~ - - rn/n eM w h i c h i s the f o r m c i t e d e a r l i e r 2 w i t h r = rn/n. S i n c e , f o r t h e c a s e s c o n s i d e r e d , eM i s a l w a y s l e s s t h a n z e r o , a m i n i m u m p o i n t m a y be e x p e c t e d . I n the u s u a l m a n n e r , i t m u s t be s h o w n t h a t d 2 x o / d x ~ i s >0 a t t h e m i n i m u m p o i n t ,s A s n o t e d e a r l i e r , t h e r e m a y be a d d i t i o n a l v a l u e s of xM a t w h i c h d x o / d x M i s z e r o . T h e y m a y be f o u n d by c o n s i d e r i n g i n m o r e d e t a i l t h e v a r i a t i o n s o f 7 o a n d VM w i t h x o a n d x M• SOLUTE INTERACTIONS
THE SOLUBILITY P R O D U C T O x i d e - m e t a l solvent equilibria m a y be considered in t e r m s of Eq. [I]. MmO n = mM + nO.
[1]
F o r t h e p u r p o s e o f the p r e s e n t d i s c u s s i o n , t h e d e p e n d e n c e of 7 0 a n d ~M on c o m p o s i t i o n w i l l be r e p r e s e n t e d by E q s . [5] a n d [6]. o M 2 yo In + °xo + + p O x L + po
+ p~,oxM, o
W h e r e t h e s t a n d a r d G i b b s e n e r g y o f f o r m a t i o n of MmOn i s m u c h l e s s t h a n t h a t of t h e o x i d e o f the p r i n c i p a l solvent metal, the oxide may be considered to be almost pure MmOn. By rearranging the equilibrium constant of Eq. [1] and replacing aM and ao with
o
I n ~M ~ I n rM +
[51
eMMXM + eOMx o + P M XM + p O x ° M 2
2
o,M xoXM" + DM
[6
T h e s e f o r m s h a v e the a d v a n t a g e of b e i n g w e l l k n o w n t h r o u g h t h e w o r k o f L u p i s a n d E l l i o t t .7 I n a d d i t i o n ,
TMXM and TOXO, respectively, Eq. [2] is obtained. m InxM +n InxO = InKlaMmon-m InYM -
n
In YO"
I000
[2]
d~M
-
- -
~M
dlny M +
m
-
-
axM
d In y O 1 +
n
TM ]
.
[3]
'
'
'
'
''1
'
'
",C
-..
*Eq. 3 may also be derived by meansof a Gibbs-Duhem treatment; however, the given method is rigorous and direct.
~o[m
% ,
'
'
'
' ' ' 1
,
,
i
, , , , _
!
%'k
IOO -
,
~
%4
D i f f e r e n t i a t i o n o f E q . [2] w i t h r e s p e c t t o x M a t c o n s t a n t v a l u e s of temperature and and further r e a r r a n g e m e n t y i e l d s E q . [3]* aMinO n
dxo
T
V
/
~"~.~
~
E Q . E 0
E q . [3] g i v e s t h e s l o p e o f a s o l u b i l i t y c u r v e a t a n y p o i n t i n t e r m s o f the a l l o y c o m p o s i t i o n (XM, x o ) a n d the variation of the activity coefficients with alloy c o m p o s i t i o n ( s o l u t e i n t e r a c t i o n s ) . S i n c e our a t t e n t i o n is d i r e c t e d at m i n i m a a n d m a x i m a a l o n g t h e c u r v e , t h e v a l u e s of XM a t w h i c h d x o / d xM i s z e r o m u s t be f o u n d f r o m E q . [4].
~_
10
I
lO.Ol GEORGE R. St. PIERRE is Professor of Metallurgical Engineering, Ohio State University, Columbus, OH 43210. Manuscript submitted July 1 5 , 1976. METALLURGICAL
TRAN
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