X-ray analysis of diffusion controlled phase transformation kinetics
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Communications X-Ray Analysis of Diffusion
80
r =0 . 0
Controlled Phase Transformation Kinetics
60'
W. H. HALSTED AND J. E. GRAGG, JR.
40,
T H E u s e of r e l a t i v e i n t e g r a t e d X - r a y diffraction i n t e n s i t i e s to d e t e r m i n e v o l u m e f r a c t i o n s in two p h a s e a l l o y s i s a c o m m o n method of a n a l y s i s in the study of p h a s e t r a n s f o r m a t i o n k i n e t i c s . In the o r i g i n a l d e r i v a t i o n of t h i s a n a l y s i s by Cohen and A v e r b a c h , ~ however, it i s a s s u m e d that the p h a s e s a r e of the s a m e c o m p o s i t i o n and the a p p l i c a t i o n of this method of a n a l y s i s to diffusion c o n t r o l l e d t r a n s f o r m a t i o n s (where both the v o l u m e f r a c t i o n and the c o m p o s i t i o n m a y v a r y ) can lead to c o n s i d e r a b l e d i s c r e p a n c y b e t w e e n the c a l c u l a t e d v o l u m e f r a c t i o n s and those a c t u a l l y p r e s e n t in the alloy. T h u s , c o n s i d e r a b i n a r y alloy of m o l a r c o m p o s i t i o n mA and mB which c o n s i s t s of two p h a s e s , a and/3, with r e s p e c t i v e m o l a r c o m p o s i t i o n s rn~, m ~ and m~A, m~B. The i n t e g r a t e d X - r a y diffraction i n t e n s i t i e s , Pa(2O) and P/3(2,*), for t h e s e two p h a s e s a r e given by the expressions Pa(2~) =
Ka(2~)(mJ~fA + m~fB)zVa
P/3(2#) = g/~(2~)(mfiAfA
+ rn~ fB )e Vfi
[la] [lb]
w h e r e Ka(2~) and K5(2~) a r e c o n s t a n t s which depend on the g e o m e t r y of the d i f f r a c t i o n m e a s u r e m e n t and the c r y s t a l s t r u c t u r e of the specific p h a s e s , a f A and fB a r e the a t o m i c s c a t t e r i n g f a c t o r s of the A and B a t o m s p e c i e s in the alloy,* and Va and VB a r e the v o l u m e *Allphases are assumedto be disorderedsubstitutionalsolidsolutions. f r a c t i o n s of the a and 13 p h a s e s . Following the Cohen= A v e r b a c h a n a l y s i s , we can define the r a t i o of t h e s e i n t e n s i t i e s , c o r r e c t e d for g e o m e t r i c and s t r u c t u r a l f a c t o r s , by the e x p r e s s i o n
Pa(2~)/K~(2~) Y = pZ(2,~)/Kz(2,~ ) (rn*~fA + m~ fB)2V~
[2a]
Va = y/(1 + y)
[3a]
V5 = 1 / ( 1 + 7)
[35]
W. H. HALSTED, formerly a Student in the Department of Metallurgy and Materials Science, Carnegie-MellonUniversity, Pittsburgh, Pa., is now Research Engineer with Standard Oil Company of California. J, E. GRAGG, Jr. is Assistant Professor, Department of Metallurgy and Materials Science, Carnegie-MellonUniversity. This paper is based upon a thesis submitted by W. H. HALSTED in partial fulfillment of the requirements of the degree of Bachelor of Science in Metallurgy and Materials Science at Carnegie-MellonUniversity. Manuscript submitted January 25, 1972.
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Fig. 1--The true volume fraction. V~ vs the volume fraction calculated from the Cohen-Averbach analysis, Va. The a phase is the precipitating
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