Interpretation of grain boundary grooving data for combined surface and volume diffusion
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Table t. Calculation of ~ and ~ from p and w/(Bt) i ir from ST Table I
Interpretation of Grain Boundary Grooving Data for Combined Surface and Volume Diffusion W. M. ROBERTSON
AND
p
I/~"4
0 t,0 3.0
4.600 5.874 7.599 oo
S. R. SRINIVASAN
~
2.5
• Mullins a n a l y z e d g r a i n b o u n d a r y g r o o v e g r o w t h k i n e t i c s u n d e r the a c t i o n of t h e s e p a r a t e p r o c e s s e s of s u r f a c e diffusion ~ and v o l u m e diffusion, z R e c e n t l y S r i n i v a s a n and T r i v e d i s ( h e r e a f t e r r e f e r r e d to a s ST) a n a l y z e d the g r o o v i n g p r o c e s s u n d e r the c o m b i n e d a c tion of the two d i f f u s i o n m e c h a n i s m s . They d e r i v e d a n a n a l y t i c a l e x p r e s s i o n f o r the d e v e l o p m e n t of a g r a i n b o u n d a r y g r o o v e p r o f i l e a s a function of t i m e and showed that the s e p a r a t e r a t e c o n s t a n t s f o r s u r f a c e and v o l u m e diffusion could be e x t r a c t e d f r o m e x p e r i mental data by a complicated data analysis process. T h e o b j e c t i v e of the p r e s e n t w o r k is to show that the ST t h e o r y c a n be u s e d to g i v e a m u c h s i m p l e r a n a l y s i s of g r o o v e g r o w t h d a t a f r o m which the s u r f a c e and volume diffusion rate constants can be easily calculated. In t h e i r a n a l y s i s of g r o o v i n g , ST o b t a i n e d an e x p r e s s i o n f o r the s u r f a c e p r o f i l e of a g r a i n b o u n d a r y g r o o v e which r e l a t e s the h e i g h t y of the s u r f a c e p r o f i l e a b o v e the i n i t i a l f l a t s u r f a c e to the d i s t a n c e x f r o m the g r a i n b o u n d a r y a f t e r a n n e a l i n g t i m e t, Eq. [10] of S T ' s p a p e r . We c a n o b t a i n a r e l a t i o n b e t w e e n the g r o o v e width w and t i m e by d i f f e r e n t i a t i n g y with r e s p e c t to x to o b t a i n the s u r f a c e s l o p e and s e t t i n g x = w / 2 w h e r e the s l o p e e q u a l s z e r o . The r e l a t i o n we o b t a i n i s :
L
/exp Y o
(-~s 4 -/3sS) sins/2 s
ds = 1,
B= A'
=
D s 7 vf~z kT
.
.....
[2]
DvYf~ kT
-
and w h e r e D s i s t h e s u r f a c e d i f f u s i o n c o e f f i c i e n t , D v the v o l u m e d i f f u s i o n c o e f f i c i e n t , y t h e s u r f a c e f r e e e n e r g y p e r unit a r e a , u t h e n u m b e r of a t o m s p e r unit a r e a , the v o l u m e p e r a t o m , k B o l t z m a n n ' s c o n s t a n t , and T the a b s o l u t e t e m p e r a t u r e . Eq, [1] g i v e s an i m p l i c i t r e l a t i o n b e t w e e n a and/3 which w a s n u m e r i c a l l y s o l v e d by ST, though in a d i f f e r e n t f o r m . ST used a v a r i a b l e p which can be e x -
W. M. ROBERTSON is with Science Center, Rockwell International, Thousand Oaks, CA 91360. S. R, SRINIVASAN :iswith Max-Ptancklnstitut fur Eisenforschung GblBH, Dusseldorf, Germany. Manuscript submitted June 3, I974. METALLURGICAL TRANSACTIONS A
0.
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