The analysis of grain-boundary groove growth data when both surface and volume diffusion contribute
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This equation is b a s e d on the a s s u m p t i o n that the total m a s s t r a n s p o r t is the s u m of the m a s s t r a n s p o r t by s u r face and v o l u m e diffusion and the r a t h e r good a p p r o x i m a t i o n that the p r o f i l e of a g r a i n - b o u n d a r y groove is i d e n t i c a l for s u r f a c e and v o l u m e diffusion. 3 Chang a s sumed, in addition, the l i n e a r i t y of a l o g - l o g plot of w v s t, thus d log w / d log t = n, a c o n s t a n t , o r dw/dt
[2]
: nw/t
Equating the r i g h t - h a n d m e m b e r s of Eqs. [1] and [2] one obtains, after r e a r r a n g e m e n t , C h a n g ' s equation [3]
w 4 / t = (f3s/n) + ( f 3 v l n ) w
J. G. EBERHART AND H. M. F E D E R
Thus C h a n g ' s equation is a h y b r i d of t h e o r y and e m p i r i c i s m . Eq. [3] r e q u i r e s that a plot of w4/t v s w should has shown that the width, w , of a g r a i n be l i n e a r , with a slope of f3v/n and an i n t e r c e p t o f / 3 s / n . b o u n d a r y groove has a dependence on t i m e , t, of w To c a l c u l a t e ~v and ~s it is then a l s o n e c e s s a r y to obtain o c t 1/4 when the growth m e c h a n i s m is s u r f a c e diffusion, 1 n f r o m the slope of the l o g - l o g plot of w vs t. and w o c t l/s when the growth m e c h a n i s m is v o l u m e difIt is i n t e r e s t i n g to note the s i m i l a r i t y b e t w e e n Eq. [3] f u s i o n f l F o r a wide v a r i e t y of s y s t e m s an e m p i r i c a l and an analogous r e s u l t for the decay of a s i n u s o i d a l 1 growth equation of the f o r m w o c t n w h e r e ~ < n < ~ , s u r f a c e undulation by c o m b i n e d s u r f a c e and v o l u m e difhas b e e n o b s e r v e d and i n t e r p r e t e d as i n d i c a t i n g the s i f u s i o n f l F o r this p r o c e s s k4K is a l i n e a r function of ~, m u l t a n e o u s action of s u r f a c e and v o l u m e diffusion. F o r w h e r e ;~ is the wavelength, K is the exponential decay such s i t u a t i o n s , M u l l i n s and Shewmon 3 developed a t h e o c o n s t a n t for the a m p l i t u d e (a function of ~ with d i m e n r e t i c a l equation which has b e e n applied in a n u m b e r of s i o n s of t i m e - l ) , and the slope and the i n t e r c e p t provide i n v e s t i g a t i o n s 4-6 to p u r e m e t a l s whose v o l u m e s e l f - d i f the v o l u m e and s u r f a c e diffusion coefficients, r e s p e c fusion coefficients, D r , were a l r e a d y known f r o m i s o tively. This r e s u l t , however, is b a s e d e n t i r e l y on thetopic t r a c e r s t u d i e s . In t h e s e a n a l y s e s the isotopic o r y and r e q u i r e s no a p p r o x i m a t i o n that the profile is t r a c e r value of D v was used in an i t e r a t t v e p r o c e d u r e independent of the m a s s t r a n s p o r t m e c h a n i s m . to d e t e r m i n e the s u r f a c e s e l f - d i f f u s i o n coefficient, D s . A t e s t of C h a n g ' s method of a n a l y s i s of g r a i n - b o u n d A s i m
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