Morphology relations during bulk-transport sintering
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w h e r e xa is the r a t i o of i n t e r p a r t i c l e neck r a d i u s to i n i t i a l s p h e r e r a d i u s , n is the m e c h a n i s m c h a r a c t e r i s tic exponent, B is a p a r a m e t e r a c c o u n t i n g for t e m p e r a t u r e , p a r t i c l e size and m a t e r i a l p r o p e r t i e s , and t is the i s o t h e r m a l s i n t e r i n g t i m e . E v a l u a t i o n s of the v a r i ous b u l k - t r a n s p o r t s i n t e r i n g m o d e s by n u m e r o u s a u t h o r s v have r e s u l t e d in c o n t r a d i c t i n g f o r m s of Eq. [1]. Thus, the i n a b i l i t y of this approach to c h a r a c t e r i z e the i n d i v i d u a l s i n t e r i n g m e c h a n i s m s with u n a m b i g u o u s v a l u e s of n has made a n a l y s e s by neck growth (and t h e r e b y s h r i n k a g e ) of q u e s t i o n a b l e v a l u e . However, it h a s long been r e a l i z e d that definite m o r p h o l o g y c h a n g e s a c c o m p a n y s i n t e r i n g by the b u l k - d i f f u s i o n p r o c e s s e s . 8 Both the i n t e r p a r t i c l e s p a c i n g and the p o r e - s o l i d f r e e s u r f a c e a r e a d i m i n i s h with advanced s i n t e r i n g . R. M. GERMAN is Staff Member in thc Metallurgyand Electroplating Divisionof Sandia Laboratories, Livermore, CA 94550. Z. A. MUNIR is Profcssor in the Materials and Devices Research Group and the Dept. of Mechanical Engineering, University of California, Davis, CA 95616. Manuscript submitted January 17, 1975. METALLURGICALTRANSACTIONS A
It would, t h e r e f o r e , s e e m that an u n d e r s t a n d i n g of the morphology changes accompanying b u lk - tr a ns port sint e r i n g is highly d e s i r a b l e . To achieve this goal, geom e t r i c m o d e l i n g of the s i n t e r i n g s p h e r e i s p r o v i d e d in which the i n t e r p a r t i c l e s p a c i n g and s u r f a c e a r e a a r e r e l a t e d to the t i m e dependent neck growth. The p r e s e n t work p r o v i d e s a u s e f u l m e t h o d for m o n i t o r ing b u l k - t r a n s p o r t c o n t r o l l e d s i n t e r i n g m o r p h o l o g y c h a n g e s . F u r t h e r , by s i m u l t a n e o u s m e a s u r e of s h r i n k a g e a n d s u r f a c e a r e a , it is shown that the r e l a tive role of s u r f a c e - t r a n s p o r t m e c h a n i s m s d u r i n g s i n t e r i n g can be a s s e s s e d . F i n a l l y , this work p r o v i d e s the f i r s t p r e c i s e m e a s u r e of the c o n t i n u o u s change for the s p h e r e - s p h e r e g e o m e t r y .
THE MODEL In this m o d e l the i n t e r p a r t i c l e neck f o r m e d between two equal s i z e d s p h e r e s is r e p r e s e n t e d by r e v o l v i n g a m o d i f i e d c a t e n a r y about the Z - a x i s as shown in Fig. 1. In this figure a is the p a r t i c l e r a d i u s , x is the neck r a d i u s , qb is the angle c i r c u m s c r i b i n g the i s o s i n t e r ing zone, Ys is the s p h e r e b o u n d a r y , YN is the m o d i f i e d c a
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