An analysis of the capillary forces in liquid-phase sintering of spherical particles
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is a well known fact that u n d e r c e r t a i n c i r c u m s t a n c e s v e r y s m a l l additions of a liquid phase can d r a s t i c a l l y i n c r e a s e s i n t e r i n g k i n e t i c s . ' - a The p u r pose of this paper is to c o r r e c t and extend the a n a l y s i s of the r e s u l t i n g c a p i l l a r y f o r c e s that i s a l r e a d y p r e s e n t in the l i t e r a t u r e . 4,s P r e v i o u s w o r k e r s t r e a t e d the case where the l i q u i d - s o l i d - v a p o r contact a n g l e , O, was z e r o and t a c i t l y made t h r e e m a j o r a s s u m p t i o n s : i) the p a r t i c l e s a r e s p h e r i c a l , ii) the total i n t e r p a r ticle force is due s o l e l y to the p r e s s u r e d i f f e r e n c e a c r o s s the liquid-vapor i n t e r f a c e , and iii) that at s m a l l v o l u m e s of liquid the c u r v a t u r e of the l i q u i d - v a p o r i n t e r f a c e in the r - z plane, see Fig. 1, is c o n s t a n t (the so-called circle approximation). In a s u b s e q u e n t paper we will show that s e v e r a l m a j o r c o n c l u s i o n s about s i n t e r i n g will be a l t e r e d if the p a r t i c l e s a r e jagged i n s t e a d of s p h e r i c a l . In this paper we will extend the s p h e r e a n a l y s i s to all 0 and show that the a s s u m p t i o n s ii) and iii) a r e v a l i d for s m a l l volume f r a c t i o n s of liquid and lead to e x t r e m e I y a c c u r a t e r e s u l t s when applied to s i n t e r i n g p r o b l e m s . F i r s t we will e x a m i n e the f o r m the force equation m u s t take, then calculate the i n t e r p a r t i c l e f o r c e s a s a function of v o l u m e - f r a c t i o n liquid and contact a n g l e , and then c o m p a r e these f o r c e s to those obtained by using the c i r c l e a p p r o x i m a t i o n .
2) V, the volume of the liquid phase in the " p e n d u l a r r i n g " , see Fig. 1. 3) R, the r a d i u s of the s p h e r e s . 4) YLV, YSV, and YSL, the s u r f a c e free e n e r g i e s of the l i q u i d - v a p o r , s o l i d - v a p o r , and s o l i d - l i q u i d i n t e r f a c e s , r e s p e c t i v e l y . The y ' s a r e not functions of the c u r v a t u r e of the i n t e r f a c e s f o r v a l u e s of c u r v a t u r e of p r a c t i c a l i m p o r t a n c e . THE FORCE EQUATION By cutting the s y s t e m through any plane z = k and m a k i n g a force b a l a n c e in the z d i r e c t i o n a c r o s s the cut, t h e r e s e e m to be three f o r c e s which m u s t be con-
z=Ry
Liquid - V a p o r Interface = f (r,z) T a n g e n t to Sphere Slope = - r c / Z c
z c =Ry c THE SYSTEM F o r s i m p l i c i t y , only the c a s e of two s p h e r e s of the s a m e r a d i u s will be c o n s i d e r e d . We a l s o a s s u m e the s p h e r e s a r e i s o t r o p i c with r e s p e c t to the s u r f a c e free e n e r g y and i n e r t with r e s p e c t to the liquid and vapor. F o r p a r t i c l e s i z e s of g e n e r a l i n t e r e s t in s i n
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