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Interaction of particles with an advancing solid/liquid interface Schvezov, Carlos Enrique 1984

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INTERACTION OF PARTICLES WITH AN ADVANCING SOLID/LIQUID INTERFACE By CARLOS ENRIQUE SCH.VEZOV Licenciado en F i s i c a , Universidad Nacional de Rosario(Argentina), 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Metallurgical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1983 (c) Carlos Enrique Schvezov, 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree th a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permission f o r ex t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date JdM U&TH i i ABSTRACT The i n t e r a c t i o n o f p a r t i c l e s w i t h an a d v a n c i n g s o l i d / l i q u i d i n t e r f a c e has been s t u d i e d c o n s i d e r i n g m e c h a n i c a l f o r c e s and f o r c e s a r i s i n g from the n a t u r e o f the p a r t i c l e , l i q u i d and s o l i d m a t e r i a l s . E x p e r i m e n t s were c a r r i e d out on l e a d and l e a d a l l o y s c o n t a i n i n g i r o n p a r t i c l e s to d e t e r m i n e whether the p a r t i c l e s were r e j e c t e d o r encompassed by an a d v a n c i n g i n t e r f a c e . The d i s t r i b u t i o n o f p a r t i c l e s i n the s o l i d as a f u n c t i o n o f the morphology o f the i n t e r f a c e was a l s o examined. I t was found t h a t p a r t i c l e s were not r e j e c t e d by an a d v a n c i n g i n t e r f a c e and were p r i m a r i l y s e g r e g a t e d i n t o c e l l w a l l s and i n t e r d e n d r i t i c zones f o r a non p l a n a r i n t e r f a c e . For a p l a n a r i n t e r f a c e the p a r t i c l e d i s t r i b u t i o n was u n i f o r m both on a m a c r o s c o p i c and m i c r o s c o p i c l e v e l . T h i s was a l s o the case f o r s o l i d i f i c a t i o n i n d i f f e r e n t d i r e c t i o n s , v e r t i c a l and h o r i z o n t a l . A water model system was examined to o b s e r v e d i r e c t l y how p a r t i c l e s i n t e r a c t w i t h a s o l i d s u r f a c e under d i f f e r e n t v e l o c i t i e s and s u r f a c e c o n d i t i o n s . The r e s u l t s c l e a r l y d e m o n s t r a t e d how s p h e r e s s e g r e g a t e to c e l l w a l l s on a s o l i d c e l l u l a r i n t e r f a c e as a r e s u l t o f m e c h a n i c a l f o r c e s . i i i The p r e s e n t r e s u l t s are compared to t h e o r i e s f o r p a r t i c l e i n t e r f a c e i n t e r a c t i o n s ini both metal and non m e t a l l i c systems. C a l c u l a t i o n s o f L i f s h i t z - V a n der Waals f o r c e s i n a metal show t h a t an a t t r a c t i v e f o r c e e x i s t s between the metal p a r t i c l e and metal s u r f a c e . As a r e s u l t p a r t i c l e s are not r e j e c t e d by an i n t e r f a c e i n a metal system i n agreement w i t h the p r e s e n t r e s u l t s . The o b s e r v e d d i s t r i b u t i o n o f p a r t i c l e s i n the s o l i d can be a c c o u n t e d f o r on the b a s i s o f buoyancy f o r c e s , c o n v e c t i o n i n the l i q u i d , and the i n t e r f a c e morphology. i v TABLE OF CONTENTS Page A b s t r a c t T a b e l o f C o n t e n t s . Iv L i s t o f T a b l e s v i i L i s t o f F i g u r e s — x Acknowledgement — ... — x i x Ch a p t e r I INTRODUCTION 1 II REVIEW 4 1. I n t e r a c t i o n of a P a r t i c l e w i t h a S o l i d / L i q u i d I n t e r f a c e ; D e f i n i t i o n o f the Problem- ..... — .,- 4 2. Comparison o f the Proposed T h e o r i e s w i t h E x p e r i m e n t a l Resul t s 13 a) E f f e c t o f the Thermal C o n d u c t i v i t y . . 19 b) The E f f e c t o f I m p u r i t i e s 22 c) Other E f f e c t s 24 i ) V i s c o s i t y o f the M e l t 24 i i ) Body F o r c e s 24 i i i ) C o n v e c t i o n 25 V Chapter Page 3. P ushing i n M e t a l s 2 5 - 4. Summary o f C h a p t e r II 26 m OBJECTIVES OF PRESENT RESEARCH 2 8 IV EXPERIMENTAL APPARATUS AND PROCEDURE• * 30 1. The M e t a l l i c PLS System 3 0 a) S e l e c t i o n and P r e p a r a t i o n o f Samples.. 3 0. b) The A p p a r a t u s and T e c h n i q u e f o r Con-t r o l l e d S o l i d i f i c a t i o n 33 i ) S o l i d i f i c a t i o n ..... 3 3 i i ) Metal 1 o g r a p h i c P r e p a r a t i o n ..... 3 6 i i i ) C o u n t i n g and P a r t i c l e S i z e D i s t r i b u t i o n 3^ c) P r o c e d u r e f o r t h e . C a s t i n g . E x p e r i m e n t . . 3 8 2. The Water Model 3 8 3 . P a r t i c l e I n t e r a c t i o n w i t h a F r e e z i n g Water I n t e r f a c e 4 2 V RESULT AND DISCUSSION 4 3 1. C o n t r o l l e d S o l i d i f i c a t i o n 4 3 a) N on-Planar I n t e r f a c e 4 3 i ) V e r t i c a l Growth 4 3 Pb 1% Sb C e l l u l a r I n t e r f a c e -Sample V-1 4 3 The Real D i s t r i b u t i o n o f P a r t i -c l e s i n the M a t r i x Chapter p a 9 e Pb 2% Sb D e n d r i t i c I n t e r f a c e -Sample V-2 5 7 i i ) H o r i zontal Growth . 7 0 b) Planar I n t e r f a c e 7 6 2. The Water-Nylon Sphere Model 7 9 a) Horizontal Mode 7 9 i ) The Motion of the Nylon Spheres 7 9 i i ) Modelling i i i ) Comparison with a S o l i d i f i c a -t i o n Process b) V e r t i c a l S o l i d i f i c a t i o n 3. The Casting Experiment . . — 4. Pushing in Water VI SUMMARY AND CONCLUSIONS REFERENCES APPENDICES I Theories of P a r t i c l e Rejection at a S o l i d / L i q u i d Interface II C a l c u l a t i o n of w 86 95 100 105 1 22 5. The L i f s h i t z - V a n der Waals Force H 1 29 1 33 1 37 167 v i i LIST OF TABLES T a b l e Page I Summary of f i n d i n g s f o r the i n t e r a c t i o n o f p a r t i c l e s w i t h an a d v a n c i n g s o l i d / l i q u i d i n t e r f a c e 7 II (a) V a l u e s of the exponent n i n the r e l a t i o n V c= C/Rn f o r the c r i t i c a l v e l o c i t y as a f u n c t i o n o f p a r t i c l e r a d i u s R , f o r the d i f f e r e n t t h e o r i e s , (b) E x p e r i m e n t a l v a l u e s of n and V. 17 I I I Raw P a r t i c l e S i z e D i s t r i b u t i o n f o r sample V-1. V - l - 1 I n t e r f a c e , V - l - l a , 200 ym from V - l - 1 ; V - l - 2 , 25 mm from I n t e r f a c e ; V - l - 2 a , 200 ym from V - l - 2 ; V - l - 3 , 5 mm from I n t e r -f a c e ; V - l - 3 a , 200 ym from V - l - 3 . 4 9 IV P a r t i c l e S i z e D i s t r i b u t i o n f o r Sample V-1 and % i n the M a t r i x . (a) counts a t the i n t e r f a c e , .(b) c o u n t s b e h i n d the i n t e r f a c e , and ( c ) o v e r a l 1 51 V S c h w a r z - S a l t i k o v c o e f f i c i e n t s f o r the c a l c u l a t i o n o f the Real S i z e D i s t r i b u t i o n .. 56 VI P a r t i c l e S i z e D i s t r i b u t i o n and % o f p a r t i -c l e s i n the m a t r i x f o r Sample V-1 ( c o u n t s on a l l s e c t i o n s ) (a) A p p a r e n t D i s t r i b u t i o n (b) Real D i s t r i b u t i o n 58 v i i i T a b l e Page VII Raw P a r t i c l e S i z e D i s t r i b u t i o n f o r s e g r e g a -t e d and M a t r i x p a r t i c l e s f o r Sample V-2. S e c t i o n V-2-1 a t the i n t e r f a c e ; V - 2 - l a , 20Q \xm from V-2-1 ; V-2-2, 2.5 mm from the i n t e r f ace 64 V I I I P a r t i c l e S i z e D i s t r i b u t i o n f o r s e g r e g a t e d and m a t r i x p a r t i c l e s , and % i n the m a t r i x f o r Sample V-2. Counts on a l l s e c t i o n s .... 65 IX P a r t i c l e S i z e D i s t r i b u t i o n f o r s e g r e g a t e d and m a t r i x p a r t i c l e s , and % i n the m a t r i x f o r Sample V-2. (a) Ap p a r e n t D i s t r i b u t i o n , (b) Real D i s t r i b u t i o n 68 X Raw P a r t i c l e S i z e D i s t r i b u t i o n f o r S e g r e g a -t e d and m a t r i x p a r t i c l e s f o r Sample H - l . H-l-1 c o u n t s at t h e i n t e r f a c e ; H - l ^ l a , 200 pm from H-l-1 ; H-l92, 2.5 mm from H - l - 1 ; and H - l - 2 a , 200 from H-l-2 7 2 XI P a r t i c l e S i z e D i s t r i b u t i o n f o r s e g r e g a t e d and m a t r i x p a r t i c l e s , and % i n the m a t r i x f o r Sample H - l . Counts on a l l s e c t i o n s .... 7^ XII C r i t i c a l V e l o c i t i e s i n f u n c t i o n o f p a r t i c l e r a d i u s p r e d i c t e d by the t h e o r i e s o f B o i l i n g and C i s s e (a) and Chernov e t a l . (b) 7 7 T a b l e i x " ' P a g e XIII C o r r e l a t i o n t a b l e f o r p a r t i c l e s i z e i n the m e t a l l i c system v e r s u s f r e q u e n c y o f r o t a t i o n ( E q u a t i o n 35) and v e r s u s p e r i o d o f r o t a t i o n . 8 8 XIV T h e o r e t i c a l and e x p e r i m e n t a l h o r i z o n t a l v e l o c i t i e s o f the n y l o n s p h e r e s as a f u n c t i o n of t i l t i n g a n g l e f o r w = 0.5 rps and V p = 1 cm s e c " 1 9 0 XV T e r m i n a l v e l o c i t y , t r a n s i e n t t i m e , t r a n s i e n t l e n g t h v e r s u s r a d i u s f o r i r o n p a r t i c l e s i n 9 9 l i q u i d l e a d XVI P a r t i c l e S i z e D i s t r i b u t i o n f o r the c a s t sample, (a) a t the bottom, (b) 5mm from bottom, (c) 15mm from bottom, (d) 25 mm from bottom,,(e) 30 mm from bottom - top o f the sample. In (a) and (b), p a r t i c l e s have been d i s c r i m i n a t e d as b e i n g s e g r e g a t e d or i n t h e d e n d r i t e s ' . In ( c ) , (d) and (e) t h r e e d i f f e r e n t p o s i t i o n s f o r the s e g r e g a -t e d p a r t i c l e s have been c o n s i d e r e d , bottom o f the d e n d r i t e o r t r a p p e d , i n the e u t e c t i c and a t the top o f the d e n d r i t e 109 XVII P a r t i c l e S i z e D i s t r i b u t i o n f o r h e i g h t s 15 mm, 25 mm and 30 mm f o r c a s t sample, and % i n the e u t e c t i c 116 LIST OF FIGURES A p a r t i c l e b e i n g r e j e c t e d by an a d v a n c i n g s o l i d / l i q u i d i n t e r f a c e ( s c h e m a t i c ) . The i n t e r f a c e s h o u l d e x h e r t a f o r c e t o a c c e l e r -a t e the p a r t i c l e and to w i t h s t a n d the drag f o r c e Free e n e r g i e s o f the PLS system v e r s u s d i s t a n c e from the i n t e r f a c e ( s c h e m a t i c ) i and i ' are two i n i t i a l s t a t e s w i t h h i g h e r and lower e n e r g i e s r e s p e c t i v e l y than the f i n a l s t a t e f . Curves a to d a r e p o s s i b l e e f f e c t s o f the c l o s e p r e s e n c e o f the p a r t i -c l e a t the i n t e r f a c e . When the p a r t i c l e i s r e p e l l e d (a-b) and a t t r a c t e d ( c - d ) . In c u r v e e the f r e e e nergy o f the system does not change u n t i l the p a r t i c l e i s i n the s o l i d Changes i n f r e e e nergy d u r i n g the e n g u l f -ment o f a cube o r s p h e r e o f u n i t s u r f a c e a r e a by a s o l i d S. The s u r f a c e e n e r g i e s i n the bul k media do not change due to the p r e s e n c e o f a t h i r d i n t e r f a c e The e f f e c t i v e s u r f a c e energy o f the PLS system as a f u n c t i o n o f d i s t a n c e . d Q i s a minimum d i s t a n c e a t which p a r t i c l e and s o l i d are c o n s i d e r e d to be i n c o n t a c t S c h e m a t i c r e p r e s e n t a t i o n o f s e p a r a t i o n v e r s u s growth v e l o c i t y f o r v a r i o u s p a r t i c l e s i z e s i n the Uhlmann t h e o r y . The upper branch are s t a b l e and the lower are u n s t a b l e a g a i n s t f l u c t u a t i o n s i n growth v e l o c i t y . The c r i t i c a l v e l o c i t y a t which the p a r t i c l e i s c a p t u r e d i s the maximum v e l o c i t y B o i l i n g and C i s s e scheme of p u s h i n g F p ( a ) i s the drag f o r c e ; F .',versus the s u r f a c e f o r c e and -r i s the c o n t a c t r a d i u s A p a r t i c l e ( l i q u i d , s o l i d or gas) i n "con-t a c t " w i t h the s o l i d . The d i s j o i n i n g p r e s -sure i n a gas bubble as a f u n c t i o n o f s e p a r a t i o n h i s measured by p r e s s i n g the bubble a g a i n s t t he s o l i d , T y p i c a l b e h a v i o r s o f the d i s j o i n i n g p r e s s u r e It(h) f o r a gas bubble i n a p o l a r l i q u i d (1) and n o n - p o l a r l i q u i d ( 2 ) , as a f u n c t i o n o f the d i s t a n c e from the s o l i d . Branches a and 8 g i v e s t a b l e f i l m s w h i l e y i s u n s t a b l e .... Three media s o l i d (1) - l i q u i d or gas (3) -s o l i d , l i q u i d or gas (2) l e a d i n g to an i m p o r t a n t i n t e r a c t i o n a t v e r y c l o s e d i s -t a n c e s ~ 10"^ cm d e s c r i b e d by the L i f s h i t z -Ven der Waal t h e o r y XI 1 F i g u r e Page 10 R e p r e s e n t a t i o n o f the symbols used i n the Chernov e t a l . t h e o r y o f p u s h i n g 162 11 C r i t i c a l growth r a t e v e r s u s p a r t i c l e r a d i u s f o r S i 0 2 and W p a r t i c l e s . (a) w i t h bump r a d i u s o f 5 ym and 10 ym r e s p e c t i v e l y ; and (b) copper p a r t i c l e s a t two t e m p e r a t u r e g r a d i e n t s and f o r c l e a n e d and non c l e a n e d p a r t i c l e s . The t h r e e d e f i n i t e r e g i o n s c o r -r e s p o n d to f o r m u l a s 18 ( a - c ) ( R e f . 23) 1 4 12 T h e o r e t i c a l and e x p e r i m e n t a l c r i t i c a l '. v e l o c i t i e s f o r water. Curves 1 and 2 are f o r s m a l l and b i g p a r t i c l e s r e s p e c t i v e l y g i v e n by the t h e o r y o f Chernov e t a l . Curve 4 c o r r e s p o n d s to the c r i t i c a l v e l o c i t i e s g i v e n f o r e l e c t r o s t a t i c i n t e r a c t i o n o n l y (Re f . 10) 1 5 13 E f f e c t o f the t h e r m a l c o n d u c t i v i t y on the p u s h i n g p r o c e s s . (a) kp < k^ , k g ; (b) k p = k s and (c) k p > k r k s . In (a) and ( c ) the i s o t h e r m s are d i s t o r t e d , heat t r a n s f e r i s enchanced and i n h i b i t e d r e s p e c -t i v e l y and so the c a p t u r e o f the p a r t i c l e . . 20 SEM m i c r o g r a p h o f the ARMCO i r o n p a r t i c l e s -400 mesh used i n the e x p e r i m e n t s 200X Diagram o f the e x p e r i m e n t a l s e t up f o r s o l i d i f i c a t i o n o f m e t a l s ( f o r e x p l a n a t i o n see t e x t ) .. The p h y s i c a l model to s t u d y the motion o f p a r t i c l e s a t a c e l l u l a r " i n t e r f a c e " . (a) The l u c i t e c y l i n d r i c a l c o n t a i n e r w i t h a b r i n e s o l u t i o n , the i n t e r f a c e and the n y l o n s p h e r e s ; a i s the t i l t i n g a n g l e form the h o r i z o n t a l f o r the h o r i z o n t a l mode, (b) The c e l l s and the p a r t i c l e s i n d e t a i l , (c) A sphere w i t h v o l i c i t y V due to buoyancy f o r c e s c o l l i d i n g w i t h a c e l l t i p , V i s the r e s u l t a n t v e l o c i t y i m m e d i a t e l y a f t e r s c a t -t e r i n g . The v e c t o r V o r i e n t e d downward-i s the v e l o c i t y a f t e r h a l f r e v o l u t i o n w i t h r e s p e c t to a system f i x e d t o the r o t a t i n g c e l l s L o n g i t u d i n a l view of Sample V-1. The quenched l i q u i d and u n i d i r e c t i o n a l l y grown s o l i d are c l e a r l y d e f i n e d . The c e l l w a l l a ppear as l o n g s t r i p s 50X ( e t c h e d ) T r a n s v e r s a l views of Sample V - l a t the i n t e r f a c e (a) 100.x;-.,. (b), and (c) 500X, (d) SEM-200X; and 12 mm b e h i n d the i n t e r -f a c e (e) 100X. Most of the p a r t i c l e s are i n the c e l l w a l l s . In (a.) a t the c e n t e r l e f t a r a m i f i c a t i o n i s shown. In (c) and (d) the s m a l l p a r t i c l e s i n the m a t r i x are shown. In (e) most of the c e l l w a l l s have d i s s o l v e d . The i r o n p a r t i c l e s are the w h i t e s p o t s P a r t i c l e S i z e D i s t r i b u t i o n f o r Sample V - l ( c e l l u l a r i n t e r f a c e ) i n t h e c e l l b o u n d a r i e s and i n the m a t r i x . (a) For c o u n t s at the i n t e r f a c e (b) b e h i n d the i n t e r f a c e and (c) t o t a l c o u n t s . The d i s t r i b u t i o n s are s i m i l a r . Only a few p a r t i c l e s appear i n the m a t r i x .. % o f p a r t i c l e s i n the m a t r i x v e r s u s s i z e a t the i n t e r f a c e and b e h i n d the i n t e r f a c e . Small p a r t i c l e s appear more f r e q u e n t l y i n the m a t r i x than l a r g e p a r t i c l e s . The change i n t h e c u r v e from c o u n t s at the i n t e r f a c e and b e h i n d i t , i s a t t r i b u t e d to d i s s o l u t i o n o f r a m i f i c a t i o n i n the m i c r o s t r u c t u r e (a.) Apparent D i s t r i b u t i o n and (b) Real D i s t r i b u t i o n o f p a r t i c l e s f o r Sample V-1 ( c e l l u l a r i n t e r f a c e ) . The shape o f the h i s t o g r a m does not change s u b s t a n t i a l l y e x c e p t t h a t p a r t i c l e s s m a l l e r than 4 ym are not p r e s e n t i n the r e a l d i s t r i b u t i o n . In diagram (b) the volume i s a r b i t r a r y % o f p a r t i c l e s i n the m a t r i x v e r s u s s i z e f o r the a p p a r e n t and r e a l d i s t r i b u t i o n . Sample V-1. The e f f e c t f o r s m a l l p a r t i -c l e s i s remarked L o n g i t u d i n a l view a t the i n t e r f a c e o f Sample V-2 ( d e n d r i t i c i n t e r f a c e ) 100X ( e t c h e d ) ( a ) , ( b ) , ( c ) T r a n s v e r s a l views o f Sample V-2 a t the i n t e r f a c e . The p a r t i c l e s are i n the i n t e r d e n d r i t i c r e g i o n s 500X ( e t c h e d ) ... P a r t i c l e s i z e d i s t r i b u t i o n s f o r p a r t i c l e s i n the m a t r i x and i n i n t e r d e n d r i t i c r e g i o n s f o r Sample V-2. Less p a r t i c l e s than i n Sample V-1 ( c e l l u l a r i n t e r f a c e ) a r e i n the % o f p a r t i c l e s i n the m a t r i x v e r s u s s i z e f o r Sample V-2 ( d e n d r i t i c i n t e r f a c e ) . As i n Sample V - l the same p a t t e r n i s o b s e r v e d a l t h o u g h f o r Sample V-2 i s l e s s pronounced., (a) A p p a r e n t D i s t r i b u t i o n and (b) Real D i s t r i b u t i o n o f p a r t i c l e s f o r Sample V-2 ( d e n d r i t i c i n t e r f a c e ) T r a n s v e r s a l views o f Sample H-l a t the i n t e r -f a c e (a) 500X showing p a r t i c l e s i n the c e l l w a l l s and (b) 200X showing p a r t i c l e s a t the edge of the sample i n c o n t a c t w i t h the c o n t a i n e r P a r t i c l e S i z e D i s t r i b u t i o n f o r Sample H-l a t the c e l l b o u n d a r i e s and i n the m a t r i x ... % o f p a r t i c l e s i n the m a t r i x v e r s u s s i z e f o r Sample H - l . A random d i s t r i b u t i o n i s o b t a i n e d . Spheres d i s t r i b u t i o n i n the p h y s i c a l model f o r the h o r i z o n t a l mode f o r d i f f e r e n t r o t a -t i o n a l s peeds. (a).\w= 0.033 r p s , (b) 0.083 r p s , ( c ) 0.2 r p s , (d) rps and (e) 1 r p s . The p a r t i c l e s p a s s e d from b e i n g c o n c e n t r a t e d a t the w a l l o f the c y l i n d e r to the c e n t e r as e x p l a i n e d i n the t e x t . . . . . x v i i o f the c y l i n d e r , v = 1 cm/sec. , P 35 S e m i - q u a l i t a t i v e d e s c r i p t i o n o f the segrega-t i o n o f n y l o n s p h e r e s d u r i n g the v e r t i c a l e x p e r i m e n t s w i t h the model. In (a) the sur-f a c e a r e a s are c a l c u l a t e d . In (b) the be-h a v i o r o f s p h e r e s f o r s t i l l and c o n v e c t i v e l i q u i d are compared w i t h the e x p e r i m e n t a l o b s e r v a t i o n s , 36 P a r t i c l e s t r a p p e d i n a w e l l d e v e l o p e d d e n d r i t e (a) 200X and (b) 500X. P a r t i c l e s a t the bottom of the d e n d r i t e s ( c ) and ( d ) . P a r t i c l e f r e e i n the e u t e c t i c ( e ) . In (d) a p a r t i c l e i s p a r t i a l l y t r a p p e d a t the top of the d e n d r i t e . In (b) a s m a l l p a r t i c l e a ppears i n the c o r e of a d e n d r i t e F i g u r e Page 32 The two c l a s s e s o f p a t t e r n s f o l l o w e d by the s p h e r e s i n the model as a f u n c t i o n o f rotation . speed; (a) the s p h e r e s r e a c h the w a l l , (b) the s p h e r e s do not r e a c h the wal 1 i n one c y c l e 35 33 Nylon s p h e r e s t r a p p e d i n the grooves be-tween c e l l s f o r the h o r i z o n t a l mode. Photograph taken from one end o f the c y l i n d e r , w = 0.5 r p s a = 4 ° . . . 92 34 Nylon s p h e r e s i n the grooves i n the v e r t i c a l e x p e r i m e n t s . Photograph ta k e n from one end 92 101 106 F i g u r e 37 P a r t i c l e S i z e D i s t r i b u t i o n f o r the c a s t sample. (a) At the bottom, (b) 0.5 cm; from the bottom, (c) 1.5 cm from bottom, (d) 2.5 cm from bottom, (d) 3.0 cm from bottom 38 D e n s i t y o f p a r t i c l e s v e r s u s h e i g h t i n the c a s t sample 39 view o f the top p a r t o f t h e c a s t sample. The h i g h c o n c e n t r a t i o n o f p a r t i c l e s t h e r e / i s a t t r i b u t e d to f l o t a t i o n . S o l i d i f i c a -t i o n began when the p a r t i c l e s were a l r e a d y t h e r e 40 P u s h i n g i n water. (a) Water w i t h Cr p a r t i c l e s , (b) The i c e - w a t e r i n t e r f a c e a d v a n c i n g to the c e n t e r . (c) Dark band o f p a r t i c l e s pushed by the i n t e r f a c e . (d) S i m i l a r to (c) but w i t h a h i g h e r d e n s i t y o f p a r t i c l e s which r e s u l t e d i n a w i d e r band ACKNOWLEDGEMENTS I would l i k e to thank Dr. F r e d Weinberg f o r h i s a d v i c e and encouragement d u r i n g the c o u r s e o f t h i s work. F i n a n c i a l a s s i s t a n c e from the N a t i o n a l Research C o u n c i l ( C a n a d a ) , U n i v e r s i d a d Nacnonal de M i s i o n e s ( A r g e n t i n a ) and Fu n d a c i o n R o t a r i a n a de Posadas ( A r g e n t i n a ) i s : g r a t e f u l l y acknowledged. the Memory o f My P a r e n t s 1 C h a p t e r I INTRODUCTION The p r e s e n c e o f p a r t i c l e s i n a m a t e r i a l can s t r o n g l y i n f l u e n c e i t s p r o p e r t i e s ; . The m a t e r i a l p r o p e r t i e s can be enhanced as i s the ca s e f o r a l l o y h a r d e n i n g w i t h v e r y f i n e p r e c i p i t a t e s , or c o m p o s i t e m a t e r i a l s . A l t e r n a t i v e l y the p r o p e r t i e s can be i m p a i r e d as i n the case o f l a r g e i n -c l u s i o n s i n s t e e l . In most c a s e s when the p r e s e n c e o f p a r t i c l e s i n the m a t e r i a l i s d e t r i m e n t a l t o the p r o p e r t i e s , the p a r t i c l e s a r e i n h e r e n t to the f a b r i c a t i o n p r o c e s s and cannot be e l i m i n a t e d . A c c o r d i n g l y the c o n t r o l o f the d e t r i -mental e f f e c t o f the p a r t i c l e s i s r e l a t e d to the c o n t r o l o f the d e n s i t y and d i s t r i b u t i o n o f the p a r t i c l e s i n the m a t e r i a l . S i m i l a r l y enhanced m a t e r i a l p r o p e r t i e s are a l s o d i r e c t l y r e l a t e d t o the p a r t i c l e d e n s i t y and d i s t r i b u t i o n . P r e c i p i t a t i o n from s o l u t i o n , e l e c t r o l y t i c d e p o s i t i o n , and s o l i d i f i c a t i o n a r e t h r e e common p r o c e s s e s i n i n d u s t r y where a s o l i d - l i q u i d i n t e r f a c e i s p r e s e n t . I t i s a t t h i s ; s t a g e t h a t t h e main i n t e r a c t i o n o f the p a r t i c l e s ; , which may be p r e s e n t i n the l i q u i d , w i t h the s o l i d w i l l t a k e p l a c e . In the case o f water and many o r g a n i c s u b s t a n c e s , 2 the s o l i d i n t e r f a c e d u r i n g s o l i d i f i c a t i o n w i l l "push" s m a l l s o l i d p a r t i c l e s ; ahead o f the i n t e r f a c e o v e r l a r g e d i s t a n c e s r e l a t i y e to the d i a m e t e r o f the p a r t i c l e s . In m e t a l s i t i s not c l e a r to what e x t e n t p a r t i c l e s ; a r e r e -j e c t e d by an a d v a n c i n g s o l i d - l i q u i d i n t e r f a c e . A s t r o n g c o r r e l a t i o n has; been o b s e r v e d between p a r t i c l e d i s t r i b u -ti'on and c a s t s t r u c t u r e ~ i n d e o x i d i z e d s t e e l i n g o t s , the p a r t i c l e s b e i n g c o n c e n t r a t e d p r i m a r i l y i n the i n t e r -d e n d r i t i c r e g i o n s . T h i s s u g g e s t s the p o s s i b i l i t y t h a t some p a r t i c l e r e j e c t i o n o c c u r s a t the s o l i d - l i q u i d i n t e r f a c e . D i f f e r e n t mechanisms; have been p r o p o s e d to a c c o u n t f o r the o b s e r v e d c o n c e n t r a t i o n of p a r t i c l e s i n the i n t e r d e n d r i t i c r e g i o n s . These range from the p r o p o s a l t h a t the p a r t i c l e s r e s u l t from homogeneous n u c l e a t i o n i n the i n t e r d e n d r i t i c 33 34 r e g i o n , w i t h no r e j e c t i o n , at one extreme ' to p a r t i c l e 32 p u s h i n g d u r i n g s o l i d i f i c a t i o n s i m i l a r to t h a t o b s e r v e d i n water and o r g a n i c m a t e r i a l s ; a t the o t h e r extreme. In t h i s i n v e s t i g a t i o n the i n t e r a c t i o n o f a s o l i d -l i q u i d i n t e r f a c e i n a m e l t , w i t h s o l i d p a r t i c l e s i n the m e l t i s examined to d e t e r m i n e i f p a r t i c l e s a r e r e j e c t e d by an a d v a n c i n g I n t e r f a c e . No c o n s i d e r a t i o n w i l l be g i v e n to the n u c l e a t i o n and growth, o f s o l i d p a r t i c l e s i n the m e lt ahead o f t h e i n t e r f a c e , and i t w i l l 6e assumed t h a t no c h e m i c a l r e a c t i o n occurs between p a r t i c l e and the m e l t . The r e j e c t i o n i s c o n s i d e r e d from two points; o f view: (.1 ) the f o r c e s t h a t o r i g i n a t e a t the p a r t i c l e , s o l i d -l i q u i d i n t e r f a c e ; ( 2 ) the forces; a r e p u r e l y m e c h a n i c a l due to t h e r e l a t i v e movement o f the s o l i d , l i q u i d and p a r t i c l e a t t h e i n t e r f a c e . The c o n d i t i o n s ; f o r p a r t i c l e r e j e c t i o n by an advanc-i n g i n t e r f a c e a r e f i r s t r e v i e w e d , f o l l o w e d by a r e v i e w o f the t h e o r i e s p r o p o s e d to a c c o u n t f o r t h e s e o b s e r v a t i o n s . T h i s i s f o l l o w e d by a d e s c r i p t i o n . o f the e x p e r i m e n t a l p r o c e d u r e and o b s e r v a t i o n s ; i n a metal a l l o y system f o r both d i r e c t i o n a l : . ; ' and normal f r e e z i n g c o n d i t i o n s . The r e s u l t s ; a r e then compared to the o b s e r v a t i o n s , on p a r t i c l e r e j e c t i o n i n water and the t h e o r e t i c a l p r e d i c t i o n s f o r a metal a l l o y system. The p r e s e n t r e s u l t s i n d i c a t e t h a t t h e r e i s no r e -j e c t i o n o f p a r t i c l e s i n a m e t a l system by a s o l i d l i q u i d a d v a n c i n g i n t e r f a c e . The o b s e r v e d p a r t i c l e d i s t r i b u t i o n i n the s o l i d can be. accounted .fbri..by a m e c h a n i c a l mechanism f o r t h e i n t e r a c t i o n o f the p a r t i c l e w i t h the i n t e r f a c e . T h i s mechanisjn i s d e v e l o p e d and v e r i f i e d i n a p h y s i c a l model o f the system. 4 C h a p t e r II REVIEW. 1. I n t e r a c t i o n o f a P a r t i c l e w i t h a Sol i d / L i q u i d I n t e r f a c e ; D e f i n i t i o n o f the Prob!em. Assume a p a r t i c l e i s a t r e s t w i t h r e s p e c t t o t h e l i q u i d , f a r away from a p l a n a r a d v a n c i n g s o l i d - l i q u i d i n t e r f a c e . What then happens when the i n t e r f a c e a p proaches t h e p a r t i c l e ? Two extreme s i t u a t i o n s may be c o n s i d e r e d : e i t h e r t h e i n t e r -f a c e w i l l be c o m p l e t e l y i n d i f f e r e n t to the p r e s e n c e o f the p a r t i c l e , o r the p a r t i c l e w i l l be r e j e c t e d by the i n t e r f a c e and move ahead o f the f r o n t i n s t e a d y s t a t e . The second a l t e r n a t i v e i s shown s c h e m a t i c a l l y i n F i g . 1 . The s t e a d y r e j e c t i o n o f the p a r t i c l e r e q u i r e s the p a r t i c l e t o be a c c e l e r a t e d from z e r o to t h e i n t e r f a c e v e l o c i t y V. When t h i s v e l o c i t y i s r e a c h e d the i n t e r f a c e must e x e r t a f o r c e on the p a r t i c l e i n the d i r e c t i o n o f s o l i d i f i c a t i o n to compensate f o r the drag f o r c e t h a t a r i s e s due t o the t r a n s p o r t o f f l u i d n e c e s s a r y f o r s o l i d i f i c a t i o n to p r o c e e d . T h i s f o r c e on the p a r t i c l e s h o u l d be g e n e r a t e d a t the l i q u i d l a y e r between the p a r t i c l e and s o l i d t h r o u g h an i n t e r a c t i o n between the p a r t i c l e , the l i q u i d l a y e r and the s o l i d i f y i n g m a t e r i a l ( r e f e r r e d to as PLS). T h i s i n t e r a c t i o n w i l l be c o n s i d e r e d l a t e r . 5 L i q u i d ( L ) V \ \ \ \ \ \ \ \ \ \ S o l i d (S ) Figure 1. A p a r t i c l e being rejected by a S/L interface.The i n t e r -face should ex ert a force to accelerate the p a r t i c l e and withstand the drag f o r c e . 6 L a v a l l e i n 1853 noted t h a t f o r e i g n p a r t i c l e s ; are r e p e l l e d by a growing c r y s t a l . However i t was not u n t i l the n i n e t e e n f i f t y ' s ; t h a t the p r o c e s s was: examined i n d e t a i l . The. r e s u l t o f i n v e s t i g a t i o n s ; o f p a r t i c l e i n t e r -a c t i o n s w i t h a s o l i d i f y i n g f a c e r e p o r t e d i n the l i t e r a t u r e to d ate a r e summarized i n T a b l e I. Pikunov,^ s o l i d i f y i n g s e v e r a l o r g a n i c l i q u i d s con-t a i n i n g a s u s p e n s i o n of p a r t i c l e s , f o u n d t h a t the m i g r a t i o n o f p a r t i c l e s ; a t the i n t e r f a c e depended on the p a r t i c u l a r p a r t i c l e and l i q u i d m a t r i x c o n s i d e r e d . For example, s a l o l does not pus;h carbon nor s t a r c h p a r t i c l e s . In d i p h e n y l a m i n e , c a r b o n i s pushed and s t a r c h i s n o t . In azobezene and b e n z y l , both, s t a r c h and carbon p a r t i c l e s a r e pushed by an a d v a n c i n g i n t e r f a c e . Th;e r e s u l t s , r e p o r t e d above by Pikunov i n s a l o l a r e 2 i n c o n t r a d i c t i o n w i t h t h o s e o f Kuo and W i l c o x who r e -p o r t e d t h a t c arbon p a r t i c l e s ; (Q. 5-2.5ym) a r e pushed by s a l o l . On the o t h e r hand UliTmann and Pikunov s o l i d i f i e d the m e l t h o r i z o n t a l l y , with., a v e r t i c a l i n t e r f a c e . Kuo and W i l c o x s o l i d i f i e d v e r t i c a l l y . S o l i d i f y i n g v e r t i c a l l y s h o u l d make t h e r e j e c t i o n o f the p a r t i c l e more d i f f i c u l t s i n c e the f o r c e s ; a t t h e i n t e r f a c e have to compensate f o r the drag f o r c e o f the p a r t i c l e as w e l l as the. g r a y i t y 7 TABLE I EXPERIMENTAL RESULTS ON PUSHING Particle Size, pm Critical Velocity, um/sec Pushing Observed Growth Direction and Comments Ref. LIQUID: WATER hollow carbon 20 5.5 yes vertical 23 spheres C(grapMte) — >20 yes horizontal 4 C(diamond) — >20 yes horizontal 4 Si02 20-50 0.89-0.66 yes vertical 23 Si — >20 yes horizontal 4 MgO — >20 yes horizontal 4 A1203 - >20 yes horizontal 4 fe2°3 — >20 yes horizontal 4 Agl — >20 yes horizontal 4 glass bead 60-120 0.34-0.03 yes vertical 21 broken glass 149-590 0.2 -0.03 yes vertical 54 calcite 149-590 0.2 -0.01 yes vertical 54 rutile 149-590 0.33-0.11 yes vertical 54 quartz 149-590 0.36-0.12 yes vertical 54 shale 149-590 0.55-0.17 yes vertical 54 mica 149 1.39 yes vertical 54 silt — >20 yes horizontal 4 Ni — >20 yes horizontal 4 Zn - >20 yes horizontal 4 Sn - >20 yes horizontal 4 Cu 2.5-62.5 '5-0.165 yes vertical 23 w 2.5-30 3.3-0.18 yes vertical 23 LIQUID: SALOL Agl — < 0.28 no horizontal 4 C(graphite) — < 0.28 no horizontal 4 C 0.5-2.5 1.94 yes vertical 2 C 0 no horizontal 1 C(diamond) Or5 2-2.1 yes horizontal 4 starch 0 no horizontal 1 1icopodium — 0.97-0.55 yes horizontal 1 silt — 0.7 yes horizontal 4 Fe2°3 — 2.5 yes horizontal 4 Fe2°3 0.45-1.8 < 1.67 no vertical 2 MgO — 3 yes horizontal 4 A12°3 — 1.94-0.83 yes horizontal 1 Cr 20 3 — 1.94-0.83 yes horizontal 1 w2c 5-12 0.35-0.17 yes vertical 24 Si <150 < 1.67 no vertical 2 Si — 0.8 yes horizontal 4 Zn — < 1.67 no vertical 2 Zn -- 7 yes horizontal 4 w 2.5-5 0.35-0.17 yes vertical 24 Sn — 1 yes horizontal 4 Ni — 2.3 yes horiiontal 4 Si02 20-40 0.35-0.17 yes vertical 24 TABLE 1 EXPERIMENTAL RESULTS -ON PUSHING (Cont.) 8 Particle Size, vm Critical Pushing Growth Direction Ref. Velocity, um/sec Observed and Comments LIQUID: NAPHTALENE carbon 9.4 yes horizontal,Zonal & rotation 55 carbon ~ 11.1 yes vertical 55 carbon 0.5-2.5 3.3-5.8 yes vertical 7 carbon 0.5-2.5 8.3-10.8 yes vertical & stirring 2 Cu 15-20 yes horizontal.zonal S rotation 55 Cu -- 6.9 yes vertical 55 Cu 8.5-5.5 5 yes vertical 2 Ag . „ 1.6-7^2 yes ? vertical 55 Fe(Ag coated) several 0.49-0.20 yes vertical 29 iron oxide 10 yes horizontal .zonal & rotation 2 acetal 10-ZOO 64-40 yes horizontal.zonal rotation 7 acetal -- « 8 yes horizontal 5,6 nylon 10-240 40-10 yes horizontal.zonal & rotation 7 nylon — « 8 yes horizontal 5,6 polyestirene -- no horizontal.zonal & rotation 7 polyestirene ; — « 8 no horizontal 5,6 teflon no horizontal.zonal S rotation 7 teflon -- « 8 no horizontal 5,6 siliconed glass no horizontal.zonal rotation 7 Fe(Ag coated) 82-160 0.42-0.09 yes vertical & magnetic f ie ld 29 Fe and steel 92-148 0.56-0.053 yes vertical S (Ag coated) magnetic f ie ld . 29 aluminium 0.46-0.0078 yes vertical & magnetic f ie ld 29 LIQUID: THYMOL F e 2 ° 3 2 yes horizontal 4 . Sn — 4 yes horizontal 4 Agl — 6 yes horizontal 4 Zn — 6 yes horizontal 4 MgO — 8 yes horizontal 4 Ni -- 8 yes horizontal 4 C(Diamond) 3-5 9 yes horizontal 4 Si — 10 yes horizontal 4 C(graphite) — 12 yes horizontal 4 s i l t 16 yes horizontal 4 LIQUID: ORTHOTERPHENYL Agl _ _ 0.28 no horizontal 4 C(graphite) 0.28 yes horizontal 4 MgO 0.5 yes horizontal 4 s i l t 0.7 yes horizontal 4 Si 0.8 yes horizontal 4 Sn 1 yes _ horizontal 4 TABLE 1 EXPERIMENTAL RESULTS ON PUSHING (Cont.) 9 P a r t i c l e Size, um C r i t i c a l V e l o c i t y , urn/sec Pushing Observed Growth Direction and Comments Ref. C(diamond) 0-5 1.4-1.3 yes horizontal 4 Ni — 2.0 yes horizontal 4 F e2°3 — 2.5 yes horizontal 4 Zn — 2.5 yes horizontal 4 LIQUID: DYPHENYLAMINE Licopodium pdr. 22.2-13.8 yes horizontal 1 carbon -- 13.9-5.55 yes horizontal 1 A1 20 3 — 16.67-9.72 yes horizontal 1 C r2°3 — 13.9-5.55 yes horizontal 1 starch -- 0 no horizontal 1 LIQUID: AZOBENZENE Licopodium pdr. — 3.33-1.38 yes horizontal 1 carbon — 0.83-0.55 yes horizontal 1 A1 20 3 — 4.17-8.33 yes horizontal 1 C r 2 0 3 — 0.5-11.1 yes horizontal 1 Starch — 0.78-8.33 yes horizontal 1 LIQUID: BENZIL Licopodium pdr. -- 0.55-0 yes horizontal 1 carbon — 0.55-0.28 yes horizontal 1 A 12°3 — 19.44-11.11 yes horizontal 1 C r 2 0 3 — 2.78-1.39 yes horizontal 1 starch — 2.78-1.39 yes horizontal 1 LIQUID: BIPHENIL acetal 10-200 38-8 yes horizontal.zonal 7 & rota t i o n acetal — « 8 yes horizontal 5,6 nylon 10-240 32-6 yes horizontal.zonal 7 & r o t a t i o n nylon — <<8 yes horizontal 5,6 polyestirene — — no . horizontal.zonal 7 & r o t a t i o n polyestirene — « 8 no ? horizontal 5,6 tef1 on — — no horizontal.zonal 7 & r o t a t i o n t e f l o n — « 8 no horizontal 5,6 te f l o n covered with s i l i c o n e o i l no horizontal.zonal 7 & r o t a t i o n LIQUID: CAMPHOR C 0.5-2.5 <0.55 no v e r t i c a l 2 LIQUID: COPPER Sn 0 2 — — yes casting 1 ZnO — no casting 1 10 f o r c e a c t i n g on the h e a v i e r p a r t i c l e . S i n c e p u s h i n g has been o b s e r v e d w i t h v e r t i c a l s o l i d i f i c a t i o n and not f o r h o r i z o n t a l s o l i d i f i c a t i o n the e f f e c t o f g r a v i t y cannot be used to a c c o u n t f o r the d i f f e r e n c e i n t h e o b s e r v e d p a r t i c l e b e h a v i o r . The e x p e r i m e n t s o f Uhlmann and Kuo and W i l c o x con-s t i t u t e s v..examples where the d i r e c t i o n o f s o l i d i f i c a t i o n can a c c o u n t f o r the d i f f e r e n c e s i n the o b s e r v a t i o n s w i t h F e 2 0 3 » S i and Zn p a r t i c l e s i n s a l o l as shown i n T a b l e I. Uhlmann w i t h h o r i z o n t a l s o l i d i f i c a t i o n found t h a t t h e s e p a r t i c l e s a r e r e j e c t e d by s a l o l and Kuo and W i l c o x d i d not ob s e r v e p u s h i n g d u r i n g v e r t i c a l s o l i d i f i c a t i o n . R e t u r n i n g to the system c a r b o n i n s a l o l , p a r t i c l e p u s h i n g i s o b s e r v e d t o depend upon the growth v e l o c i t y and p a r t i c l e s i z e . I t w i l l be shown l a t e r t h a t t h e s e two v a r i a b l e s a re r e l a t e d : the v e l o c i t y a t which a p a r t i c l e (which i s i n i t i a l l y pushed by the i n t e r f a c e ) i s t r a p p e d by the i n t e r f a c e i s i n v e r s e l y p r o -p o r t i o n a l to the r a d i u s o f the p a r t i c l e to some power. T h i s s u g g e s t s t h a t the s i z e o f the p a r t i c l e s used by Uhlmann and Pikunov may have been l a r g e r than the c i r i t c a l r a d i u s f o r pu s h i n g a t t h e growth v e l o c i t i e s they used; however i t can be shown t h i s was not the c a s e . Comparing growth v e l o c i t y f i r s t , P ikunov and Uhlmann used v e l o c i t i e s o f I mm/hr, which i s lower than t h a t used by Kuo and Wi l c o x o f 7 mm/hr. The sl o w e r v e l o c i t i e s p e r m i t l a r g e r p a r t i c l e s to be pushed which f a v o u r s the Pikunov p a r t i c l e s f o r p u s h i n g . Pikunov 11 d i d not r e p o r t h i s p a r t i c l e s i z e d i s t r i b u t i o n . N e v e r t h e l e s s , the p a r t i c l e s i z e may be e s t i m a t e d assuming t h a t the same c a r -bon p a r t i c l e s were r e j e c t e d by an azobenzene i n t e r f a c e a t 3 growing r a t e s o f 16 mm/hr. A c c o r d i n g to t h e o r y p a r t i c l e s o f the o r d e r o f m i c r o n s a r e t r a p p e d a t t h e s e v e l o c i t i e s . T h e r e -f o r e i t may be assumed t h a t t h i s i s the o r d e r o f the p a r t i c l e s i z e employed by Piku n o v . Kuo and W i l c o x used c a r b o n p a r t i -c l e s w i t h s i z e s between 0.5 ym and 2.5 ym and found p u s h i n g o c c u r r e d a t 7 mm/hr f r e e z i n g r a t e s . Pikunov w i t h a p p r o x i -m a t e l y t h e same s i z e p a r t i c l e s and f r e e z i n g r a t e s of 1 mm/hr o b s e r v e d no p u s h i n g . T h e r e f o r e , the r e l a t i o n v e l o c i t y v e r s u s p a r t i c l e s i z e cannot be the rea s o n f o r the d i s c r e p a n c y be-tween the r e p o r t e d o b s e r v a t i o n s . A n o t h e r mechanism proposed to a c c o u n t f o r the d i f -f e r e n c e i s a s s o c i a t e d w i t h the r e j e c t i o n o f gases a t the moving f r o n t , as r e p o r t e d by Kuo and W i l c o x and o b s e r v e d by P i k u n o v . The r e j e c t e d gases were a b s o r b e d by carbon r e s u l t i n g i n the n u c l e a t i o n o f b u b b l e s . The bu b b l e s may h e l p the p u s h i n g p r o c e s s or even r e p l a c e i t by a f l o t a t i o n p r o c e s s . I f t h i s does o c c u r then i t may be c o n c l u d e d t h a t carbon i s not pushed by s a l o l . More examples i n which p a r t i c l e s are r e j e c t e d by an a d v a n c i n g i n t e r f a c e i n one s u b s t a n c e but not i n o t h e r can be c i t e d . Kuo and W i l c o x c o n c l u d e d t h a t carbon i s not 12 pushed by camphor but i t i s pushed by n a p h t a l e n e . Uhlmann showed A g l p a r t i c l e s a r e not moved by o r t h o t e r p h e n y 1 and s a l o l i n t e r f a c e s , but they a r e d i s p l a c e d by t h y m o l . One v e r y i n t e r e s t i n g o b s e r v a t i o n i s t h a t i n s a l o l g r a p h i t e i s not pushed by diamond (0-2y and 3 - 5 y) i s r e j e c t e d . T h i s means t h a t the c r y s t a l s t r u c t u r e o f the p a r t i c l e i s the c r i t i c a l component i n t h i s case d e t e r -m i n i n g whether the p a r t i c l e i s pushed. A n o t h e r example 2 i l l u s t r a t i n g t h a t a s t r u c t u r a l e f f e c t may be s i g n i f i c a n t i s the o b s e r v a t i o n i n t h a t d i f f e r e n t c r i t i c a l v e l o c i t i e s a p p l y to f a c e t e d and n o n - f a c e t e d i n t e r f a c e s o f the same m a t e r i a l . In summary, p a r t i c l e s may o r may not be pushed ahead o f an a d v a n c i n g s o l i d - l i q u i d i n t e r f a c e , depending on the p a r t i c l e , s o l i d and l i q u i d , p o s s i b l y gas e v o l u t i o n and p a r t i c l e and s o l i d s t r u c t u r e . I t i s t h e r e f o r e d e s i r -a b l e t o p r e d i c t when p u s h i n g w i l l o c c u r i n a g i v e n system on t h e b a s i s o f a t h e o r e t i c a l model. Models have been p r o p o s e d and a r e p r e s e n t e d i n Appendix I. Formulas 1 t o 28; f i g u r e s 2 to 10; and r e f e r e n c e s 3 to 22, 59 and 60 are i n t r o d u c e d t h e r e . 13 2. Comparison o f the Proposed T h e o r i e s w i t h E x p e r i m e n t a l  R e s u l t s A c o m p a r i s o n o f f o r m u l a 18(.a-c) w i t h the e x p e r i m e n t a l c r i t i c a l v e l o c i t i e s f o r a v a r i e t y o f p a r t i c l e s i n water 23 due t o C i s s e and B o i l i n g i s shown i n F i g . 11. In s a l o l the r e s u l t s were r e p o r t e d to be s i m i l a r to t h o s e i n F i g . 11' Chernov and Temlkin c o l l e c t e d the r e s u l t s a v a i l a b l e u n t i l 1976 and p l o t t e d a l l the d a t a as shown i n F i g . 12. In F i g . 12 l i n e s 1 and 2 c o r r e s p o n d to f o r m u l a s 27 and 28 r e s p e c t i v e l y . The s c a t t e r i s almost one o r d e r o f magnitude and may be a t t r i b u t e d to the d i f f e r e n t e x p e r i m e n t a l t e c h n i -ques. Curve (4) c o r r e s p o n d s to the c r i t i c a l v e l o c i t i e s when the r e p u l s i v e f o r c e i s o f e l e c t r o s t a t i c o r i g i n o n l y and r e m a r k a b l y i t i s s t r o n g enough to s o l e l y a c c o u n t f o r the r e j e c t i o n o f p a r t i c l e s . An a l t e r n a t i v e and l e g i t i m a t e method to prove t h e s e t h e o r i e s i s to luse a PLS system to d e t e r m i n e the dependence o f c r i t i c a l v e l o c i t y i n f u n c t i o n o f p a r t i c l e s i z e ; and to o b t a i n n. and C from and r e l a t i o n o f the type V = ...29 c R n n f o r the s e t o f t h e o r i e s p r e s c r i b e d here v a r i e s from 0.5 to 2 as shown i n T a b l e 1 1 ( a ) . Omenyi and Neumann f i t t e d e x p e r i m e n t a l c r i t i c a l v e l o c i t i e s f o r a v a r i e t y o f 14 Figure 11. C r i t i c a l growth rates vs. p a r t i c l e radius for Si^O and W p a r t i c l e s with bump radius of 5 fx m and 10 /Urn r e s p e c t i v e l y (a)_; and Cu p a r t i c l e s , c l e a n e d and nonn-cleaned,for two tmperature gradients.(from r e f . 23). 15 Figure 12. Comparison of c r i t i c a l v e l o c i t i e s given i n the Chernov theory with the experimental c r i t i c a l v e l o c i t i e s f o r water.Curves 1 and 2 correspond to small and large p a r t i c l e s . Curve 4 represents the c r i t i c a l v e l o c i t i e s when there i s e l e c t r o s t a t i c i n t e r a c t i o n only (from r e f . 10). 16 PLS s y s t e m s , to f o r m u l a s l i k e 29 and found t h a t n v a r i e s between 0.27 to 1.51 as shown i n T a b l e 11(b). In a d d i t i o n , i t can be seen t h a t i n a l l the c a s e s n was s m a l l e r i n the range o f p a r t i c l e r a d i u s from 10 to 100 ym than i n the range 100 t o 200 ym. T h i s s t r o n g e r dependence f o r l a r g e r p a r t i c l e s was a l s o e v i d e n t i n the C c o n s t a n t s which were al m o s t one o r d e r o f magnitude g r e a t e r . T h i s b e h a v i o u r i s o p p o s i t e to t h a t p r e d i c t e d by the Chernov e t a l . t h e o r y which f o r b i g g e r p a r t i c l e s g i v e s a s m a l l e r power, 1.0, than f o r s m a l l e r p a r t i c l e s , 1.33. On the o t h e r hand, when c o n s i d e r i n g c o n s t a n t C, the d i v e r -s i t y . o f v a l u e s f o r d i f f e r e n t p a r t i c l e s i n the same m a t r i x , shows the l i m i t a t i o n o f t h e s e t h e o r i e s t h a t g i v e o n l y one c o n s t a n t r e g a r d l e s s o f the n a t u r e o f the p a r t i c l e . In a d d i t i o n the d i f f e r e n c e i n C c o n s t a n t s f o r the d i f f e r e n t systems r e i n f o r c e s the s t a t e m e n t t h a t the p u s h i n g p r o c e s s depends on an i n t e r a c t i o n where the p r o p e r t i e s o f the t h r e e components P-L-S must be i n v o l v e d . I f o t h e r p a r a m e t e r s are a n a l y z e d from the s c a r c e e x p e r i m e n t a l work a v a i l a b l e the problem seems to be f a r from s o l v e d and no t h e o r y i s good enough to e x p l a i n the e x p e r i m e n t a l f i n d i n g s . C i s s e and B o i l i n g r e p o r t e d e x p e r i m e n t a l v a l u e s from TABLE l i r a VALUES QF n FROM EQUATION 29 FOR THE DIFFERENT THEORIES Theory P a r t S i z e n Uhlmann e t a l . smal 1 0. 5 Chernov e t a l . -^500 ym 1 . 0 Chernov e t a l . < 500 ym 1 . 33 B o i l i n g &;;Ci.sse ' s m a l l 1 . 5 Uhlmann et a l . smal 1 2 . -TABLE I I - b EXPERIMENTAL VALUES OF n and C FROM  EQUATION 29.7 P a r t i c l e S i z e 10-100 um 100-200 ym System n C n G; B i p h e n y l / a c e t a l 0.61 202.1 f . 5 ! 11887.0 Bi p h e n y l / n y l o n 0.54 107.8 1 .49 8210.5 N a p h t h a ! e n e / a c e t a l 0.27 124.7 0. 31 143.5 Naphtha!ene/ny1 on 0.42 99.8 0.62 232.5 19 w h i c h i t i s i n f e r r e d t h a t t h e c r i t i c a l v e l o c i t y i n c r e a s e s f o r l o w e r t e m p e r a t u r e g r a d i e n t s , w h e r e a s no t h e o r y p r e -d i c t s any r e l a t i o n w i t h t h i s p a r a m e t e r f o r s m a l l p a r t i c l e s . On t h e c o n t r a r y C h e r n o v e t a l . p r e d i c t s an o p p o s i t e b e h a v i o r f o r l a r g e p a r t i c l e s ( s e e E q u a t i o n 2 8 ) . So f a r i n t h e t h e o r i e s p r e s e n t e d t h e c r i t i c a l v e l o c i t i e s have been o b t a i n e d a s s u m i n g i d e a l c o n d i t i o n s e . g . s p h e r i c a l p a r t i c l e s , no s o l u t e i n t h e l i q u i d and same t h e r m a l c o n d u c t i v i t y f o r p a r t i c l e and s o l i d i f y i n g m a t e r i a l , e t c . T h e s e f a c t o r s w i l l now be c o n s i d e r e d i n r e l a t i o n t o t h e p a r t i c l e p u s h i n g p r o c e s s . a) E f f e c t o f t h e T h e r m a l C o n d u c t i v i t y I f t h e t h e r m a l c o n d u c t i v i t y o f a p a r t i c l e i s h i g h e r t h a n t h a t o f t h e l i q u i d and s o l i d , k-j * k g t h e c a p t u r e o f t h e p a r t i c l e i s e n h a n c e d . T h i s i s v i s u a l i z e d 3 t h r o u g h t h e f o l l o w i n g q u a l i t a t i v e l y r e a s o n i n g . I f kp* k-j k t h e i n t e r f a c e w i l l be d e f o r m e d by a n o n - u n i d i r e c t i o n a l s h e a t f l o w i n w h i c h h e a t i s p r e f e r e n t i a l l y c o n d u c t e d t h r o u g h t h e p a r t i c l e . T h i s w i l l p r o d u c e a d e p r e s s i o n b e h i n d t h e p a r t i c l e as i t i s shown i n F i g . 13 t r a p p i n g t h e p a r t i c l e i n i t . The o p p o s i t e c a s e , when kp< k-j , k g w i l l e n h a n c e p u s h i n g by c r e a t i n g a bump b e h i n d t h e p a r t i c l e ( F i g . 1 3 ) . Figure 13. E f f e c t of the thermal conductivity on the pushing process, a) k < , k ; b) k ~ ,k and c) k > ^ ,k g . In a) and.c) ?he isotherms a r e P d i s t o r t e l , heat transfer i s i n h i b i t e d and enhanced,respectively and so the capture of the p a r t i c l e . V c = 2. ( l - 2 x ) " 3 / 4 (1+X) ...30 21 The f a c t t h a t t h e s e e f f e c t s s h o u l d be s t r o n g e r f o r h i g h e r heat f l o w s may e x p l a i n the o b s e r v e d dependence o f 23 p u s h i n g on t e m p e r a t u r e g r a d i e n t by Cisse" and B o i l i n g . They used Cu p a r t i c l e s which have a much h i g h e r thermal c o n d u c t i v i t y than water and t h e r e f o r e under a s t e e p g r a d i e n t the c r i t i c a l v e l o c i t y w i l l be l o w e r . N e v e r t h e -l e s s , t h i s i s not i n agreement w i t h the o n l y f o r m a l t r e a t -ment a v a i l a b l e ^ i n which the thermal e f f e c t s are i n c l u d e d to g i v e the f o l l o w i n g e q u a t i o n f o r the c r i t i c a l v e l o c i t y ^3 B 3 GAS 1 / 4 24nR B 3AS where x =::(k1 - k ) / ( 2 k 1 + k ). I t i s e v i d e n t t h a t the above f o r m u l a c o n t r a d i c t s the former a n a l y s i s . Moreover f o r x 1 > i - e . f o r kp>>k-j as i n the case o f Cu or any o t h e r m e t a l l i c p a r t i c l e i n w a t e r , the c r i t i c a l v e l o c i t y tends t o z e r o . T h i s i s not shown i n the C i s s e and B o i l i n g d a t a where the c r i t i c a l v e l o c i t i e s f o r Cu and W a r e o f t h e same o r d e r as t h o s e f o r Si 0^ p a r t i c l e s d e s p i t e the v e r y d i f f e r e n t t hermal c o n d u c t i v i t y . On the o t h e r hand, t h e s e f i n d i n g s : b r i n g i n t o doubt the 25 e x p e r i m e n t a l o b s e r v a t i o n s o f Zubko and co-workers who employed m e t a l l i c p a r t i c l e s (W, Ta, Mo, Fe, N i , C r ) , 2-3 mm i n d i a m e t e r and m e t a l l i c m a t r i c e s (Zn, Bi and Sn) t h a t 22 i s w i t h s i m i l a r v a l u e s o f k p , and k s > They v e r i f i e d t h a t f o r kp/k-j > 1 t h e r e i s c a p t u r e and f o r k^/k^< 1 the p a r t i c l e s are pushed. Two p;e:c:u;T.i.ar f a c t s must be n o t e d ; f i r s t , t h e growth v e l o c i t i e s a t which the o b s e r v a t i o n s were made v a r i e d i n the range 5-100 cm/hr w h i l e the t h e o r i e s p r e d i c t c r i t i c a l v e l o c i t i e s o f the o r d e r o f _ 3 10 cm/hr f o r t h e s e p a r t i c l e s . Second, f o r the system Cr ( p a r t i c l e s ) - Sn ( m a t r i x ) the r e l a t i o n k p/k-j was assumed to be l e s s than 1 (0.07/0.157) and the e x p e r i -ments gave r e p u l s i o n w h i l e u s i n g ASM t a b l e s the c a l c u l a t e d v a l u e o f kp/k-j i s 76/32.6 >1 and a c c o r d i n g l y t h i s s h o u l d have l e d to t r a p p i n g . b) The E f f e c t o f I m p u r i t i e s The i n t e r a c t i o n i n the PLS system changes d r a s t i -c a l l y when the 1 i q u i d c o n t a i n i m p u r i t i e s . Depending on the v a l u e o f the p a r t i t i o n c o e f f i c i e n t t h e r e w i l l be a c c u m u l a t i o n or d e p l e t i o n o f s o l u t e i n f r o n t o f the advanc-i n g i n t e r f a c e . The p r e s e n c e o f a f o r e i g n body i n f r o n t o f the i n t e r f a c e w i l l o b s t r u c t t h e d i f f u s i o n o f s o l u t e away from the i n t e r f a c e i n c r e a s i n g the c o n c e n t r a t i o n i n the i n t e r m e d i a t e l i q u i d l a y e r . As a r e s u l t a d e p r e s s i o n a t the i n t e r f a c e w i l l appear f a v o r i n g the c a p t u r e o f a p a r t i c l e . Tembin e t a l . i n t r o d u c e d t h i s f a c t o r i n the t h e o r y 23 o f Chernov e t a l . and a r r i v e d a t the f o l l o w i n g e q u a t i o n f o r the c r i t i c a l v e l o c i t y V a, DftK 1 s ~T " R ASmC. I n 30-|sn DfiK B 0AS m C -, - l r 1 I n l n A In A -,-1 .31 where A D m K 3t&. n^DK/B-AS m C,„ d i f f u s i o n c o e f f i c i e n t l i q u i d u s s l o p e p a r t i t i o n c o e f f i c i e n t c o n c e n t r a t i o n o f i m p u r i t y i n the b u l k l i q u i d It can be seen i n t h i s e x p r e s s i o n t h a t the c r i t i c a l v e l o c i t y has a s t r o n g e r dependence w i t h p a r t i c l e r a d i u s . Whose exponent p a s s e s from -4/3 to -2. T h i s t r a n s i t i o n o c c u r s a t a v a l u e o f R ~ 2 . 5 x l 0 ~ 4 cm f o r m Cro/ K.: = 1 deg. at c o n s t a n t Cc , , and t h i s R d e c r e a s e s as m 0^/^:1 i n c r e a s e s . CO -J r ; i • On the o t h e r hand f o r a g i v e n p a r t i c l e r a d i u s , f o r i n s t a n c e R = 1 ym t h i s change o f b e h a v i o r happens a t a c o n c e n t r a -t i o n o f C > 3x10" 2 wt % f o r m = 3 deg/wt % and K = 10 . At t h e s e p o i n t s the c a p t u r e p r o c e s s i s c o n t r o l l e d by the d i f f u s i o n o f s o l u t e to the bul k l i q u i d r a t h e r than the t r a n s p o r t o f l i q u i d t o the s o l i d i f i c a t i o n f r o n t or the d i s j o i n i n g p r e s s u r e , whose parameter B^ appears i n a weak T!o.g;a<rn"ttunic r e l a t i o n . 24 I t i s n e c e s s a r y to mention t h a t t h e r e a r e no e x p e r i -mental r e s u l t s r e p o r t e d on the e f f e c t o f i m p u r i t y on p a r t i c l e t r a p p i n g . c) O t h e r E f f e c t s i ) V i s c o s i t y o f the m e l t . A l l the t h e o r i e s p r e d i c t lower c r i t i c a l v e l o c i t i e s f o r l i q u i d s w i t h h i g h e r 24 v i s c o s i t i e s . C i s s e and B o i l i n g found t h i s q u a l i t a t i v e dependence comparing the c r i t i c a l r a t e s f o r s a l o l and water. i i ) Body f o r c e s . I t i s o b v i o u s t h a t the d i f f e r -ence i n d e n s i t y between the p a r t i c l e and the l i q u i d w i l l enhance t r a p p i n g depending on whether the r e l a t i o n dp/d-j i s e i t h e r g r e a t e r or l e s s than 1. The d e n s i t y d i f f e r e n c e w i l l c o n t r i b u t e to s i n k i n g o r f l o a t i n g o f the p a r t i c l e . T h i s e f f e c t w i l l be g r e a t e r f o r l a r g e p a r t i c l e s as con-- 3 s i d e r e d by B o i l i n g and C i s s e . The r o l e o f the body f o r c e i s not c l e a r when i t i s 29 produced by a m a g n e t i c f i e l d . Cheu and W i l c o x u s i n g ^ f e r r o m a g n e t i c p a r t i c l e s and inaphithalene: as m a t r i x found t h a t t h i s body f o r c e iinh'ibiit's: c a p t u r e but a t the same time when the p a r t i c l e s were used s e v e r a l times the c r i t i c a l v e l o c i t y i n c r e a s e d w i t h the run number s u g g e s t i n g t h a t a,, h y s t e r e s i s e f f e c t may o b s c u r e t h e i r o b s e r v a t i o n s . 25 i i i ) C o n v e c t i o n . I t has not been e s t a b l i s h e d i n what way movement of the melt may a f f e c t the c a p t u r e 3 28 p r o c e s s . I t i s b e l i e v e d ' c o n v e c t i o n s h o u l d i n h i b i t t r a p p i n g o f b i g p a r t i c l e s which may be washed o f f from the i n t e r f a c e ; a t l e a s t f o r p a r t i c l e s l a r g e r than the -3 -1 l i m i t l a y e r (10 -10 cm). -3. P u s h i n g i n M e t a l s O b s e r v a t i o n s r e l a t e d to p a r t i c l e p u s h i n g i n an a l l metal system a r e r e l a t i v e l y r a r e . There i s no c o n c l u s i v e e v i d e n c e i n the l i t e r a t u r e o f p a r t i c l e p u s h i n g a t a m e t a l l i c s o l i d - l i q u i d i n t e r f a c e . P i k u n o v 1 i n t r y i n g t o f i n d the o r i g i n o f the e q u i a x e d zone i n i n g o t s performed some e x p e r i m e n t s i n c o p p e r -oxygen a l l o y s i n which " r a d i o a c t i v e Zn^^ or S n 1 1 3 were i n t r o d u c e d as d e o x i d i z e r . A f t e r s o l i d i f i c a t i o n was com-p l e t e samples were a n a l y z e d . A u t o r a d i o g r a p h s o f the sectioned samples::: d i d not show any p r e f e r e n t i a l m a c r o s c o p i c d i s t r i -113 65 113 b u t i o n o f Sn or Zn.*..- However, Sn; was found to be s e g r e g a t e d i n s t r i p s a l o n g the g r a i n b o u n d a r i e s and a l s o 65 i n the i n t e r d e n d r i t i c l i q u i d , whereas Zn was found to be homogeneously d i s t r i b u t e d . T h i s l e d to the c o n c l u s i o n 113 65 t h a t Sn 0 2 i s pushed w h i l e Zn 0 i s n o t . The same 11 3 o b s e r v a t i o n on the Sn 0 p a r t i c l e s , as w i l l be shown i n the p r e s e n t i n v e s t i g a t i o n may be due to p h y s i c a l , v entrapment and not p u s h i n g . 26 On the o t h e r hand, i n d e o x i d i z e d i n g o t s or ESR 33-36 i n g o t s ~ i n c l u s i o n s were o b s e r v e d to be p r e f e r e n t i a l l y s i t u a t e d i n i n t e r d e n d r i t i c p o s i t i o n s . In the l a t t e r r e -p o r t s the a u t h o r s a s s o c i a t e d t.heise o b s e r v a t i o n s to a homogeneous n u c l e a t i o n from the i n t e r d e n d r i t i c e u t e c t i c . With a d i s t i n c t view F r a n k l i n and Evans assumed the dend-r i t e s sweep o f f the i n c l u s i o n s , i - ' M y e r s and F l e m i n g a t t r i b u t e the s e g r e g a t i o n o f the i n c l u s i o n between d e n d r i t e s to a p u s h i n g mechanism. „ 3 B o i l i n g and C i s s e r e p o r t e d t h a t some Cu p a r t i c l e s _ 5 10 cm i n r a d i u s were found to be c l u s t e r e d i n Pb i n which Cu i s s o l u b l e . F i n a l l y , the o n l y r e l a t e d work where p a r t i c l e s were.; i n t r o d u c e d i n m e t a l l i c m e l t s to s t u d y t h e i r r e j e c t i o n i s 25 t h a t o f Zub&o e t a l . which has been d i s c u s s e d i n the p r e v i o u s s e c t i o n 4. Summary o f C h a p t e r II The r e j e c t i o n o f p a r t i c l e s by an a d v a n c i n g s o l i d -l i q u i d i n t e r f a c e has been e x t e n s i v e l y s t u d i e d both t h e o -r e t i c a l l y ; ' ! and e x p e r i m e n t a l l y . At the p r e s e n t time t h e r e 27 i s no mechanism a v a i l a b l e to p r e d i c t when r e j e c t i o n o f p a r t i c l e s w i l l o c c u r by an a d v a n c i n g i n t e r f a c e . N e v e r t h e -l e s s , the c a l c u l a t i o n o f o>, the L i f s h i t z - Van der Waals', c o n s t a n t , may g i v e a l e g i t i m a t e way to e s t a b l i s h whether pushing, w i l l o c c u r under s p e c i f i e d c o n d i t i ons. (see Appendix I ) . When p u s h i n g o c c u r s s e v e r a l t h e o r i e s have been d e v e l o p -ed to p r e d i c t the c r i t i c a l v e l o c i t y a t which the p a r t i c l e s a r e c a p t u r e d by t h e i n t e r f a c e . However, t h e r e i s no a g r e e -ment among the t h e o r i e s i n the dependence o f the c r i t i c a l v e l o c i t y on the r a d i u s . o f , t h e p a r t i c l e . For t h i s c r i t i c a l v e l o c i t y r e l a t i o n s o f the form V « R~n have been o b t a i n e d w i t h n r a n g i n g from 0.5-2. E x p e r i m e n t a l v e r i f i c a t i o n o f t h e s e t t h e o r i e s i s c o m p l i c a t e d due to the p r e s e n c e o f o t h e r v a r i a b l e s i n the system such as thermal c o n d u c t i v i t y , shape o f the p a r t i c l e s , i m p u r i t i e s and c o n v e c t i o n i n the m e l t which a r e i n a r e a l system. These f a c t o r s can a f f e c t the c a p t u r e p r o c e s s . Some o f t h e s e n o n - i d e a l s i t u a t i o n s have been s t u d i e d but have not been v e r i f i e d e x p e r i m e n t a l l y . Most o f the work has been done i n n o n - m e t a l l i c systems where the e l e c t r o s t a t i c f o r c e may be i m p o r t a n t . The e l e c t r o s t a t i c f o r c e , which i s not p r e s e n t i n m e t a l s , may i n i t s e l f be the overwhelming f a c t o r i n d e t e r m i n i n g whether p a r t i c l e p u s h i n g may o c c u r i n non m e t a l l i c systems. 28 C h a p t e r I I I OBJECTIVES OF PRESENT RESEARCH The o b j e c t i v e o f t h i s programme i s to e s t a b l i s h whether s o l i d p a r t i c l e s i n an a l l metal system a r e r e j e c t e d by an a d v a n c i n g s o l i d - l i q u i d i n t e r f a c e and i f so under what con-d i t i o n s r e j e c t i o n o c c u r s . The p r o c e d u r e to be f o l l o w e d i s l i s t e d below. 1) A s u i t a b l e p a r t i c l e , l i q u i d , s o l i d system w i l l be s e l e c t e d and t h e d i s t r i b u t i o n o f p a r t i c l e s ahead o f an a d v a n c i n g s o l i d - l i q u i d i n t e r f a c e d e t e r m i n e d . I n t e r f a c e s w i t h p l a n a r , c e l l u l a r and d e n d r i t i c m o r p h o l o g i e s w i l l be c o n s i d e r e d . S o l i d i f i c a t i o n w i l l be both u n i d i r e c t i o n a l and n o n d i r e c t i o n a l . 2) The r e s u l t s from (1) w i l l be c o n s i d e r e d i n terms o f a p h y s i c a l model o f p a r t i c l e i n t e r a c t i o n s w i t h an a d v a n c i n g i n t e r f a c e . 3) A water and n y l o n s p h e r e model s i m u l a t i n g the a l l metal system w i l l be examined to d e t e r m i n e by d i r e c t o b s e r v a -t i o n how p a r t i c l e s move a g a i n s t a s o l i d i n t e r f a c e . 4) The o b s e r v a t i o n s o f p a r t i c l e movement i n the a l l 29 metal system, w i l l be compared to the o b s e r v a t i o n s i n the water and n y l o n sphere model and to the p r e d i c t i o n s o f the t h e o r i e s p r e s e n t e d i n the p r e v i o u s c h a p t e r . 30 C h a p t e r IV EXPERIMENTAL APPARATUS AND PROCEDURE 1. The M e t a l l i c PLS System a) S e l e c t i o n and P r e p a r a t i o n o f Samples In an i d e a l PLS system, the p a r t i c l e s s h o u l d have the same d e n s i t y and t h e r m a l c o n d u c t i v i t y as the m e l t . In a d d i t i o n the p a r t i c l e s s h o u l d be s p h e r i c a l and must not d i s s o l v e i n the melt but s h o u l d be wetted by the m e l t . A number o f systems were i n v e s t i g a t e d u s i n g e i t h e r Sn, Pb o r Bi as the l i q u i d m a t r i x and C r , W, SnO, Fe o r Cu as p a r t i -c l e s . The major problem e n c o u n t e r e d was t h a t i f a p a r t i c l e had l i m i t e d s o l u b i l i t y i n the m e l t , or was o x i d i z e d , i t was not p o s s i b l e to i n t r o d u c e and keep the p a r t i c l e s i n the m e l t d u r i n g s o l i d i f i c a t i o n . To overcome o x i d a t i o n e f f e c t s s i l i c o n o i l , Zn C12 a n d C1 2 were used as f l u x e s on the melt s u r f a c e s when the p a r t i c l e s were added to the m e l t . T h i s was i n i t i a l l y t h o u g h t to be s u c c e s s f u l i n i n t r o d u c i n g p a r t i c l e s i n the m e l t , but s u b s e q u e n t l y i t was found t h a t the p a r t i c l e s , even w i t h the f l u x , were not i n the m e l t . One s y s t e m , Fe p a r t i c l e s i n a Pb m a t r i x was found to be r e a s o n a b l y s a t i s f a c t o r y , i n t h a t the p a r t i c l e s were wet t e d by the Pb but were not d i s s o l v e d , and c o u l d be i n t r o d u c e d 31 in the melt. However the density and thermal conduct iv i ty of the Fe pa r t i c l es d i f f e red s i g n i f i c a n t l y from the Pb matrix. In spi te of these di f ferences the Fe pa r t i c l e Pb matrix system was adopted as the best a l t e rna t i ve . Master samples of lead of 99.99% and 99.999% puri ty were cast containing a high density of ARMCO iron spherical p a r t i c l e s . The 99.99% pur i ty lead was used with a l loys and the higher purity lead when pure lead matrix was used. The ARMCO powder was screened and the . r-400: : mesh powder was introduced in the lead. F ig . 14 shows a SEM micrograph of the spherical p a r t i c l e s . Lead was melted in a graphite mold 3.5 cm in diameter and 6.5 cm height, ZnCl^ added to the melt surface, and then the Fe p a r t i c l e s . The f lux deoxidized the Fe resu l t ing in the pa r t i c l es being wetted by the lead. The pa r t i c l e s were introduced into the melt by vigorous s t i r r i n g of the melt. The amount of par t i c l es that entered into the lead was found to depend on the total number of pa r t i c l es and the degree of s t i r r i n g of the melt. Lead antimony a l loys were prepared in graphite con-tainers s imi la r to pure lead using a part of the master F i g u r e 14. SEM micrograph of the ARMCO i r o n p a r t i c l e s ,-400 mesh, used i n the experiments. 200X. 33 sample and a d d i n g pure l e a d and pure antimony (99.99%) to o b t a i n the d e s i r e d c o n c e n t r a t i o n o f antimony and p a r t i c l e s . The m e l t was then c a s t i n t o a 11 mm i n s i d e d i a m e t e r V y c o r t u b e , 20 cm l o n g , p a i n t e d w i t h g r a p h i t e . The m e l t was r a p i d l y s o l i d i f i e d to a v o i d f l o t a t i o n o f the Fe p a r t i c l e s and o b t a i n homogeneous a x i a l d e n s i t y o f p a r t i c l e s . The p a r t i c l e d i s t r i b u t i o n was checked by c u t t i n g the c a s t samples and o b s e r v i n g the p a r t i c l e d i s t r i b u t i o n w i t h an o p t i c a l m i c r o s c o p e i n the p o l i s h e d s u r f a c e s e c t i o n . Samples were a l s o a n a l y s e d q u a n t i t a t i v e l y u s i n g x - r a y a n a l y s i s i n the SEM. No t r a c e ; o f Fe was found i n the m a t r i x even a f t e r r e m e l t i n g the same sample. b) The A p p a r a t u s and T e c h n i q u e s f o r C o n t r o l l e d  S o l i d i f i c a t i on i ) S o l i d i f i c a t i o n . To examine the i n t e r a c t i o n o f an a d v a n c i n g i n t e r f a c e w i t h p a r t i c l e s i n the m e l t i t i s n e c e s s a r y t h a t the p a r t i c l e s remain u n i f o r m l y d i s t r i b u t e d i n the m e l t d u r i n g s o l i d i f i c a t i o n . S i n c e the d e n s i t y o f Fe (7.87 gr cm ) i s a p p r e c i a b l y l e s s than l i q u i d l e a d 3 (10.6 gr cm ) the p a r t i c l e s w i l l f l o a t to the top o f the melt d u r i n g s o l i d i f i c a t i o n which i s u n s a t i s f a c t o r y . To overcome t h i s d i f f i c u l t y a zone r e f i n i n g s o l i d i f i c a t i o n p r o c e s s was adopted i n which the l i q u i d zone was p r o g r e s -s i v e l y moved a l o n g the r a p i d l y s o l i d i f i e d rod c o n t a i n i n g the To g r oph re c o rd er TC Chrom eI-Alumel OJ or V solid Sample CO - r otation 0 V A C rMullit e tube - I-5 5 cm I D Vycor tube - Mem ID -liquid . i i i i i • i i i i i i i i I T T Furnace V = hori zo ntg mot ion • • £ f • t I f I M t f l « * f i f « k W W W W W l heating coils-Chromel 24 '////////////. in su Io t io n - Fiberflax -shield V = upward -» f i xed motion cm S C A L E h i Figure 15. Diagram of the apparatus used for the s o l i d i f i c a t i o n of al l o y s and pure metals.(For explanation see t e x t ) . co 35 p a r t i c l e s . Two s o l i d i f i c a t i o n modes were used, v e r t i c a l l y downward and h o r i z o n t a l w i t h r o t a t i o n u s i n g the system shown s c h e m a t i c a l l y i n F i g . 15. With t h e s e t e c h n i q u e s the problems due to buoyancy f o r c e are overcome i n the f o l l o w -i n g ways. In the v e r t i c a l mode the f u r n a c e was kept f i x e d and the sample moved upwards, t h a t i s , the s o l i d i f i c a t i o n f r o n t advanced downwards. As the zone moved, m a t e r i a l m e l t e d a t the bottom r e l e a s e d p a r t i c l e s and e n s u r e d the c o n t i n u o u s p r e s e n c e o f p a r t i c l e s a t the s o l i d - l i q u i d i n t e r -f a c e . In the h o r i z o n t a l mode the sample was rotated: on --its axis a t c o n s t a n t a n g u l a r v e l o c i t y and the f u r n a c e was/ moved hori-z o n t a l l y a t c o n s t a n t speed. The'-minimum a n g u l a r v e l o c i t y employed was e s t a b l i s h e d by the time t a k e n f o r the b i g g e s t p a r t i c l e to t r a v e l a d i s t a n c e equal to the d i a m e t e r o f the sample. To c a l c u l a t e t h i s d i s t a n c e the c o r r e c t e d S t o k es 37 v e l o c i t i e s and t r a n s i e n t times were used. The f u r n a c e c o n s i s t e d o f a m u l l i t e tube 1.55 cm I.D. and 3.5 cm l o n g w i t h Chromel 24 w i n d i n g cemented and i n -s u l a t e d w i t h f i b e r f l a x . A t h e r m o c o u p l e was p l a c e d i n the c e n t e r o f t h e i n n e r p a r t i n o r d e r to c o n t r o l ^ t h e tempera-t u r e . To do t h i s , the c u r r e n t t h r o u g h the h e a t i n g c o i l was c a l i b r a t e d so as to o b t a i n an adequate zone which was o f the o r d e r o f 2.5 cm l o n g . Once t h i s zone was formed the t e m p e r a t u r e was m a i n t a i n e d c o n s t a n t by m a n u a l l y a d j u s t i n g 36 the c u r r e n t to a v a l u e t h a t compensated f o r .the heat l o s s e s . The maximum v a r i a t i o n o f t e m p e r a t u r e i n the f u r n a c e was not g r e a t e r than 0.5%. In t h e v e r t i c a l mode the t e m p e r a t u r e of the sample was r e c o r d e d by a t h e r m o c o u p l e p o s i t i o n e d i n the c e n t r e o f the r o d . The measured t e m p e r a t u r e s showed t h a t the t e m p e r a t u r e v a r i a t i o n s a s s o c i a t e d w i t h c o n v e c t i o n i n the m e l t were much g r e a t e r than t h o s e a s s o c i a t e d w i t h changes i n the t e m p e r a t u r e o f the f u r n a c e . The t e m p e r a t u r e g r a d i e n t i n the melt was d e t e r m i n e d to be 30°C c m ~ \ and remained c o n s t a n t d u r i n g s o l i d i f i c a t i o n . Samples i n both s o l i d i f i c a t i o n modes were grown as: ! few c e n t i m e t e r s and the l i q u i d was quenched. T h i s growth was sometimes r e p e a t e d a l o n g t h e sample i n c r e a s i n g the number of i n t e r f a c e s and t h e r e f o r e the number of o b s e r v a t i o n s f o r a g i v e n s o l i d i f i c a t i o n c o n d i t i o n To change the shape o f the i n t e r f a c e from c e l l u l a r t o d e n d r i t i c the c o n c e n t r a t i o n o f antimony was i n c r e a s e d r a t h e r than c h a n g i n g the t e m p e r a t u r e g r a d i e n t o r growth v e l o c i t y . The l a t t e r were m a i n t a i n e d c o n s t a n t i n o r d e r t o m i n i m i z e changes i n c o n v e c t i o n f l o w p a t t e r n s i n the m e l t . i i ) Metal 1 o g r a p h i c P r e p a r a t i o n . The samples were c u t l o n g i t u d i n a l l y and h o r i z o n t a l l y w i t h r e s p e c t to 37 the growth d i r e c t i o n both a t the i n t e r f a c e and be h i n d i t a t v a r i o u s l e v e l s . Sample p r e p a r a t i o n c o n s i s t e d o f s t a n d a r d p o l i s h i n g w i t h paper and alu m i n a up to 0.05 urn. When n e c e s -s a r y diamond p a s t e 1 ym was used. The samples were e t c h e d to r e v e a l the m i c r o s t r u c t u r e o f s e g r e g a t i o n . Two e t c h a n t s were used, one w i t h 3 p a r t s o f a c e t i c a c i d and 1 p a r t o f h i d r o g e n p e r o x i d e , and the second e t c h a n t c o n t a i n i n g m o l y b d i c a c i d , 100 g r ; NH^OH, 140 ml; H 20, 240 ml and HN0 3, 60 ml. i i i ) C o u n t i n g and P a r t i c l e S i z e D i s t r i b u t i o n . The d i s t r i b u t i o n o f t h e p a r t i c l e s was determined, by c o u n t i n g the p a r t i c l e s w i t h the o p t i c a l m i c r o s c o p e a t 500X m a g n i f i c a t i o n s c a n n i n g t h e e n t i r e p o l i s h e d sample a r e a . The a p p a r e n t s i z e o f t h e p a r t i c l e s was measured by s u p e r i m p o s i n g a t r a n s -p a r e n t g r i d , d i v i d e d i!mm;i 11 i m e t e r s , onto the p r o j e c t i o n s c r e e n o f the m i c r o s c o p e . At t h i s m a g n i f i c a t i o n t h e b i g g e s t p a r t i c l e (~35 ym) had a s i z e o f 1.75 cm and the s m a l l e s t 0.1 cm. For c o u n t i n g p u r p o s e s the p a r t i c l e s i z e s were grouped i n mm i n t e r v a l s i n 15 p a r t i c l e c l a s s e s . These c l a s s e s were l a t e r r e g r o u p e d f o r a n a l y s i s o f the d a t a . When samples c o n t a i n e d a low p a r t i c l e d e n s i t y on the p o l i s h e d s u r f a c e , i n s u f f i c i e n t to g i v e a s t a t i s t i c a l l y r e p r e -s e n t a t i v e d i s t r i b u t i o n , they were r e s e c t i o n e d and ; p o l i s h e d and co u n t e d a g a i n . 38 The r e a l p a r t i c l e d i s t r i b u t i o n was c a l c u l a t e d u s i n g the 3 8 S c h w a r z - S a l t t k o v method which e n a b l e s the number o f p a r t i -c l e s o f a g i v e n r e a l s i z e to be d e t e r m i n e d u s i n g the ap-p a r e n t d i s t r i b u t i o n o n l y ; t h a t i s , the c a l c u l a t i o n o f the r e a l number o f p a r t i c l e s o f a g i v e n s i z e does not r e q u i r e the r e a l number o f p a r t i c l e s o f o t h e r s i z e s . c) P r o c e d u r e f o r the C a s t i n g E x p eriment A 50% l e a d - 50% t i n m e l t w i t h i r o n p a r t i c l e s was p r e p a r e d i n a manner s i m i l a r to t h a t used i n the c o n t r o l l e d s o l i d i f i c a t i o n e x p e r i m e n t s . The melt was then c a s t i n t o a g r a p h i t e mould. A f t e r 2 min the a l l o y was c o m p l e t e l y s o l i d i f i e d and the s t r u c t u r e r e v e a l e d f u l l y d e v e l o p e d den-d r i t e s . The m e t a l l o g r a p h y was s i m p l e r because i t was not n e c e s s a r y t o e t c h the sample. C o u n t i n g was p e r formed a t s e v e r a l h e i g h t s i n the sample u s i n g the method d e s c r i b e d p r e v i o u s l y . 2. The Water Model A water model was c o n s t r u c t e d c o n s i s t i n g o f a l u c i t e c y l i n d e r 17 cm l o n g and 9.5 cm I.D. c l o s e d a t both ends. At one end a c e l l u l a r i n t e r f a c e was s i m u l a t e d w i t h hexagons w i t h rounded t i p s p l a c e d 4 mm a p a r t as shown i n F i g . , 1 6 . Nylon p a r t i c l e s 0.375 cm i n d i a m e t e r were used, s e l e c t i n g the d i a m e t e r to c o r r e s p o n d to the a v e r a g e p a r t i c l e s i z e 39 I c m 40 Figure 16. The p h y s i c a l model to study the motion of nylon spheres at a c e l l u l a r i n t e r f a c e , a) The l u c i t e c y l i n d r i c a l container with the brine solution,the spheres and the"interface" ; 06 i s the t i l t i n g angle.b) The c e l l t i p s and the spheres i n d e t a i l . c ) A sphere with v e l o c i t y V due to buoyancy force c o l l i d i n g with a c e l l t i p ; v i s the re s u l t a n t v e l o c i t y immediately a f t e r c o l l i s i o n ; t h e downward vector V i s the v e l o c i t y a f t e r h a l f revolution.The v e l o c i t i e s are with respect to a system of reference f i x e d to the c e l l s . 41 i n t h e r e a l Fe-Pb s y s t e m (~18 ym) when c o m p a r i n g w i t h t h e c e l l s i z e (~100 ym). A b r i n e s o l u t i o n was u s e d t o s i m u l a t e t h e m e l t . By a d j u s t i n g t h e d e n s i t y of; theesojuti.o.n by c h a n g i n g t h e s a l t c o n c e n t r a t i o n » a r a n g e o f p a r t i c l e v e l o c i t y was o b t a i n e d . To s i m u l a t e h o r i z o n t a l s o l i d i f i c a t i o n w i t h r o t a t i o n t h e c y l i n d e r was t i l t e d f r o m t h e h o r i z o n t a l i n - o r d e r t o g i v e t h e s p h e r e s a component o f v e l o c i t y p a r a l l e l t o t h e a x i s s i m u l a t i n g an a d v a n c i n g i n t e r f a c e . The h o r i z o n t a l v e l o c i t y was d e t e r m i n e d by m e a s u r i n g t h e t i m e t a k e n by t h e p a r t i c l e s t o move between two marks on t h e t u b e . The t i l t i n g a n g l e was c a l c u l a t e d f r o m t h e h e i g h t and l e n g t h o f t h e f r a m e s u p p o r t i n g t h e c y l i n d e r . The a b s o l u t e v e l o c i t y w i t h r e s p e c t t o t h e s t i l l l i q u i d was m e a s u r e d on s e v e r a l p a r t i c l e s w h i c h were r e l e a s e d i n t h e l i q u i d by means o f a j - t u b e i n t r o d u c e d i n t h e b o t t o m o f t h e s o l u t i o n . The m o t i o n and d i s t r i b u t i o n o f p a r t i c l e s were o b s e r v e d f o r a r a n g e o f f r e q u e n c i e s o f r o t a t i o n o f t h e t u b e and t i l t i n g a n g l e s . In t h e v e r t i c a l " s o l i d i f i c a t i o n " mode t h e d e n s i t y o f t h e s o l u t i o n was v a r i e d and t h e d i s t r i b u t i o n o f p a r t i c l e s as a f u n c t i o n o f p a r t i c l e v e l o c i t y was e x a m i n e d . Two t e c h n i q u e s were u s e d t o r e l e a s e t h e p a r t i c l e s . In one t h e c y l i n d e r , w i t h a l l t h e p a r t i c l e s a t t h e t o p end o p p o s i t e 42 the c e l l u l a r i n t e r f a c e , was r a p i d l y r o t a t e d t h r o u g h 180 deg r e e s about a h o r i z o n t a l a x i s . In the second p r o c e d u r e the p a r t i c l e s were r e l e a s e d i n the s t i l l l i q u i d , one a t a t i m e , w i t h the j - s h a p e d t u b e . 3. P a r t i c l e I n t e r a c t i o n w i t h a F r e e z i n g Water I n t e r f a c e A pyrex tube 10 mm i n d i a m e t e r and 25 cm l o n g w i t h one c l o s e d t e n d was f i l l e d w i t h d i s t i l l e d w ater. Cr p a r t i c l e s were added to t h e w a t e r , mixed w e l l and the tube then p l a c e d i n a f r e e z i n g bath a t -10°C. The d i s t r i b u t i o n o f the Cr p a r t i c l e s i n the water was o b s e r v e d as t h e water i n the tube s o l i d i f i e d . The d i s t i l l e d water was degassed p r i o r t o use by b o i l i n g the water and f r e e z i n g s l o w l y . Cr p a r t i c l e s were s e l e c t e d as t h e y a r e wetted by water. 43 C h a p t e r V RESULTS AND DISCUSSION 1 . C o n t r o l l e d S o l i d i f i c a t i o n a) N o n - P l a n a r I n t e r f a c e i ) V e r t i c a l Growth. Two l e a d - a n t i m o n y a l l o y s { ] % and 2%) c o n t a i n i n g s p h e r i c a l p a r t i c l e s o f i r o n were grown a t the same v e l o c i t y i n the v e r t i c a l mode. The Pb 1% Sn a l l o y had a c e l l u l a r s o l i d l i q u i d i n t e r f a c e and the Pb 2% Sb a l l o y s o l i d i f i e d d e n d r i t i c a l l y . Pb 1% Sb C e l l u l a r I n t e r f a c e - Sample V-1 F i g . 17 shows the e t c h e d l o n g i t u d i n a l s e c t i o n o f t h i s sample i n c l u d i n g the quenched r e g i o n i n the upper p a r t o f the sample. I t can be c l e a r l y seen t h a t t h e r e i s a sharp change i n m i c r o s t r u c t u r e due to the q u e n c h i n g p r o c e s s . The a p p e a r a n c e o f t r a n s v e r s e s e c t i o n s o f the sample a f t e r e t c h -i n g , a t d i f f e r e n t d i s t a n c e s b e h i n d the quenched i n t e r f a c e i s shown i n F i g . 18. Note t h a t a l m o s t a l l of the p a r t i c l e s ( t h e w h i t e s p o t e s ) - i n F i g . 18 a r e s i t u a t e d i n the i n t e r -c e l l u l a r r e g i o n s . The d e n s i t y o f p a r t i c l e s was low so t h a t t h e r e was n e g l i g i b l e i n t e r a c t i o n among them. The r e s u l t s o f the measurements o f p a r t i c l e s i z e , 44 Figure 17. L o n g i t u d i n a l view of sample V - l . The quenched l i q u i d and the u n i d i r e c t i o n a l l y grown s o l i d are c l e a r l y d e f i n e d by the sharp change i n microstructure.Etched,50X. 46 47 (e) F i g u r e 18. T r a n s v e r s a l views of sample W I at the i n t e r f a c e (a t o d) and 12 mm from the i n t e r f a c e ( e ) . In d) the i r o n p a r t i c l e s are darker than the m a t r i x . I n a,b,c and e the p a r t i c l e s are the b r i g h t p spots.Most of the p a r t i c l e s are i n the c e l l w a l l s . I n a at the c e n t e r l e f t a r a m i f i c a t i o n w i t h a p a r t i c l e i s shown.In c and d the s m a l l p a r t i c l e s appear i n the m a t r i x . In e most of the c e l l w a l l s have d i s s o l v e d . a)100X; b)and c)500X; d)SEM 200X and e ) 1 0 0 X . A l l the s e c t i o n s are etched. 48 d e n s i t y , and d i s t r i b u t i o n on sample V -1 are t a b u l a t e d i n T a b l e I I I . Column V - l - 1 l i s t s the number o f p a r t i c l e s on a p l a n e p e r p e n d i c u l a r t o the growth d i r e c t i o n a t the quenched i n t e r f a c e . The d i a m e t e r s c o f the p a r t i c l e s a r e i n d i c a t e d on the l e f t , then the p a r t i c l e p o s i t i o n as s e g r e g a t e d a t the c e l l w a l l s o r i n the m a t r i x . Column V - l - l a l i s t s the number of p a r t i c l e s on a p l a n e 200 um b e h i n d the p l a n e V - l - 1 . Column V - l - 2 i n d i c a t e s the p a r t i c l e c o n f i g u r a t i o n on a p l a n e 2.5 mm behi n d the i n t e r f a c e and V - l - 2 a 200 ym b e h i n d t h i s p l a n e . Column V - l - 3 i s 5 mm b e h i n d the quenched i n t e r f a c e w i t h V - l - 3 a 200 ym b e h i n d t h i s . Measurements were not made beyond 5 mm b e h i n d the i n t e r f a c e as the c e l l s b e g i n t o homogenize and some c e l l 39 wal 1 S ; d i s a p p e a r . . At 2.5 mm beh i n d the i n t e r f a c e the s o l i d i s c l o s e to the m e l t i n g t e m p e r a t u r e f o r a p e r i o d o f 5 min. I f s i g n i f i c a n t h o m o g e n i z a t i o n o c c u r r e d the number o f p a r t i c l e s i n the m a t r i x s h o u l d i n c r e a s e as c e l l w a l l s d i s -a ppear. Examining T a b l e I I I the t o t a l number o f p a r t i c l e s i n the m a t r i x i n c r e a s e s from 25 to 42 from V - l - 1 to V - 1-2a then drops t o 25 i n V - l - 3 a the same as the i n t e r f a c e . A c c o r d i n g l y the d e c r e a s e a t V - l - 2 a i s a t t r i b u t e d t o s c a t t e r and not h o m o g e n i z a t i o n o f the c e l l w a l l s . G r o u p i n g the p a r t i c l e s i n l a r g e r s i z e i n c r e m e n t s and TABLE I I I Raw P a r t i c l e S i z e D i s t r i b u t i o n f o r Sample V - l Se c t i o n •+ V - l - 1 V - l - l a V - l - 2 V-l-2 1 V-lM 3 V-l-3a Size : um Segregated Matrix Segregated Matrix Segregated Matrix Segregated Matrix Segregated Matrix Segregated Matrix 4 5 1 4 3 3 1 4 0 3 1 2 1 6 7 4 8 5 8 4 7 2 7 5 5 0 8 16 4 17 3 19 6 10 2 9 4 10 4 10 25 4 17 0 17 6 20 10 17 6 9 5 12 25 5 9 1 15 1 12 4 17 2 7 4 14 23 2 15 0 18 5 23 6 17 4 7 0 16 15 3 21 3 20 3 21 2 18 3 15 1 18 15 0 09 1 7 9 19 4 14 0 13 4 20 17 0 19 3 19 2 25 7 12 0 16 3 22 11 2 05 0 7 0 14 2 5 0 5 1 24 10 0 07 0 3 0 17 1 1 0 11 1 26 5 0 7 0 1 1 7 0 1 0 6 l 28 1 0 4 0 0 0 4 0 1 1 5 0 30 0 0 7 0 0 0 6 1 1 0 6 0 32 0 0 1 0 0 0 1 1 0 0 1 0 TOTAL 175 25 150 19 137 38 190 42 123 26 118 25 50 a d d i n g t h e c o u n t s f o r a number o f s e c t i o n e d p l a n e s g i v e s the r e s u l t shown i n T a b l e IV. The same d a t a p l o t t i n g the number of p a r t i c l e s a g a i n s t p a r t i c l e s i z e isf'shown i n F i g . 19, a, b and c. The t.hree.hystograms show s i m i l a r p a r t i c l e d i s t r i -b u t i o n s w i t h an average o f 11.4% o f the t o t a l p a r t i c l e s i n the m a t r i x a t the i n t e r f a c e and 18.7% b e h i n d i t due t o homo-g e n i z a t i o n . I t i s emphasized here t h a t the r e l a t i v e number o f p a r t i c l e s i n the m a t r i x i s h i g h e r f o r s m a l l s i z e s . T h i s i s e v i d e n t from T a b l e IV a, b and c and F i g . 20.where the p e r c e n t o f p a r t i c l e s i n the m a t r i x as a f u n c t i o n o f s i z e i s shown. In t h i s f i g u r e the s o l i d l i n e c o r r e s p o n d s t o the p a r t i c l e s i z e d i s t r i b u t i o n a t t h e i n t e r f a c e - v a n d t h e broken l i n e to d i s t r i b u t i o n s b e h i n d the quenched i n t e r f a c e . In both c a s e s the p e r c e n t o f p a r t i c l e s i n the m a t r i x d e c r e a s e s w i t h s i z e , i n o t h e r words, s e g r e g a t i o n a t the c e l l w a l l s i s h i g h e r f o r b i g g e r p a r t i c l e s . T h i s i s an i m p o r t a n t e x p e r i m e n t a l r e s u l t because i t shows t h a t t h e o b s e r v e d d i s t r i b u t i o n i s due t o phenomenon o t h e r than p u s h i n g . In t h e r e s u l t s r e p o r t e d i n the l i t e r a t u r e i t was o b s e r v e d t h a t t h e r e i s a c r i t i c a l v e l o c i t y a t which p a r t i -c l e s a r e t r a p p e d and t h a t t h i s c r i t i c a l v e l o c i t y d e c r e a s e s w i t h p a r t i c l e s i z e . T h e r e f o r e , i f p u s h i n g was the mechanism l e a d i n g t o the o b s e r v e d s e g r e g a t i o n o f p a r t i c l e s a t the c e l l w a l l s , s m a l l p a r t i c l e s s h o u l d have been p r e f e r e n t i a l l y TABLE IV Parti ele Si ze Di s t r i bution and % on Matri x for Sample V - l Section -> (a) 1+lA (b) 2+2a+3+3a (e) 1 + la+2+2a+ 3+ 3a Size, ym Segregated Matrix % Matrix Segregated Matrix %: Matrix Segregated Matrix % Matrix 3-7 : 24 11 31.4 39 14 26.4 63 27 30.0 7-11 75 11 12.8 111 43 27.9 186 54 22.5 11-15 72 8 10.0 116 26 ' 18.3 188 34 15.3 15-19 60 7 10.4 127 26 \ 17.0 177 33 15.7 19-23 : 52 5 8.8 103 15 12.7 155 20 15.7 23-27 29 0 0 47 5 9.6 76 5 11.4 27-31 12 0 0 23 1 4.7 35 1 6.2 :31-34 1 0 0 2 1 3 1 TOTAL 325 42 11:4 568 131 18.7 893 173 16.2 80 6 0 4 CP - 4 0 + Q. -£204 — cell boundaries in matrix nT= 367 1 H h + II (a) 19 27 S i ze , fjLm 35 5 0 -«/> a> - 1004 p 5 0 4 o i L. 1 — cell boundaries — in matri x n T = 6 99 1 L, 1 3-19 27 (b) S i z e ^ m 35 150 + — intercel lulo r in matr ix o o QL o 50 + 0 4 — 4 -i I II 1 I 19 i = 4 — 2 7 35 ( c ) S ize m Figure 19. P a r t i c l e s i z e d i s t r i b u t i o n for sample V-1 ( c e l l u l a r i n t e r f a c e ) for p a r t i c l e s i n the c e l l walls and i n the matrix. a)At the interface.b)Behind the i n t e r f a c e . c ) T o t a l d i s t r i b u t i o n . 54 30--20 £ 10 0 Q i n t e r f a c e • behind interface Figure 20. % of p a r t i c l e s i n the matrix vs. p a r t i c l e s i z e , at the i n t e r f a c e and behind the interface.Small p a r t i c l e s are more l i k e l y to appear i n the matrix than large p a r t i c l e s . T h e d i f f e r e n c e of the two curves i s a t t r i b u t e d to d i s s o l u t i o n of r a m i f i c a t i o n s . 55 r e j e c t e d i n s t e a d o f l a r g e p a r t i c l e s . ; !-., The l a r g e p a r t i c l e s s h o u l d have been found i n t h e m a t r i x which i s not the c a s e . The d i f f e r e n c e i n t h e p e r c e n t o f p a r t i c l e s i n the m a t r i x on t h e i n t e r f a c e and b e h i n d t h e i n t e r f a c e shown i n F i g . 2 0 , i s c o n s i s t e n t w i t h h o m o g e n i z a t i o n o c c u r r i n g . Some o f the l a r g e r p a r t i c l e s , i n i t i a l l y i n t h e c e l l w a l l s , a f t e r homogeni-z a t i o n , a r e now counte d i n the m a t r i x i n c r e a s i n g the p e r c e n t o f p a r t i c l e s i n the m a t r i x . A h i g h e r p e r c e n t a g e o f sm a l l p a r t i c l e s ds> i n i t i a l l y i n the m a t r i x so t h a t h o m o g e n i z a t i o n has a s m a l l e r e f f e c t . The Real D i s t r i b u t i o n o f P a r t i c l e s i n the M a t r i x . P o l i s h i n g a s e c t i o n c o n t a i n i n g p a r t i c l e s e c t i o n s does not„ g i v e : . the f u l l p a r t i c l e d i a m e t e r . The r e a l d i s t r i b u t i o n o f p a r t i c l e s per volume i s c a l c u l a t e d u s i n g the S c h w a r z - S a l t i k o v method r e f e r r e d t o p r e v i o u s l y . T h i s method e n a b l e s the o b s e r v e d d a t a to be c o r r e c t e d f o r the random p l a n e o f s e c t i o n o f t h e p a r t i c l e . The Schwarz-S a l t i k o v method g i v e s the f o l l o w i n g f o r m u l a f o r the number of p a r t i c l e s N v ( J ) o f s i z e J : a m N (i)-a(i + l )N (i+.l )- a ( i +2 ) .N . ( i +2 )-... -a(k)H (k) N ( J ) = -lii-i § § a v where A = D m a x/k = interval size D = maximum size :.max k = number of groups a ( i ) = Schwartz-Saltikov c o e f f i c i e n t s Table V Shwartz-Saltikov Coefficients 1 a ( l ) o(2) a(3) a(4) a(5). a(6) a(7) a(8) J=l 1 0.1547 0.036 0.013 0.0061 0.0033 0.0020 0.009 J=2 0.5774 0.1529 0.042 0.0171 0.0087 0.0051 0.0031 J=3 0.4472 0.1382 0.0408 : 0.0178 0.0093 0.0057 J=4 0.3779 0.1260 0.0386 0.0174 0.0095 J=5 0.3333 0.1161 0.0366 0.0168 J=6 0.3015 0.1081 0.0346 J=7 0.2773 0.1016 J-8 : 0.2582 57 For the d i s t r i b u t i o n o f p a r t i c l e s o b t a i n e d i n t h i s e x p e r i -ment D = 36 um. The i n t e r v a l s i z e r e g r o u p i n g the raw i n d A data i s A= 4 ym. T h e r e f o r e the number o f groups i s k = 8. The c o e f f i c i e n t s a ( i ) a r e g i v e n i n T a b l e V. For c o m p a r i s o n p u r p o s e s the a p p a r e n t d i s t r i b u t i o n i s r e a r r a n g e d because the S c h w a r z - S a l t i k o v method r e q u i r e s a d i s t r i b u t i o n o f s i z e s b e g i n n i n g w i t h z e r o . T h i s new g r o u p i n g as w e l l as the r e a l d i s t r i b u t i o n are g i v e n i n T a b l e s VI (a) and (b) r e s p e c t i v e l y and i l l u s -t r a t e d i n F i g . 21 (a) and ( b ) . I t can be n o t e d t h a t t h e r e i s no s i g n i f i c a n t changes i n the d i s t r i b u t i o n e x c e p t t h a t p a r t i c l e s s m a l l e r than 4 ym a r e not p r e s e n t i n t h i s sample and a l s o t h a t t h e number o f p a r t i c l e s i n the m a t r i x sub-s t a n t i a l l y i n c r e a s e s w i t h r e s p e c t t o the a p p a r e n t d i s t r i -b u t i o n f o r s m a l l e r s i z e s . T h i s can be seen i n F i g . 22 where the p e r c e n t o f p a r t i c l e s i n the m a t r i x f o r both s i z e d i s t r i b u t i o n s i s p l o t t e d . Bb 2% Sb D e n d r i t i c I n t e r f a c e - Sample V-2 A l o n g i t u d i n a l s e c t i o n o f a d e n d r i t i c a l 1 y s o l i d i f i e d and quenched sample which has been p o l i s h e d and e t c h e d i s shown i n F i g . 23. The quenched m a t e r i a l i n the upper p a r t of the f i g u r e has a f i n e r s t r u c t u r e than the lower p a r t as TABLE VI Particle Size Distribution and % in the Matrix for Sample V-l (a) Apparent Distribution (b) Real Distribution for Total Counts. (a) Apparent Distribution (b) Real D i s t r i b u t i o n * Size, ym Segre-gated;-; 1 Matrix';••-<.; % Matrix Segre-gated;- . .Matrix % Matrix 0-4 21 7 25.0 0 0 4-8 123 43 25.9 7.2 3.9 35.0 8-12 190 48 20.2 11.4 3.9 25.4 12-16 213 32 13.1 13.1 1.9 12:6 16-20 185 33 15.1 •12.1 2.5 17.1 20-24 96 7 6.8 5.9 0.4 7.0 24-28 42 3 6.7 2.3 0.2 8.1 28-32 22 0 0.0 1.4 0.0 0.0 *Arbitrary Volume 200-— intercel lu lar — in matr ix 100-0 I 0 8 16 (a) 24 Size 32 fjLm 50-a 30 QL 10+ 0 0 — intercel lular — in m a t r i x i r i i i 1 8 16 (b) 1 i 24 Size 32 /JLvr\ Figure 21. a)Apparent and b)Real d i s t r i b u t i o n of p a r t i c l e s f o r sample V-1.The d i s t r i b u t i o n s are s i m i l a r except for the f a c t that p a r t i c l e s smaller than L- yO- m are not present i n the r e a l d i s t r i b u t i o n . I n b) the volume i s a r b i t r a r y . 60 Figure 22. % of p a r t i c l e s i n the matrix vs. p a r t i c l e s i z e for the apparent and r e a l d i s t r i b u t i o n of p a r t i c l e s . T h e e f f e c t for small p a r t i c l e s i s remarked i n the r e a l d i s t r i b u t i o n . 61 e x p e c t e d . The p a r t i c l e s a r e t h e w h i t e s p o t s s i t u a t e d i n t h e i n t e r d e n d r i t i c r e g i o n . T r a n s v e r s e s e c t i o n s o f t h e s a m p l e c u t a t and b e h i n d t h e i n t e r f a c e a r e shown i n F i g . 24. The p a r t i c l e s a p p e a r w h i t e and a r e a t i n t e r d e n d r i t i c p o s i t i o n s . The number o f p a r t i c l e s o b s e r v e d on t h e t r a n s v e r s e s e c t i o n a t i n t e r d e n d r i t i c ( s e g r e g a t e d ) and i n t h e m a t r i x as a f u n c t i o n o f p a r t i c l e s i z e i s g i v e n i n T a b l e V I I . S e c t i o n V-2-1 i s a t t h e i n t e r f a c e , V - 2 - l a i s 200 um f r o m V-2-1 and V-2-2 i s 2.5 mm b e h i n d t h e i n t e r f a c e . The t a b l e shows a h i g h p r o p o r t i o n o f t h e p a r t i c l e s o f a l l s i z e s a r e i n t h e i n t e r d e n d r i t i c r e g i o n . The t h r e e c o l u m n s i n T a b l e V I I a r e c o m b i n e d i n T a b l e V I I I and p l o t t e d i n F i g . 25. C o m p a r i n g t h e s e r e s u l t s w i t h t h e c o r r e s p o n d i n g p a r t i c l e d i s t r i b u t i o n f o r a c e l l u l a r i n t e r f a c e ( F i g . 19) t h e o n l y d i f f e r e n c e i n t h e d e n d r i t i c c a s e i s t h a t t h e a v e r a g e number o f p a r t i c l e s i n t h e m a t r i x i s l o w e r , more p a r t i c l e s b e i n g s e g r e g a t e d t o t h e i n t e r d e n d r i t i c r e g i o n . A p p l y i n g t h e S c h w a r z - S a l t i k o v c o r r e c t i o n t o t h e o b s e r v e d p a r t i c l e s i z e s f o r d e n d r i t i c g r o w t h g i v e s t h e r e a l d i s t r i b u t i o n g i v e n i n T a b l e IX ( b ) . The d a t a i s a l s o p l o t t e d i n F i g . 27j. Comparing:; F i g . 27 a , t h e a p p a r e n t d i s t r i b u t i o n , w i t h b, t h e r e a l d i s t r i b u t i o n , shows t h e two d i s t r i b u t i o n s a r e s i m i l a r . However w i t h t h i s a r r a n g e -ment f o r t h e s i z e i n t e r v a l , t h e d i s t r i b u t i o n i n t h e ; m a t r i x d oes n o t c l e a r l y f o l l o w t h e same p a t t e r n o b s e r v e d i n 62 Figure 23. L o n g i t u d i n a l view of sample V-2 at the i n t e r f a c e . T h e sharp change i n s t r u c t u r e i s due to the quenching proeess.Etched. 100X. Figure 24. Transversal views of sample V-2 ( d e n d r i t i c i n t e r f a c e ) . The b r i g h t c i r c l e s are the particles.The p a r t i c l e s are i n the i n t e r d e n d r i t i c regions, a),b) and c) 500X.Etched. TABLE VII Raw Particle Size Distribution for Segregated and Matrix P a r t i c l e s for Sample 1 Section re- V-2-11 V-2-la ' V-2-•2 si z e , um Segregated Matrix Segregated Matrix Segregated Matrix 4 8 1 8 1 7 0 6 20 4 18 3 18 2 8 31 1 35 5 27 2 10 49 10 52 13 47 10 12 33 5 47 4 36 5 14 44 5 50 3 45 12 16 '. 38 0. 45 3 34 7 18 : 24 2 50 0 32 3 20 38 2 56 8 34 2 22 : 19 0 32 7 19 1 24 : 14 0 26 2 12 0 26 9 0 17 0 7 0 28 7 0 9 0 3 0 30 0 0 16 0 2 0 32 1 0 4 0 0 0 TOTAL 334 30 465 49 323 44 65 TABLE VIII P a r t i c l e Size Distribution and % of Particles in the Matrix for Sample V-2. Counts on a l l sections V-2: (l+la+2) Size, um Segre-gated .. Matrix •.: % Matrix 3-7 79 11 12.2 7-11 241 41 14.5 11-15 255 34 11.76 15-19 223 15 6.3 19-23 198 20 9.17 23-27 85 2 2.3 27-31 37 0 0 31-35 6 0 0 TOTAL 1122 123 9.87 66 300 if 200 — interdendritic — in matrix Uj= 1247 o CL o d 00-0 i i 1 1 - ! — -II = i — 27 35 S i z e , /JL m Figure 25. P a r t i c l e s i z e d i s t r i b u t i o n for sample W2 (.dendritic i n t e r f a c e ) for p a r t i c l e s i n the matrix and i n i n t e r d e n d r i t i c regions.Less p a r t i c l e s than f o r sample V - l ( c e l l u l a r i n t e r f a c e ) are present i n the matrix. Figure 26. % of p a r t i c l e s i n the matrix vs. p a r t i c l e s i z e for sample V-2 ( d e n d r i t i c interface).The same pattern as i n sample V-1 ( c e l l u l a r i n t e r f a c e ) i s observed although insample V-2 i s l e s s pronounced. TABLE IX Particle Size Distribution for Segregated and Matrix P a r t i c l e s and % in the Matrix, (a) Apparent Distribution (b) Real Distribution. Counts on a l l Sections i V-2-(l- la-2) Size, utT (a) Apparent Distribution 1 (b) Real Di s t r i b u t i o n Segregated.. Matrix % Matrix Segregated., Matrix % Matrix QT4 23 2 8.0 0 0 -4^8 149 17 10.2 7.4 0.6 7.9 8-12 264 37 12.3 17.6 2.9 14.1 12-1.6 256 30 10.5 15.3 2.2 12.5 16-20 234 17 6.8 15.4 1.1 6.8 20-24 233 10 7.5 7.6 0.7 9.0 24-28 52 0 0 3.0 0 0 28-32 23 0 0 1.5 0 0 co 300 J200-o Q. H— o d ioo4 o-o 69 — interdendriti c — in ma t r i x i 1 "L__. 8 16 (a ) '3-24 32 Size , /JLm 70+ — inte rdendr i t i c — in ma t r i x ^ 50+ o Q. ° 30-b 10+ 0-0 r • i i 8 16 (b) 24 32 Size, [JL.™ Figure 27. a)Apparent an b)Real d i s t r i b u t i o n of p a r t i c l e s f o r sample V- 2 ( d e n d r i t i c i n t e r f a c e ) . P a r t i c l e s smaller than 4/U. m are not present i n the r e a l d i s t r i b u t i o n . I n b) the volume i s a r b i t r a r y . 70 F i g . 26 and the d i s t r i b u t i o n o f s i z e s i n the m a t r i x appears to be random as i t can be seen i n T a b l e IX (a) and ( b ) . i i ) H o r i z o n t a l Growth. An a l l o y Pb 1.5% Sb c o n t a i n i n g Fe p a r t i c l e s was grown i n an h o r i z o n t a l mould _3 r o t a t i n g at 0.5 rps and a t a growth v e l o c i t y o f 17x10 cm.s"^. T r a n s v e r s e s e c t i o n s o f t h i s sample a r e shown i n F i g . 28. As b e f o r e most of the p a r t i c l e s a r e found i n the c e l l w a l l s . The number and p o s i t i o n o f the p a r t i c l e s as a f u n c t i o n o f s i z e i s g i v e n i n T a b l e X and the r e g r o u p e d d a t a i n T a b l e XI. S e c t i o n H-l-1 i s a t the i n t e r f a c e , H - l - l a i s a t 200 um from s e c t i o n H - l - 1 , H-l-2 i s a t 2.5 mm from the i n t e r f a c e and H -l-2a i s a t 200 um from H - l - 2 . A h i s t o g r a m o f the d i s t r i b u t i o n i s shown i n F i g . 29. The p e r c e n t o f p a r t i c l e s i n the m a t r i x as a f u n c t i o n o f p a r t i c l e s i z e i s shown i n F i g . 30. In t h i s f i g u r e t h e r e i s no c l e a r t r e n d o f m a t r i x p a r t i c l e s w i t h p a r t i c l e s i z e , d i f f e r i n g from the v e r t i c a l s o l i d i f i c a t i o n r e s u l t s . Many o f the p a r t i c l e s were found to be s i t u a t e d a t the o u t s i d e s u r f a c e o f the sample a d j a c e n t to the c o n t a i n e r w a l l . T h i s i s shown i n the l e f t hand s i d e o f F i g . 28 (b) where a h i g h c o n c e n t r a t i o n o f w h i t e p a r t i c l e s . i s p r e s e n t at the edge o f the sample. 71 (b) Figure 28. Transversal views of sample H-l.a)A s e c t i o n at the i n t e r f a c e showing p a r t i c l e s i n the c e l l walls.500X.Etched.b)A section at the i n t e r f a c e showing p a r t i c l e s near the w a l l of the container.200X.Etched. TABLE X Raw Particle Size Distribution for Sample H-l. •Section -> H-l- 1 H-l- la H-l- 2 H-l--2a Size, um Segregated Matrix Segregated ! Matrix Segregated Matrix Segregated Matrix 2 2 1 0 0 0 0 1 0 4 4 5 7 2 4 0 7 1 6 15 2 10 1 11 1 16 3 8 20 10 10 6 18 5 16 1 10 12 6 20 2 23 4 23 5 12 16 6 13 3 21 5 24 6 14 22 2 16 4 25 7 28 7 16 22 4 15 6 25 3 33 0 18 25 1 13 0 13 0 13 5 20 23 1 10 2 22 3 23 2 22 11 3 4 1 10 2 12 1 : 24 8 0 7 1 9 2 7 2 26 8 0 3 0 7 1 12 - 3 : 28 3 .: 0 3 0 5 : 0 7 0 : 30 0 0 1 0 1 0 0 0 TOTAL 191 41 132 28 194 33 222 36 73 TABLE XI P a r t i c l e S i z e D i s t r i b u t i o n f o r S e g r e g a t e d and M a t r i x P a r t i c l e s , and % i n the M a t r i x f o r Sample H - l . Counts on a l l S e c t i o n s . S i z e , ym . S e g r e g a t e d M a t r i x % i n M a t r i x 3-7 74 14 16 7-11 142 35 19.8 11-15 165 44 21 .0 15-19 159 20 11.2 19-23 115 13 10.15 23-27 61 12 16.4 27-31 20 0 0 31-35 0 0 0 TOTAL 736 138 15.8 74 160+ to ~I20-o Q. 80+ — interce 11 u I a r — in matrix n T = 87 7 40+ rm i i i i i 1 0: 19 + 27 Size , 35 m Figure 29. P a r t i c l e s i z e d i s t r i b u t i o n f o r sample H-l for p a r t i c l e s at the c e l l walls and p a r t i c l e s i n the matrix. 75 O E II 19 27 35 Size , fjLrn Figure 30. % of p a r t i c l e s i n the matrix vs. p a r t i c l e s i z e for sample H-l.The d i s t r i b u t i o n i s random. 76 b) - P l a n a r I n t e r f a c e S e v e r a l samples o f pure l e a d (99.999% p u r i t y ) c o n t a i n i n g f e p a r t i c l e s were gfown v e r t i c a l l y and h o r i z o n -t a l l y . The p r e s e n c e o f a p l a n a r i n t e r f a c e d u r i n g growth can be e s t a b l i s h e d u s i n g the c r i t e r i o n 6/R-(m C /D)(l-k )k »» o o o f o r c o n s t i t u t i o n a l s u p e r c o o l i n g . In t h i s e x p r e s s i o n m i s the l i q u i d u s s l o p e , C Q i s the c o n c e n t r a t i o n o f s o l u t e , D i s the d i f f u s i o n c o e f f i c i e n t o f s o l u t e atoms i n the melt and k Q i s the p a r t i t i o n c o e f f i c i e n t . I f G/R i s l e s s than the r i g h t hand p a r t o f t h e e q u a t i o n , no c o n s t i t u t i o n a l s u p e r c o o l i n g i s p r e s e n t and the i n t e r f a c e i s p l a n a r . The measured t e m p e r a t u r e g r a d i e n t i n t h i s group o f e x p e r i m e n t s -1 -4 was 30°e cm and the growth v e l o c i t y was l e s s than 1.5x10 -1 5 - 2 cm.s . Under t h e s e c o n d i t i o n s G/R i s 2x10 °C cm .sec -2 40 o b t a i n e d f o r l e a d w i t h 10 wt % Sn a l l o y . E x a m i n a t i o n o f the quenched samples showed no a c c u m u l a t i o n o f p a r t i c l e s ahead o f t h e quenched i n t e r f a c e , and no d i f f e r e n c e i n the number o f p a r t i c l e s o f a l l s i z e s was e v i d e n t between the grown and quenched m a t e r i a l . T h i s a p p l i e d to both v e r t i c a l and h o r i z o n t a l s o l i d i f i c a t i o n . From t h e t h e o r i e s o f p a r t i c l e r e j e c t i o n d e s c r i b e d p r e -v i o u s l y , p r e d i c t i o n s o f c r i t i c a l v e l o c i t i e s f o r t h e t r a n s i -t i o n from t r a p p i n g to r e j e c t i o n a r e g i v e n . The B o i l i n g and C i s s e t h e o r y ( f o r m u l a 1 8 ( a ) ) g i v e s f o r l e a d 2 3 -18 5 -? V^R J = 4.71x10 '° cm 0 sec c ...32 77 TABLE XII C r i t i c a l V e l o c i t i e s Predicted by the Theories 1 R, um C r i t i c a l V e l o c i t i e s , V cm sec" 1 XlO 4 Boiling and Cisse Theory Chernov e t a l . Theory I 21.67 26.94 5 1.94 3.14 10 0.69 1.25 78 i n w h i c h t h e f o l l o w i n g v a l u e s f o r t h e c o n s t a n t s were u s e d : v = 0.34, k = 1 . 3 8 x l 0 " 1 6 e r g / d e g " 1 , T = 6 0 0 ° K , <? l s = -2 41 * o 33.3 e r g cm , a Q = 3.5 : A , n = 1.67 mPa.s. ( T h i s l a s t two were t a k e n f r o m M e t a l s Handbook V o l . l i , 9 t h Ed.) The C h e r n o v t h e o r y g i v e s t h e f o l l o w i n g e x p r e s s i o n f o r t h e c r i t i c a l v e l o c i t i e s f o r s m a l l p a r t i c l e s i n l e a d ( E q u a t i o n 2 7 ) . V C R 4 / 3 = 1 - 2 5 X 1 0 " 8 C M 7 / 3 S E C _ 1 • • • 3 3 To o b t a i n t h i s e q u a t i o n t h e same v a l u e s o f n and a . Is -1 4 were u s e d and was t a k e n as 10 e r g . The c r i t i c a l v e l o c i t i e s v e r s u s r a d i u s g i v e n by E q u a t i o n s 32 and 33 a r e shown i n T a b l e X I I . The a b s e n c e o f p u s h i n g a t g r o w t h v e l o c i t i e s p r e d i c t e d by t h e t h e o r i e s s u g g e s t s t h a t , r e g a r d l e s s o f t h e e x i s t e n c e o f any f o r c e t h a t can l e a d t o t h i s k i n d o f r e j e c t i o n , t h e s e g r e g a t i o n o f p a r t i c l e s t o i n t e r s t r u c t u r a l p o s i t i o n s i s g o v e r n e d by a mec h a n i s m i n w h i c h t h e b u o y a n c y f o r c e and c o n v e c t i o n i n t h e l i q u i d a r e t h e d r i v i n g f o r c e s . On t h e o t h e r hand t h e s e e x p e r i m e n t a l r e s u l t s show t h e l i m i t a t i o n o f t h e s e t h e o r i e s i n p r e d i c t i n g p a r t i c l e p u s h i n g . N o t e : Ewing r e p o r t e d e x p e r i m e n t a l and t h e o r e t i c a l v a l u e s -2 o f a - | S w h i c h a r e 55 and 53 e r g cm r e s p e c t i v e l y . The l o w e r v a l u e g i v e n i n r e f 41 was a d o p t e d i n o r d e r t o c a l c u l a t e t h e l o w e r v e l o c i t i e s . 79 2. The M a t e r - N y l o n Sphere Model The purpose o f s t u d y i n g t h i s model i s t o v i s u a l i z e the motion o f t he p a r t i c l e s a t a c e l l u l a r i n t e r f a c e i n o r d e r to d e v e l o p the mechanism t h a t i s p r o p o s e d i n the next s e c t i o n s . a) H o r i z o n t a l Mode i ) The Motion o f the Nylon S p h e r e s . The d e n s i t y of the water was a d j u s t e d by a d d i n g s a l t (NaCT) to a v a l u e t h a t gave the s p h e r e s a t e r m i nal v v e l o c i t y o f 1 cm sec"''. Comparing t h i s v e l o c i t y w i t h the c e l l s i z e , t h i s c o r r e s p o n d s to the s m a l l p a r t i c l e s i n the m e t a l l i c system which t r a v e l a d i s t a n c e o f a c e l l d i a m e t e r per s e c o n d . R e l a t i n g the s p h e r e t e r m i n a l v e l o c i t y o f 1 cm s e c " ' w i t h the d i a m e t e r o f the c y l i n d e r s e l e c t e d , t h i s c o r r e s p o n d s to a p a r t i c l e o f a p p r o x i m a t e l y 16 um i n ^ r a d i u s t r a v e r s i n g the d i a m e t e r o f the tube (11 mm) i n 10 s e c o n d s . The most i m p o r t a n t m o d e l l i n g f a c t o r i s t h e r e l a t i o n o f the f i n a l v e l o c i t y o f the s p h e r e s to the a n g u l a r v e l o c i t y w'. For a f i x e d v a l u e o f Vjp and v a r y i n g w', i t i s p o s s i b l e to o b t a i n a s i t u a -t i o n i n which the s p h e r e s t r a v e l the d e s i r e d number o f c e l l d i a m e t e r s per r e v o l u t i o n and can be compared to the r e a l m e t a l l i c system. The f i r s t s t e p i s t o s t u d y the motion o f the s p h e r e s i n t h i s f l u i d i n r o t a t i o n a l m o t i o n . T h i s was done i n a range o f p e r i o d s o f r o t a t i o n between 30 s e c . to 0.5 s e c . per c y c l e . 80 A f t e r a t r a n s i e n t p e r i o d the l i q u i d r o t a t e d as a s o l i d 42 as e x p e c t e d . The i n i t i a l t r a n s i e n t was o b s e r v e d to depend on the r o t a t i o n speed and i t was found t o be l o n g e r f o r low v e l o c i t i e s , i . e . the t r a n s i e n t time i s p r o p o r t i o n a l to t o ! . N e g l e c t i n g the c e n t r i p e t a l f o r c e , the t r a j e c t o r y o f the n y l o n s p h e r e s may be p r e d i c t e d by a p p l y i n g the s u p e r p o s i t i o n p r i n c i p l e . I f the d e n s i t y o f l i q u i d and sphere a r e the same, the t r a j e c t o r y s h o u l d be c i r c u l a r and the v e l o c i t y i d e n t i c a l t o the v e l o c i t y o f the l i q u i d . In t h i s c ase the d e n s i t y o f the s p h e r e s i s lower than t h a t o f the l i q u i d ; t h e r e f o r e the v e l o c i t y due to buoyancy f o r c e s has a v e r t i c a l component t h a t d i s t o r t s the c i r c u l a r path o f the s p h e r e s . The e c c e n t r i c i t y o f t h i s d i s t o r t e d c i r c l e depends on the r a t i o o f the t e r m i n a l v e l o c i t y to the f r e q u e n c y o f r o t a t i o n -V.j/to' . For h i g h w ' the path i s a l m o s t c i r c u l a r , and f o r low co' the path i s e l o n g a t e d i n the v e r t i c a l d i r e c t i o n , and w i t h l e s s c u r v a t u r e i n the r e g i o n i n which the v e l o c i t y due to buoyancy f o r c e has the same d i r e c t i o n a>s; the t a n g e n t i a l v e l o c i t y due t o r o t a t i o n . The t y p i c a l paths and d i s t r i b u t i o n o f s p h e r e s i n the c y l i n d e r f o r v a r i o u s v a l u e s o f co' are shown i n the sequence of p h o t o g r a p h s i n F i g u r e 31. Photographs a t o e were taken from one end o f the r o t a t i n g c y l i n d e r ; the w h i t e p a r t i c l e s i n the p h o t o g r a p h s are the n y l o n s p h e r e s . F i v e speeds o f 81 r o t a t i o n a r e shown u = 0.033 r p s ( a ) , 0.083 r p s ( b ) , 0.2 r p s ( c ) , 0.5 r p s (d) and 1 r p s ( e ) . A t t h e l o w e s t r o t a t i o n a l s p e e d 0.033 r p s t h e t a n g e n t i a l v e l o c i t y o f t h e l i q u i d a t t h e c y l i n d e r r a d i u s i s o f t h e o r d e r o f t h e f l o t a t i o n v e l o c i t y . As a r e s u l t t h e s p h e r e s a r e p o s i t i o n e d a t t h e r a d i u s and e s s e n t i a l l y a t r e s t w i t h r e s p e c t t o t h e s y s t e m o f r e f e r e n c e , t h i s i s shown i n F i g . 3 1 ( a ) . The f l u i d v e l o c i t y a t t h e w a l l i s V ^ = 2 T r r c u ' - 6.28x4.75 cm 0.033 s e c - 1 = 0.995 cm s e c " 1 , v e r y c l o s e t o t h e t e r m i n a l v e l o c i t y o f t h e s p h e r e s = 1 cm s e c . At a r o t a t i o n s p e e d o f 0.5 r p s u s e d i n t h e s o l i d i f i c a -t i o n o f l e a d t h e l i q u i d moved a t t h e t u b e w a l l a t a s p e e d V.p-| = 6.28x1.1 cmx0.5 s e e - 1 = 3.45 cm s e c - 1 . An i r o n p a r t i c l e h a v i n g t h e same t e r m i n a l v e l o c i t y i n l e a d s h o u l d be 177 ym i n r a d i u s . T h a t i s , a t t h a t f r e q u e n c y o f 0. 5 r p s t i h n s ' k i n d o f movement d o e s n o t e x i s t i n t h e m e t a l l i c s y s t e m b e c a u s e t h e p a r t i c l e s u s e d were t e n t i m e s s m a l l e r . As t h e f r e q u e n c y w ' i n c r e a s e s t h e t i m e a l l o w e d f o r v e r t i c a l d i s p l a c e m e n t d e c r e a s e s . The s p h e r e s do n o t have t i m e t o r e a c h t h e c y l i n d e r w a l l s i n e a c h r o t a t i o n a nd t h e s p h e r e s r e m a i n i n t h e l i q u i d m o v i n g i n l o o p s . T h i s happens a t a r o t a t i o n a l s p e e d o f 0.5 r p s , t h e d i s t r i b u t i o n o f 82 83 84 Figure 31. D i s t r i b u t i o n of spheres i n the p h y s i c a l model f o r the h o r i z o n t a l mode at d i f f e r e n t r o t a t i o n speeds. a)0,033rps. b)0.083rps c)0.2rps,d)0.5rps. e)lrps.The p a r t i c l e s f o r low speeds are concentrated i n the wall,and f o r h i g h speeds they are concentrated i n the center of the container as explained i n the t e x t . 85 (b) Figure 32. The two classes of patterns followed by the spheres i n the model as a function of r o t a t i o n speed.In a) the spheres reach the w a l l of the container and i n b) the spheres do not reach 86 s p h e r e s i n the l i q u i d i s shown i n F i g . 3 1 ( d ) . At v a l u e s o f to' g r e a t e r than 0.5 r p s the c e n t r i p e t a l f o r c e s become i m p o r t a n t and the s p h e r e s t e n d to c o n c e n t r a t e a l o n g the c y l i n d e r a x i s as shown i n f i g . 31(e) f o r to = 1 r p s . The s p h e r e s moving i n the l i q u i d due to r o t a t i o n and g r a v i t y f o r c e s may be s e p a r a t e d i n t o two g r o u p s , one i n which the s p h e r e s r e a c h the c y l i n d e r w a l l i n one r o t a t i o n , and a second where the s p h e r e s do n o t . In the f i r s t group, the l o o p d i a m e t e r t a k e n by the s p h e r e s tends to expand w i t h to' as shown s c h e m a t i c a l l y i n F i g . 3 2 ( a ) ; i n the second group the l o o p d i a m e t e r s h r i n k s as shown i n F i g . 3 2 ( b ) . The l o o p s were o b s e r v e d to change a l i t t l e w i t h f u r t h e r r o t a t i o n so t h a t i n some c a s e s s p h e r e s may t o u c h t h e ' w a l l even at h i g h v a l u e s o f to' . i i ) M o d e l l i n g . The parameter employed to model the system i s the number o f c e l l s t r a v e l l e d by a p a r t i c l e i n the r e a l system and by a sphere i n the model. T h i s q u a n t i t y i s ' e x p r e s s e d as V f • V f N = — = N = - f r . . .32 C. to C . ID where i s the t e r m i n a l v e l o c i t y , C i s t h e c e l l s i z e and <o i s the f r e q u e n c y o f r o t a t i o n . The dashed terms r e f e r to a model v a r i a b l e . In the a l l metal system G and to a r e f i x e d , C ~ 100 ym and to = 0.5 r p s . In the model V f and C 87 " _ i i are f i x e d , = 1 cm sec and C ~ 2 cm. T h i s g i v e s -2 -1 V = C.or; v' _ 10 cm x 0.5 rps 1 cm.sec f • >' ' f ' CO: 2 cm co y = 0.25 x TO" 2 ...33 v f CO U s i n g the f o r m u l a f o r t e r m i n a l v e l o c i t y g i v e n by the Stokes Law = 2 a ^ M P = 3 . 7 8 x l 0 " 4 a 2 ...34 9-u where V.J- i s g i v e n i n cm s e c " ' and a ( t h e p a r t i c l e r a d i u s ) i n ym. In t h i s c a l c u l a t i o n the f o l l o w i n g v a l u e s were used = - 3 - 3 10.6 gr.cm , p p = 7.7 gr cm , n = 1.67 m Pa.sec and g = -2 981 cm sec . The S t o k e s t e r m i n a l v e l o c i t y g i v e s a c c u r a t e 42 v a l u e s when t h e Reynolds number i s l e s s than one. The Reynolds number i s g i v e n by P*e =?- p ^ d U / n , where i s the d e n s i t y o f the l i q u i d , d i s the p a r t i c l e d i a m e t e r , U i s the t e r m i n a l v e l o c i t y and n i s the v i s c o s i t y . S u b s t i t u t i n g the a p p r o p r i a t e v a l u e s f o r l e a d g i v e s Re = 3.4x10 4 f o r the smal-l e s t , p a r t i c l e . o f y.2 ym r a d i u s .and;Re .= 2.57x10"' f o r the l a r g e s t o f 17.5 ym r a d i u s p r e s e n t i n the m e l t . Combining E q u a t i o n s 34 w i t h 33 g i v e s 6.61 w = —2~ ... 35 a V w i t h a i n um to i s g i v e n i n r p s . U s i n g t h i s e q u a t i o n the 88 TABLE X I I I C o r r e l a t i o n T a b l e f o r P a r t i c l e S i z e i n the M e t a l l i c System V e r s u s Frequency o f R o t a t i o n "11 i n the Model. a, ym u>' = 6.61/a 2 ( r p s ) i T., sec 2 1 .65 0.6 4 4.13-10" 1 2.42 6 1.84x10" 1 5.44 8 1 . 0 3 x l 0 - 1 9.68 10 6 . 6 x l 0 ~ 2 15.1 15 2 . 9 4 x l 0 " 2 34.0 17 2.29-10" 2 43.1 89 corre la t ion of the pa r t i c l e s ize in the meta l l i c system i with oi is l i s t e d in Table XI TI. In the tab le , i t is evident that the range of frequencies from 0.5 to 0.033 rps used to study the motion of spheres covers the range of pa r t i c l e s ize in the metal system between 2 ym and 17 ym. However, due to the wall e f fec t i t was necessary to r e s t r i c t the observations in the model to those frequencies for which the type of loops shown in F ig . 32(b) are obtained, that is for frequencies higher than 0.2 rps. In the meta l l i c system this corresponds to par t i c l es smaller than 6 um in radius. In th is respect , th is is the most interest ing range of s izes because outside this range larger par t i c l es t r a ve l l i ng tens of ce l l diameters per cycle are more l i k e l y to be entrapped among the ce l l t i p s . For a given frequency <D the axis of the cy l inder was t i l t e d at d i f fe rent angles to the horizontal in order to vary the pa r t i c l e ve loc i t y in the axial d i rec t ion and simulate an advancing f ront . This axial component is given by V , = V' sin a . . . 36 ph f where a is the t i l t i n g angle. Table XIV gives the i _ i values of ve loc i t y , as a function of a for V .^ = i c m sec" . Note, from Table XIV, that the measured values are always smaller than those ca lculated using Equation 36. Sometimes 90 TABLE XIV T h e o r e t i c a l and E x p e r i m e n t a l H o r i z o n t a l V e l o c i t i e s as a F u n c t i o n o f T i l t i n g AngTe f o r oo = 0.5 sec 1 .oh the Model f o r V = 1 cm/sec = p a r t i c l e v e l o c i t y . 0 a V p h ( t h ) V p h ( e x p ) cm/mi n cm/min 4..Q 4.2 1.4-1.0 2.6 2.7 1.14-0.9 1 .1 1 .2 0.75-0.69 91 the measured values are less than half the ca lculated values. This may be accounted for by the large s ize of the spheres in the model. In this case the average ve loc i ty of the spheres for a rotat ion period is less than the terminal ve loc i ty due to a large t rans ient per iod. In the observations where the cy l inder axis was t i l t e d greater than 1° above the horizontal the spheres reaching the " i n te r f ace " were observed to enter into the grooves between the c e l l s . A typ ica l resu l t is shown in F ig . 33 where most of the nylon spheres are found "segregated" and even p i led up in i n t e r c e l l u l a r pos i t ions . For t i l t angles smaller than 1° the spheres tended to come out of the channels with further rotat ion and move to another channel temporar i ly . From the observations and the analysis of the model the mechanism leading to i n t e r c e l l u l a r trapping can be formulated with the aid of F ig . 16(b) and (c). With respect to a system of reference f ixed to a " c e l l u l a r in ter face" the spheres oscxllTate ve r t i ca l ly with a frequency equal to the rotat ional frequency c o , and with an average pos i t i on , projected onto the " i n t e r f a c e " , that is nearly f i xed . In addit ion the spheres move towards the " i n t e r -face" at a constant average axial ve loc i t y . The d i rec t ion 92 F i g u r e 33. The nylon spheres i n the grooves between the c e l l s i n the h o r i z o n t a l mode. Photograph taken from the opposite s i d e of the " i n t e r f a c e " . oJ =0.5 r p s ; = 4°. Fi g u r e 34. Nylon spheres i n the grooves between the " c e l l s " i n the v e r t i c a l mode.Photograph taken from the opposite end to the " i n t e r f a c e " , v = 1 cm.sec -!. 93 o f the a x i a l v e l o c i t y may i n t e r s e c t the " i n t e r f a c e " a t t h r e e d i s t i n c t r e g i o n s : the g r o o v e s between the c e l l s , the f l a t p o r t i o n a t the top o f the c e l l , and the c u r v e d s u r f a c e j o i n -i n g the f l a t r e g i o n to the g r o o v e s . The i n t e r a c t i o n of the s p h e r e s w i t h the i n t e r f a c e depends on the a r e a s o f the s u r f a c e i t a p p r o a c h e s . As a s p h e r e a p p r o a c h e s the s u r f a c e i t has a v e r t i c a l v e l o c i t y V which changes to a s m a l l e r v a l u e v i m m e d i a t e l y a f t e r an i n e l a s t i c c o l 1 i s i o n , as shown i n F i g . 1 6 ( c ) . In the l e s s f a v o u r a b l e s i t u a t i o n the sphere r e a c h e s t he v e l o c i t y V upward i m m e d i a t e l y . In any case due to t h i s c o l l i s i o n the s p h e r e v e l o c i t y has d e c r e a s e d and the p a r t i c l e i s u n a b l e t o move ov e r the normal a m p l i t u d e o f o s c i l l a t i o n . In the next s t e p , and due t o the r o t a t i o n o f the c y l i n d e r , a f t e r half a r e v o l u t i o n the v e l o c i t y w i t h r e s p e c t to the c e l l t i p i s downward as shown by the v e c t o r o r i e n t e d downward i n F i g . 1 6 ( e ) . In both c a s e s , w i t h V up or down, . t h e r e i s a component o f v e l o c i t y p e r p e n d i c u l a r to the " i n t e r -f a c e " t h a t s i m u l a t e s the " i n t e r f a c e " movement. T h i s v e l o c i t y d r i v e s the s p h e r e s i n t o t h e g r o o v e s . On the o t h e r hand the s c a t t e r e d v e l o c i t y v a l s o has a component which i s i n the o p p o s i t e d i r e c t i o n . T h i s tends t o d r i v e t h e s p h e r e s out of the g r o o v e s . When t h i s outward component i s s m a l l e r than the inward v e l o c i t y component, the sphere w i l l t e n d to e n t e r i n t o t h e g r o o v e s by r e p e a t i n g the same p r o c e s s each r e v o l u -t i o n . 94 The d i r e c t i o n and magnitude o f the v e l o c i t y v depends on many f a c t o r s . Three f a c t o r s are the most r e l e v a n t f o r the model and t h e a l l metal system. The s c a t t e r i n g d i r e c t i o n i s always a f u n c t i o n o f both, the i n c i d e n t d i r e c t i o n and the p o s i t i o n on the t i p a t which c o l l i s i o n o c c u r s . The magnitude o f v i s a f u n c t i o n o f the i n e l a s t i c o r r e s t i t u t i o n c o e f f i c i e n t . A f t e r c o l l i s i o n the magnitude ->-o f V a l s o d e c r e a s e s by the damping due t o the v i s c o u s f o r c e . These w i l l be c o n s i d e r e d i n more d e t a i l when t h e r e s u l t s o f the water model are compared t o the metal system. In the water model, f o r s m a l l a n g l e s o f t i l t , the -> component o f V t h a t compensates the outward component o f v i s s m a l l . In a d d i t i o n , as the p a r t i c l e i s e n t e r i n g i n t o a groove by s u c c e s s i v e c o l l i s i o n s , the i n c i d e n t a n g l e , measured w i t h r e s p e c t t o t h e normal t o the t i p s u r f a c e a t the c o l l i s i o n p o i n t , d e c r e a s e s from 90 degrees a t the f l a t s u r f a c e t o z e r o i n the gr o o v e . There i s a r e g i o n i n between where t h e c o l l i s i o n a n g l e i s 45 d e g r e e s . At t h i s p o i n t the s c a t t e r e d v e l o c i t y v has a maximum component i n the outward d i r e c t i o n . T h e r e f o r e , when a s p h e r e r e a c h e s a p o s i t i o n i n which the outward component o f v i s g r e a t e r than the inward o f V t h i s s p h e r e comes out a g a i n . T h i s i s what i s o b s e r v e d f o r low t i l t i n g a n g l e s i n the water model. I f the s t r i k i n g d i r e c t i o n c o i n c i d e s w i t h a f l a t r e g i o n 95 on the c e l l , a p a r t i c l e can s t i l l r e a c h a c u r v e d p o s i t i o n or a groove because the a m p l i t u d e o f o s c i l l a t i o n i s o f the <-ovder o f the c e l l d i a m e t e r i n t h e water model and i n the metal system f o r the s m a l l e s t Fe p a r t i c l e . i i i ) Comparison w i t h a S o l i d i f i c a t i o n P r o c e s s . When comparing the model w i t h the r e a l s i t u a t i o n i n the m e t a l l i c system the main n e g a t i v e a s p e c t o f the mechanism d e r i v e d from the model i s t h a t i t has not c o n s i d e r e d t h a t the v e l o c i t y of the i n t e r f a c e i n the model and i n the r e a l s i t u a t i o n are d i f f e r e n t . In the model the "growth" v e l o c i t y i s ten times f a s t e r than i n the m e t a l l i c a l l o y s . T h i s means t h a t i n the metal system the d r i v i n g f o r c e p u s h i n g t h e p a r t i c l e s i n t o the grooves i s much s m a l l e r and, at f i r s t , i t may be t h o u g h t i t i s unable to compensate any outward component o f s c a t t e r e d v e l o c i t y v. However i n the r e a l p r o c e s s t h i s v e l o c i t y s h o u l d be s m a l l e r because o f the f o l l o w i n g t h r e e f a c t o r s (1) the p r e s e n c e o f drag f o r c e ( E q u a t i o n 1) p u s h i n g the p a r t i c l e a g a i n s t the i n t e r -f a c e , (2) the r e s t i t u t i o n c o e f f i c i e n t i n the metal system i s s m a l l e r than i n t h e model and (3) i n the metal system the p a r t i c l e s a r e s e v e r a l o r d e r s o f magnitude s m a l l e r and t h e r e f o r e the damping f o r c e i s much more i m p o r t a n t . A q u a n t i t a t i v e e v a l u a t i o n o f the e f f e c t o f the drag f o r c e i s d i f f i c u l t to o b t a i n because i t i s i m p o r t a n t o n l y 96 d u r i n g the c o l l i s i o n time when the p a r t i c l e i s v e r y c l o s e t o the i n t e r f a c e . The r e s t i t u t i o n c o e f f i c i e n t f o r both model and metal system has been d e t e r m i n e d u s i n g v /V , • n n where v n and are the components o f r e s u l t a n t and i n c i d e n t p a r t i c l e v e l o c i t y normal to the s u r f a c e onto which the p a r t i -c l e c o l l i d e s . The r e s u l t s o f the c a l c u l a t i o n s , g i v e n i n detail below, g i v e s v n / V n = 0.88 f o r n y l o n s p h e r e s imping-i n g on a l u c i t e s u r f a c e . The c o r r e s p o n d i n g v a l u e f o r i r o n - 2 s p h e r e s s t r i k i n g a l e a d s u r f a c e i s V n / V n - 1 0 . In l e a d t h e r e f o r e the p a r t i c l e l o s e s 99% o f i t s v e l o c i t y when s t r i k i n g the l e a d s u r f a c e , whereas the n y l o n s p h e r e l o s e s o n l y 12% a g a i n s t l u c i t e . The n e a r l y p l a s t i c b e h a v i o r of p a r t i c l e s i n l e a d d u r i n g c o l l i s i o n i s s i m i l a r to the be-h a v i o r o f l i q u i d p a r t i c l e s - c o l l i d i n g w i t h any s o l i d s u r f a c e . The r e s t i t u t i o n c o e f f i c i e n t was c a l c u l a t e d by measur-i n g the en e r g y l o s t a f t e r c o l l i s i o n . In f r e e f a l l i n g from a h e i g h t h^ a p a r t i c l e o f mass m g a i n s v e l o c i t y V . U s i n g the c o n s e r v a t i o n o f e n e r g y , t h e s e q u a n t i t i e s are r e l a t e d i n the f o l l o w i n g way 1 2 mgh, = T m V ... 37 • 1 2 n A f t e r c o l l i d i n g p e r p e n d i c u l a r t o a p l a n a r i n t e r f a c e the r e s u l t a n t v e l o c i t y i s v n , the k i n e t i c e nergy t r a n s f o r m s to p o t e n t i a l and the p a r t i c l e r e a c h e s a maximum h e i g h t h 9 97 l e s s than h 1 - A g a i n , by c o n s e r v a t i o n o f energy mgh 2 = j m . . .38 D i v i d i n g E q u a t i o n s 37 and 38 and t a k i n g square r o o t i t i s o b t a i n e d t h a t With t h i s e x p r e s s i o n the r e s t i t u t i o n c o e f f i c i e n t can be d e t e r m i n e d by mea s u r i n g the h e i g h t s h 2 a p a r t i c l e o r s p h e r e rebounds a f t e r s t r i k i n g a s u r f a c e o r t h o g o n a l l y . Rebound ; h e i g h t s were measured f o r both n y l o n s p h e r e s s t r i k i n g l u c i t e and Fe p a r t i c l e s s t r i k i n g l e a d . The n y l o n s p h e r e s were dropped from h e i g h t s o f 10 and 20 cm a g a i n s t l u c i t e . For the case o f i r o n a g a i n s t l e a d , s t e e l s p h e r e s 0.32, 0.4 and 0.48 cm were dropped from the same h e i g h t s as the n y l o n s p h e r e s a g a i n s t a p o l i s h e d l e a d s u r f a c e . In t h i s case the v a l u e o f h 2 / h ^ appeared to be i n d e p e n d e n t o f sphe r e d i a m e t e r and i n i t i a l h e i g h t . The damping f o r c e on the Fe p a r t i c l e s r e s u l t s i n the p a r t i c l e s s t o p p i n g much f a s t e r and i n a s h o r t e r d i s -t a n c e than the n y l o n s p h e r e s . T h i s d i s t a n c e and time f o r i r o n p a r t i c l e s i n l e a d can be a n a l y s e d i n the f o l l o w i n g way. . . .39 A f t e r a c o l l i s i o n has o c c u r r e d the dynamics o f the 98 p a r t i c l e i s governed by the e q u a t i o n m ^ = —6TTTI av" ... 40 where n i s the v i s c o s i t y , a i s the p a r t i c l e r a d i u s , m i s 3 the p a r t i c l e mass = 4 / 3 H P where p i s the p a r t i c l e d e n s i t y The r i g h t hand s i d e o f the e q u a t i o n i s the S t o k e s v i s c o u s drag v a l i d f o r s m a l l p a r t i c l e s . I t i s assumed t h a t the v e l i c i t y v i s h o r i z o n t a l and t h e r e f o r e g r a v i t y does not a c t i n t h i s d i r e c t i o n . E q u a t i o n 40 has the s i m p l e s o l u t i o n _ i t 2 v = v.e 2 p a ...41 where v. i s the i n i t i a l v e l o c i t y and the q u a n t i t y 2 T = TT -— i s a c h a r a c t e r i s t i c time at which the v e l o c i t y 9 n d e c r e a s e s to 1/e = 36.7% o f i t s i n i t i a l v a l u e . I n t e g r a t i n g a g a i n E q u a t i o n 40, the d i s t a n c e t r a v e l l e d by the p a r t i c l e a f t e r c o l l i s i o n on t h i s T i s X = v. §- (1 - h . . .42 x i 9 n e In the l e s s f a v o u r a b l e s i t u a t i o n the s c a t t e r i n g i s com-p l e t e l y e l a s t i c and v^ i s the v e l o c i t y j u s t b e f o r e c o l -l i s i o n ; i . e . the t e r m i n a l v e l o c i t y g i v e n by E q u a t i o n 34. TABLE XV Terminal Velocity, Transient Time, Transient Length for Fe P a r t i c l e s in Liquid Lead Assuming Stokes Drag Force Applies. 1 i Radius ( p i ) V f(cm/sec) = 3.78xl0 - 4a 2 T(sec) = 1.0246xl0" 6a 2 X = 2.45xl0" 1 0a 4 T 2 : T.51xl0" 3 4.1xl0" 6 0.39 A 0 4 6.06xl0" 3 1.64xl0~ 5 6.27 A° 6 1.36xl0" 2 3.7xl0" 5 31.7 A° 8 2.42x10"2 6.5xl0" 5 100.4 A° 10 3.78xl0" 2 l.OxlO" 4 245.2 A° 12 5.45xlQ - 2 1.5xl0" 4 508.4 A 0 14 7.42xl0" 2 2.0xl0" 4 941:9 A° 16 9.69xl0~ 2 2.6xl0" 4 0.16 ym 18 1.23X10"1 3.3xlQ" 4 0.26 ym 100 Subst i tut ing this ve loc i ty in Equation 42 gives 4 X T 81 9 • n.2: u e J ' * ' 4 3 Calculat ing X t from Equation 43 for l i qu id lead and iron _ 2 par t i c l es with g = 981 cm see , n = 1.67 mPa.s, p = 7.7 -3 -3 gr cm > P-J = 10-. 3. gir cm , this distance is X = 2 .45x l0~ 1 0 a 4 . . .44 T with a . i n ym X is given in cm. This function is tabu-T lated in Table XV together with the y values. For instance for a pa r t i c l e of radius a = 4 ym,' the' t ransient time and _ o length are X' = 6.3x10" cm and x = 16.4 yisec. From these resul ts i t may be concluded that once a pa r t i c l e moves into the i n t e r c e l l u l a r band by the mechanism described previously i t is very d i f f i c u l t for i t to come out again. The c o l l i s i o n stops the pa r t i c l e almost ins tant ly and at the same c o l l i s i o n s i t e . b) Ver t i ca l S o l i d i f i c a t i o n In th is mode of " s o l i d i f i c a t i o n " the resul ts are more obvious but not less important than in horizontal " s o l i d i f i c a t i o n " . Changing the density of the solut ion by addit ion of NaP.lt the ve loc i t i e s of the nylon spheres were varied in the range from 1 cm sec " 1 to 0.05 cm s e c " 1 . This lower l im i t was very d i f f i c u l t to adjust because of the small density di f ference between nylon spheres and so lu t ion . Any casual perturbation disturbed the system, (bj Particle Velocity F i g u r e 35 . S e m i - q u a l i t a t i v e d e s c r i p t i o n of the s e g r e g a t i o n of n y l o n spheres fo r the v e r t i c a l experiments w i t h the model . I n a) the r e l a t i o n i n areas i s c a l c u l a t e d . In b) the behaviour o f the spheres fo r s t i l l andconvec t ive l i q u i d are compared w i t h the e x p e r i m e n t a l o b s e r v a t i o n s . 102 t h e f l o w p a t t e r n made t h e m e a s u r e m e n t o f t h e v e l o c i t y d i f -f i c u l t . A t y p i c a l d i s t r i b u t i o n o f s p h e r e s o b t a i n e d i n t h e s e e x p e r i m e n t s i s shown i n F i g . 34. The s p h e r e s a r e f n 1 t h e g r o o v e s . The r e s u l t s a r e q u a l i t a t i v e l y p r e s e n t e d i n F i g . 35. To u n d e r s t a n d them,two o p p o s i n g s i t u a t i o n s a r e a n a l y s e d . F o r a l i q u i d a t r e s t i t i s r e a s o n a b l e t o assume t h a t t h e p a r t i c l e d i s t r i b u t i o n , d i s c r i m i n a t i n g o n l y b e t w e e n t h e i n t e r c e l l u l a r g r o o v e s and c e l l f a c e s , w i l l be d i c t a t e d by t h e r e l a t i o n b e t w e e n t h e g r o o v e and c u r v e d p a r t o f t h e t o p s u r f a c e a r e a s , and f a c e s u r f a c e a r e a s . In o t h e r w o r d s , a p a r t i c l e a p p r o a c h i n g a f l a t a r e a w i l l s t o p t h e r e w h i l e t h o s e c o m i n g t o a c u r v e p a r t o f a c e l l o r a c h a n n e l w i l l f i n a l l y end up a t t h i s p o i n t on t h e s u r f a c e . The p r o p o r t i o n o f f l a t f a c e d a r e a o f t h e c e l l t o t o t a l a r e a , a c c o r d i n g t o t h e d i m e n s i o n s o f t h e m o d e l , i s 0.37. T h i s means t h a t i n a s t i l l l i q u i d t h e p e r c e n t o f p a r t i c l e s f o u n d i n t h e c e l l s s h o u l d be 37%. T h i s was o b s e r v e d e x p e r i m e n t a l l y by r e l e a s -i n g p a r t i c l e s a t t h e b o t t o m o f t h e c y l i n d e r . The i m p o r t a n c e o f c o n v e c t i o n was d e m o n s t r a t e d by i n -t r o d u c i n g a v e r y s l o w c o n v e c t i v e m o t i o n i n t h e l i q u i d w i t h a s m a l l t e m p e r a t u r e g r a d i e n t when t h e s p h e r e s were r e s t i n g a g a i n s t t h e l u c i t e c e l l s u r f a c e . The s m a l l c u r r e n t d i s -p l a c e d t h e s p h e r e s t o p o s i t i o n s where t h e g r a v i t y f o r c e c o u l d move t h e s p h e r e s i n t o t h e g r o o v e s . The r e s u l t s o f 103 a d e t a i l e d s t u d y o f the e f f e c t o f c o n v e c t i o n on the s e g r e g a -t i o n o f the n y l o n s p h e r e s i n the model cannot be e x t r a -p o l a t e d t o the metal system f o r two main r e a s o n s . One r e a s o n , which may be the most i m p o r t a n t , i s r e l a t e d to the s i z e o f the p a r t i c l e s and the boundary l a y e r . The t h i c k n e s s -1 -3 o f the boundary l a y e r i s o f the o r d e r o f 10 to 10 cm which i s always s m a l l e r than the sphere s i z e i n the model. The s p h e r e s w i l l always be a f f e c t e d by c o n v e c t i o n i n the l i q u i d . In the metal system the p a r t i c l e s a r e o f the o r d e r o f the t h i c k n e s s o f the boundary l a y e r . Depending on the f l o w c o n d i t i o n s t h e r e may be p a r t i c l e s t h a t a r e not a f f e c t e d by c o n v e c t i o n . The e s t i m a t i o n o f the t h i c k n e s s o f the boundary l a y e r f o r the metal system i s d i f f i c u l t . The e x p r e s s i o n f o r t h i s l e n g t h i s g i v e n as 6^ = 1.7 (^p)' 2 4 2 f o r a boundary l a y e r d e v e l o p e d o v e r a f l a t s u r f a c e or a body i n a f l u i d o f k i n e m a t i c v i s c o s i t y v moving at v e l o c i t y U. - 3 For l e a d v = 1.57x10 cgs u n i t s and U may be e s t i m a t e d from the p e r i o d i c v a r i a t i o n i n t e m p e r a t u r e r e c o r d e d d u r i n g the v e r t i c a l growths as - 1 cm s e c " ' . For a p l a n a r i n t e r -f a c e i t may be assumed t h a t the boundary l a y e r d e v e l o p s a l o n g the d i a m e t e r o f the mould (~1 cm); t h i s g i v e s a maximum v a l u e o f 680 ym f o r the t h i c k n e s s w i t h i n which the p a r t i c l e s are not a f f e c t e d by c o n v e c t i o n . For a c e l l u l a r i n t e r f a c e t h e boundary l a y e r may be assumed to d e v e l o p o v e r the c e l l d i a m e t e r X~10 cm, i n t h i s case S, = 68 ym. 104 However, t h i s t h i c k n e s s may be s m a l l e r because the c e l l t i p i s c u r v e d and as a r e s u l t the f l a t p a r t o f the t i p i s s m a l l e r than 10 cm. In any e v e n t i t i s e x p e c t e d t h a t some p a r t i c l e s w i l l be a f f e c t e d by c o n v e c t i o n w h i l e o t h e r s w i l l not depending on t h e p a r t i c l e s i z e and i t s p o s i t i o n i n the t i p . The second r e a s o n p r e v e n t i n g a d i r e c t c o m p a r i s o n o f the model t o the metal system i s the d i f f e r e n c e i n the P r a n d t l number between the two systems. For a g i v e n d r i v i n g f o r c e f o r thermal c o n v e c t i o n the f l o w o b t a i n e d i n each case d i f f e r s s u b s t a n t i a l l y . ^ " ^ With the t e c h n i q u e o f t u r n i n g the c y l i n d e r w i t h the sp h e r e s i n i t i a l l y a t the t o p , the d i s t r i b u t i o n o f s p h e r e s as a f u n c t i o n o f v e l o c i t y i s g i v e n by the f u l l l i n e i n F i g . 3 5 ( b ) . The t r a n s i t i o n e x h i b i t e d i n t h i s c u r v e , from s t i l l t o c o n v e c t i v e , o c c u r s at v e r y low r a t e s o f a p p r o x i -m a t e l y 0.08-0.1 cm s e c " 1 . At t h i s speed the time taken by the s p h e r e s to r e a c h the o t h e r end ( t h e " i n t e r f a c e " ) was l o n g enough f o r c o n v e c t i o n to decay i n the l i q u i d to a minimum due t o v i s c o u s f o r c e s . From the r e s u l t s i n the metal system and the o b s e r v a -t i o n s made i n the model,the mechanism f o r the s e g r e g a t i o n 105 o f p a r t i c l e s i n the metal system i s proposed to o p e r a t e i n the f o l l o w i n g way: p a r t i c l e s s t r i k e the c e l l u l a r o r d e n d r i t i c i n t e r f a c e . I f the s t r i k i n g d i r e c t i o n c o i n c i d e s w i t h the c u r v e d p a r t o f a t i p the p a r t i c l e moves f u r t h e r i n t o the i n t e r s t r u c t u r a l p o s i t i o n as f a r as the g r a v i t y f o r c e i s not e q u i l i b r a t e d . I f the s t r i k i n g d i r e c t i o n c o i n -c i d e s w i t h the f a c e o f the t i p the p a r t i c l e w i l l s t o p t h e r e . T h i s s t o p may be temporary f o r p a r t i c l e s l a r g e r than the t h i c k n e s s o f the boundary l a y e r . In t h i s case the p a r t i c l e s are dragged to p o s i t i o n s on which the g r a v i t y f o r c e s c o u l d move the p a r t i c l e s to i n t e r s t r u c t u r a l p o s i t i o n s . Small p a r t i c l e s a r e 1 i k e l y t r a p p e d where they s t r i k e the i n t e r f a c e because t h e y a r e not a f f e c t e d by c o n v e c t i o n and a l s o because of t h e i r low m o b i l i t y . T h i s mechanism f o r the s e g r e g a t i o n o f p a r t i c l e s s h o u l d be more e f f e c t i v e when the i n t e r f a c e has s h a r p e r t i p s l i k e d e n d r i t e s . T h i s was noted p r e v i o u s l y when the average number o f p a r t i c l e s i n the m a t r i x dropped from 16.7 to 10% by c h a n g i n g the i n t e r f a c e shape from c e l l u l a r to d e n d r i t i c . 3. The ^ C a s t i n g E x p e r i m e n t The c a s t i n g e x p e r i m e n t produced v e r y w e l l d e v e l o p e d d e n d r i t e s . The d e n d r i t e s were l e a d r i c h and the i n t e r -d e n d r i t i c r e g i o n c o n t a i n e d l e a d - t i n e u t e c t i c . The p a r t i c l e s were s e g r e g a t e d p r i m a r i l y i n the i n t e r d e n d r i t i c r e g i o n s as shown i n F i g . 3 6 ( a ) . The s i z e and p o s i t i o n o f the (b) 107 (d) 108 F i g u r e 36. C a s t i n g experiment, a) and b) P a r t i c l e s trapped i n w e l l developed dendrites.200X and 500X resp. c) and d ) P a r t i c l e s at the bottom part of the dendrites.500X. d)A p a r t i c l e at the top of the dendrite.500X. And e)a p a r t i c l e moving f r e e i n the eutectic.500X. 109 Table XVI P a r t i c l e Size Distribution for Cast Sample. (a) Bottom, (b) 5 mm form the bottom, \. (c) 15 mm'from bottom, (d)-25 mm from bottom, (e) top of sample. (a) Bottom (b) 5 mm'from Bottom Size, u Segregated Matrix Segregated Matrix 4 3 i i ! 2 2 6 11 2 \ j 13 2 8 23 1 1' 22 1 10 37 1 23 1 12 23 0 i 29 2 14 36 0 ! 42 2 16 15 0 i 35 0 18 19 i o : 25 0 20 29 0 29 0 22 16 0 : 13 0 24 14 0 ; 16 0 26 8 0 12 0 28 3 0 3 0 30 4 0 3 0 32 1 0 0 0 TOTAL 242 6 267 10 Density 45 mm 50 mm % in matrix 2.4% 3 / 6 % n o TABLE XVI (cont.) / ( c) 15'..' mm from'Bottom Bottom or Size , v Trapped Eutectic Top Total 1 Matrix Total 4 0 0 0 0 0 6 5 1 1 7 1 8 8 6 5 1 12 12 10 21 2 2 25 25 12 26 2 3 31 31 14 20 11 0 31 31 16 18 8 2 28 28 18 22 4 1 28 V 2- 19 1 4 23 23 22 .8 5 1 14 14 24 <9 1 0 10 10 26 5 1 0 6 ';6 28 5 0 0 5 5 30 2 0 0 2 2 32 4 0 0 4 4 TOTAL 170 41 !4 225 1 226 % bottom or trapped 75.2% % on eutectic 18.1% density % on top 6.2% 41 mm-2 % dendrite 0.4% TABLE XVI (cont.) I l l (d) 25 mm from Bottom S i z e , ^ Bottom or Trapped Eutectic Top Total 1 Dend 4 0 1 1 6 13 2 15 8 8 3 11 10 18 2 20 12 8 4 2 14 14 11 3 1 15 16 10 1 0 11 18 6 1 0 7 20 15 4 1 20 22 5 2 7 24 6 1 7 26 5 0 5 28 3 0 3 30 1 1 2 32 0 Q 0 TOTAL 109 25 4 138 0 % bottom or trapped 79.0% % eutectic 18.1% density "I" top 2.9% 25 mm"2 % dendrite 0% 112 TABLE XVI (cont.) (e) 3 mm from Bottom - Top of Sample S i ze j u Bottom or Eutectic Top Total T Matrix Total Trapped 4 0 0 0 0 0 6 5 1 0 6 6 8 5 1 0 6 1 7 ID 14 4 0 18 18 12 13 7 0 20 1 21 14 19 2 0 21 21 16 10 3 2 15 15 18 9 1 1 11 11 20 10 3 2 15 15 22 7 1 0 8 8 24 6 2 1 9 9 26 5 2 0 7 7 28 3 0 0 3 3 30 2 0 0 2 2 32 1 0 0 1 1 TOTAL 109 27 6 142 144 % bottom or trapped % eutectic % top % dendrite 75.7% 18.7% density 4.2% _c -2 26 mm 1.4% 604 — to ta l n T = 248 n/A = 45 m 113 to OJ O Q. O 40-2 0-0: II 19 27 j q j Size, jJLm 35 to o o Q-o o' 604 CO ^ 4 0 + o CL — t o t a l in eutect ic n T =226 T -2 n/A = 41 mm o d 20 + i — r " I i i _ , i 1 19 27 S i z e , yU,m 35 (c) CO QJ u 40+ O CL — to ta l - - in eutec tic n T = 138 _ 2 n/A = 26 mm o 2 0 + 0: l — S n r - n I , 19 27 S i z e , jM m ( d ) 35 115 £40+ o CL 20-o d I 1 L 0 4= 4 — totol in eutectic n T = 144 1 -2 n/A = 26 mm 19 27 35 Size , / m m (e) F i g u r e 37. P a r t i c l e s i z e d i s t r i b u t i o n f o r the cast sample, a) and b) t o t a l number of p a r t i c l e s at the bottom of the sample and 0.5 cm from the bottom. c),d) and e) t o t a l namber of p a r t i c l e s and i n the e u t e c t i c at 1.5; 2.5; and 3.0 cm from the bottom of the sample r e s p e c t i v e l y . 116 TABLE XVII P a r t i c l e Size Distribution for Heights 15 mm, 25 mm and 30 mm for Bottom or Trapped P a r t i c l e s and Par t i c l e s in the Eutectic. Size, y Bottom or . Trapped Eutectic % in Eutectic 3-7 30 5 14.3 7-11 93 17 15.4 11-15 133 29 17.9 15-19 99 18 15.4 19-23 87 16 15.5 23-27 44 7 13.7 27-31 17 1 16.7 31-35 5 0 TOTAL 508 93 15.5 117 p a r t i c l e s were measured at the bottom o f the s o l i d and a t 0.5, 1.5, 2.5 and 3 cm from the bottom, the l a s t p o s i t i o n b e i n g c l o s e to the top o f the sample. In the f i r s t two r e g i o n s the p a r t i c l e s were s e p a r a t e d i n t o two p o s i t i o n g r o u p s , t h o s e i n the d e n d r i t i c c o r e or b r a n c h and those i n the e u t e c t i c . The r e s u l t s a r e g i v e n i n T a b l e s XVI (a) and ( b ) . The r e s u l t s i n the t a b l e s show t h a t o n l y a few p a r t i -c l e s a r e p r e s e n t i n the d e n d r i t i c m a t r i x , s p e c i f i c a l l y 2.4% and 3.6% at the bottom arid a t 0.5 cm above the bottom. Examining the p a r t i c l e s i n the i n t e r d e n d r i t i c r e g i o n i n more d e t a i l , the p a r t i c l e p o s i t i o n s were s u b d i v i d e d i n t o t h r e e groups t a k i n g c a r e to note the v e r t i c a l p o s i t i o n s o f the p a r t i c l e s , as w e l l as the p o s i t i o n s on the p o l i s h e d f a c e s : (a) P a r t i c l e s t r a p p e d between the d e n d r i t e arms and i n the bottom p a r t o f the d e n d r i t e s as shown i n F i g . 36 ( a - d ) . (b) P a r t i c l e s t h a t were at the top o f the d e n d r i t e s , some of them p a r t i a l l y e n g u l f e d as shown i n p hotograph 3 6 ( d ) . (c) P a r t i c l e s i n e u t e c t i c r e g i o n s not i n c o n t a c t w i t h the d e n d r i t e s ( F i g . 3 6 ( e ) ) . These p a r t i c l e s were f r e e to move p r e v i o u s to the e u t e c t i c s o l i d i f i c a t i o n . D i v i d i n g the number of p a r t i c l e s o b s e r v e d i n the i n t e r d e n d r i t i c r e g i o n i n t o t h e s e t h r e e groups g i v e s the r e s u l t s shown i n T a b l e s XVI (c) , (d) and ( e ) . The c o r r e s p o n d i n g d i s t r i b u t i o n o f p a r t i c l e s as a f u n c t i o n o f s i z e from the t a b l e s a r e p l o t t e d i n F i g u r e s 37 ( a - e ) . 118 The r e s u l t s i n d i c a t e s t h a t the p a r t i c l e s e n t r a p p e d by the d e n d r i t e arms were caught t h e r e a t an e a r l y s t a g e o f s o l i d i f i c a t i o n , F i g . 3 6 ( a ) . The p a r t i c l e s t h a t managed to s t a y i n the l i q u i d moved w i t h i t , due t o c o n v e c t i o n , and w i t h r e s p e c t t o i t , due t o buoyancy f o r c e s . Some o f the p a r t i c l e s are s t o p p e d by the d e n d r i t e s and remain i n t h i s p o s i t i o n , i n the bottom o f the d e n d r i t e . Other p a r t i c l e s , about 15% o f the t o t a l , can s u r v i v e and c o n t i n u e ; t o move w i t h i n the e u t e c t i c l i q u i d . Once a p a r t i c l e i s s t o p p e d by a d e n d r i t e arm i t may be d e t a c h e d from the d e n d r i t e by the c o n v e c t i v e l i q u i d and move to a new p o s i t i o n . There are s e v e r a l s o u r c e s f o r c o n v e c t i o n : f o r c e d c o n v e c t i o n i s c r e a t e d when p o u r i n g the m e l t . N a t u r a l con-v e c t i o n a r i s e s by n o n - s t a b l e d e n s i t y g r a d i e n t s due to t e m p e r a t u r e g r a d i e n t s and to the s o l u t e g r a d i e n t b u i l t up, as s o l u t e i s r e j e c t e d d u r i n g growth o f a d e n d r i t e . A n o t h e r cause of c o n v e c t i o n may be the d i s p l a c e m e n t o f l i q u i d as d e n d r i t e s (more dense) f a l l down. T h i s may have caused n e g l i g i b l e c o n v e c t i o n s i n c e i f a l a r g e number o f d e n d r i t e s f a l l t o t h e bottom d u r i n g s o l i d i f i c a t i o n , the top of the i n g o t s h o u l d m a i n l y be o f e u t e c t i c c o m p o s i t i o n which was not o b s e r v e d i n the p r e s e n t c a s t i n g . Moreover, i f d e n d r i t e s f a l l i n g to the bottom o f the c a s t i n g was i m p o r t a n t the number o f p a r t i c l e s s h o u l d be h i g h e r at to c QJ Q o-l— 1 1 1 — 0 10 2 0 3 0 Height , mm Figure 38. Density of p a r t i c l e s vs. height i n the cast sample. 120 the bottom of the sample. S u r p r i s i n g l y , t h i s was o b s e r v e d , t h a t i s t h e r e was a l a r g e r number of p a r t i c l e s i n the bottom h a l f o f the c a s t i n g than i n the upper h a l f , as shown i n F i g . 38. However, t h i s d e p l e t i o n can be a t t r i b u t e d to f l o t a t i o n o f p a r t i c l e s from t h i s r e g i o n t o the top r a t h e r than the f a l l i n g o f d e n d r i t e s . F i g . 39 i s a view o f the top p a r t o f the i n g o t showing a v e r y h i g h d e n s i t y o f p a r t i c l e s . F l o t a t i o n took p l a c e when the sample was s t i l l c o m p l e t e l y l i q u i d s i n c e t h i s top l i q u i d s o l i d i f i e d when the p a r t i c l e s were a l r e a d y t h e r e -as can be c o n c l u d e d from F i g . 39. The p r e s e n c e o f 15.5% o f the t o t a l p a r t i -c l e s i n the e u t e c t i c r e g i o n s i s c o n c l u s i v e p r o o f t h a t the l i q u i d was moving. I f t h i s was not the s i t u a t i o n a l l the p a r t i c l e s s h o u l d have been i n the bottom p a r t o f the d e n d r i t e s . The phenomenon of p u s h i n g cannot be c o n s i d e r e d as the f a c t o r l e a d i n g to t h i s s e g r e g a t i o n because o f the f o l l o w -i n g f a c t s : (1) the random d i s t r i b u t i o n o f s i z e s i n the e u t e c t i c ; (2) t h e s m a l l amount (4.5%) o f the p a r t i c l e s i n the top o f the d e n d r i t e s and (3) the p r e s e n c e o f a few s m a l l p a r t i c l e s i n the d e n d r i t e s ( F i g . 3 6 ( b ) ) . The second f a c t i n d i c a t e s t h a t the c a p t u r e i s s t r o n g enough to compen-s a t e the buoyancy f o r c e . The most p r o b a b l e cause f o r the t h i r d o b s e r v a t i o n may be the low m o b i l i t y o f the s m a l l p a r t i c l e s w i t h r e s p e c t to the l i q u i d which enhanced t h e i r 1 2 1 Figure 39. Photograph of the top part of the cast sample. The high concentration of p a r t i c l e s i s a t t r i b u t e d to f l o t a t i o n . This part of the sample s o l i d i f i e d when the p a r t i c l e s were already there. 122 e n g u l f m e n t by the growing d e n d r i t e s . In a d d i t i o n the d e n d r i t e s may have n u c l e a t e d on the p a r t i c l e s . 4. P u s h i n g i n Mater When water c o n t a i n i n g Cr p a r t i c l e s was s o l i d i f i e d the b e h a v i o r of the p a r t i c l e s was c o m p l e t e l y d i f f e r e n t than t h a t o b s e r v e d p r e v i o u s l y f o r p a r t i c l e s i n l e a d . The r e s u l t s f o r water which was f r o z e n i s g i v e n i n F i g . 40. F i g . 40(a) shows the i n i t i a l q u a s i - s u s p e n s i o n o f Cr p a r t i -c l e s i n water. The u n i f o r m g r e y appearance means t h a t the p a r t i c l e s were d i s t r i b u t e d homogeneously i n the m e l t . F i g . 40(b) c l e a r l y shows the a d v a n c i n g i n t e r f a c e by a d i f -f e r e n c e i n g r e y between s o l i d and l i q u i d due t o the h i g h e r d e n s i t y o f p a r t i c l e s i n the l i q u i d . A f t e r the e n t i r e sample had s o l i d i f i e d the p a r t i c l e s pushed by the i n t e r f a c e were i n a dark band on the c e n t e r as shown i n F i g . 4 0 ( c ) . The r e s u l t o f a s i m i l a r e x p e r i m e n t but w i t h a h i g h e r d e n s i t y o f p a r t i c l e s i s shown i n F i g . 40(d) which shows a w i d e r dark band i n the c e n t e r . The f a c t t h a t the i n t e r f a c e moved h o r i z o n t a l l y and the p a r t i c l e s are f a l l i n g down enhanced p u s h i n g . S i m i l a r o b s e r v a t i o n s o f p a r t i c l e p u s h i n g have never been o b s e r v e d i n m e t a l l i c systems under s i m i l a r c o n d i t i o n s and a t v e r y low s o l i d i f i c a t i o n r a t e s . T h e r e f o r e the water f r e e z i n g (a) (b) Figure 40. Pushing i n water, a) The l i q u i d with the suspension of Cr p a r t i c l e s , b) The ice-water i n t e r f a c e advancing to the center, c) Dark band of p a r t i c l e s pushed to the center by the advancing f r o n t , d) Similar to c) but with a higher density of p a r t i c l e s i n the l i q u i d which r e s u l t s i n a wider band. 124 e x p e r i m e n t s c l e a r l y i n d i c a t e t h a t p a r t i c l e s i n w a t e r b e h a v e e n t i r e l y d i f f e r e n t l y t h a n p a r t i c l e s i n a m e t a l s y s t e m . A s o l i d i f y i n g w a t e r i n t e r f a c e r e j e c t s p a r t i c l e s , a m e t a l l i c i n t e r f a c e does n o t . 5. The L i f s h i t z - V a n d e r Waals F o r c e In p r e v i o u s s e c t i o n s i t has been d e m o n s t r a t e d t h a t t h e s e g r e g a t i o n o f p a r t i c l e s a t a n o n - p l a n a r m e t a l l i c s o l i d -l i q u i d i n t e r f a c e i s due t o a m e c h a n i s m i n w h i c h t h e b u o y a n c y f o r c e s and c o n v e c t i o n i n t h e l i q u i d a r e t h e d r i v i n g f o r c e s r e s p o n s i b l e f o r p a r t i c l e s e g r e g a t i o n , r a t h e r t h a n a p u s h i n g m e c h a n i s m . M o r e o v e r , t h e p u s h i n g m e c h a n i s m has been shown t o be a b s e n t f o r t h e Fe-Pb s y s t e m e m p l o y e d . However, t h e two main t h e o r i e s p r e s e n t e d i n C h a p t e r I I p r e d i c t e d r e j e c t i o n o f p a r t i c l e s a t g r o w t h v e l o c i t i e s a t w h i c h t h e s a m p l e s have a p l a n a r f r o n t . The s e a r c h f o r an e x p l a n a t i o n o f t h i s a p p a r e n t c o n t r a d i c t i o n l e d t o t h e o n l y p h y s i c a l f o r c e t h a t can be r e s p o n s i b l e f o r t h e r e j e c t i o n o f p a r t i c l e s . T h i s i s t h e L i f s h i t z - V a n - d e r W a l l s f o r c e . A t s h o r t d i s -t a n c e s t h e s i g n o f t h i s f o r c e i s d e t e r m i n e d by t h e c o n s t a n t to g i v e n by e x p r e s s i o n 19. In o r d e r t o c a l c u l a t e to, t h e Drude f o r m u l a e f o r t h e d i e l e c t r i c c o n s t a n t a r e u s e d . T h e s e f o r m u l a e have been 43-4 d e m o n s t r a t e d t o f i t r e a s o n a b l y w e l l f o r many l i q u i d m e t a l s 125 46 and p a r t i c u l a r l y f o r l e a d . The Drude e x p r e s s i o n s f o r the r e a l and i m a g i n a r y p a r t o f the c o n s t a n t s are 2 2 2 1 - T 4-rrne T 11 _ 4Trne T . A K £ I _ 2 2 £ — 2 2 • • • ^ v m(l+ o o x ) mco(1 +o) T ) (a) (b) where n - number o f v a l e n c e e l e c t r o n s per u n i t volume e = e l e c t r o n charge m = e l e c t r o n mass T = r e l a x a t i o n time i n d e p e n d e n t o f f r e q u e n c y . To f i t e x p e r i m e n t a l measurements the c a r r i e r d e n s i t y n i s u s u a l l y r e p l a c e d by an e f f e c t i v e d e n s i t y . 48 49 The same f o r m u l a e a r e used f o r s o l i d s ' w i t h the d i f f e r e n c e t h a t the p r o d u c t C O T ~ 1 , w i t h co i n the v i s i b l e f r e q u e n c i e s f o r l i q u i d and i n the I.R. f o r s o l i d s . In a d d i t i o n , f o r s o l i d m e t a l s the e f f e c t i v e e l e c t r o n mass m* sh o u l d be used. The v a l u e cop g i v e n by 2 2 4irne » c co _ = — — . . . 4 6 p m i s the pi asma f r e q u e n c y of the metal:. At t h i s v a l u e the metal becomes t r a n s p a r e n t to r a d i a t i o n . 126 S u b s t i t u t i n g t h i s v a l u e o f tOp and r e p l a c i n g to f o r H , where .5 i s the pure i m a g i n a r y p a r t o f the f r e q u e n c y , the c o n s t a n t i s 2 2 CO T e(ic-) = 1 + - p 2 z ...47 1 + £ T which f o r h i g h f r e q u e n c i e s , such t h a t £x>> 1 E q u a t i o n 45 t a k e s the form to 2 e ( U ) = 1 + - 4 - ...48 Us i n g the same f o r m u l a f o r the t h r e e media and — 19 e v a l u a t i n g to t h r o u g h the i n t e g r a l the f o l l o w i n g e x p r e s s i o n i s o b t a i n e d : u -2 .IT p l 3 p23 8 _ A 2 2 " r P 2 3 ~ r P i 3 A Q. to = A t o „ 1 0 Ato„oo — — c ' ...49 2 .2 ,,2 2 , c op23 "plS- U p 2 3 " p ^ where A 2 2 2 Ato . . = to . - to . P 1 J PI PJ and 2 _ 1 ,. 2 . ? . tO . . - TT I CO • + C 0 . 1 P I J 2 pi p;r The f r a c t i o n , or second f a c t o r i n f o r m u l a 49 i s always p o s i t i v e . T h e r e f o r e the s i g n o f to i s g i v e n by the f i r s t f a c t o r i n A. U s i n g the v a l u e s o f u>p. g i v e n i n Appendix I, f p r the system F e - P b ( s ) - P b ( l )„ to i s g r e a t e r than z e r o , i . e . ii = 1. 90x1 0 1 5 s e c " 1 127 which g i v e s f o r the c o n s t a n t Bg the v a l u e B 3 = 2 . 5 4 x l 0 " 1 4 e r g . T h i s i s o f the o r d e r o f the v a l u e s which appear i n the l i t e r a t u r e / 1 0 ' 1 7 ' 1 8 o b t a i n e d t h r o u g h c a l c u l a t i o n s or e x p e r i m e n t a l measurements. However, the most i m p o r t a n t r e s u l t i s t h a t t h e s e c o n s t a n t s co and Bg are p o s i t i v e ; i . e . the L i f s h i t z - V a n der Waals f o r c e i s a t t r a c t i v e . T h i s , t h e r e f o r e , i s i n a g r e e -ment w i t h the e x p e r i m e n t a l o b s e r v a t i o n which showed no s i g n o f p u s h i n g a t v e r y low s o l i d i f i c a t i o n r a t e s w i t h a p l a n a r i n t e r f a c e . I t i s n e c e s s a r y to note t h a t the v a l u e s o f plasma f r e q u e n c i e s f o r the s o l i d s were c a l c u l a t e d u s i n g d a t a o b t a i n e d a t room t e m p e r a t u r e because t h e r e i s no d a t a a v a i l -a b l e f o r l e a d a t the m e l t i n g p o i n t nor f o r i r o n a t the same t e m p e r a t u r e . However f o r N i , a t r a n s i t i o n metal l i k e i r o n , the a b s o r p t i o n spectrum changes o n l y s l i g h t l y 51 on h e a t i n g t h r o u g h the X u r i e p o i n t , which may s u g g e s t the same b e h a v i o r f o r i r o n . C o n s e q u e n t l y s l i g h t changes on i t s o p t i c a l p r o p e r t i e s are e x p e c t e d . The p r i n c i p a l problem to e v a l u a t e the i n t e r a c t i o n a t a s o l i d - l i q u i d i n t e r -f a c e r e s i d e s i n the f a c t t h a t the o p t i c a l p r o p e r t i e s o f the 128 52 so l id at the melting point are not known. M i l l e r re -ported values of the opt ica l conduct iv i ty a ('to-) for l i qu id and so l i d Co. This o is related to the imaginary part of the d i e l e c t r i c constant by a ( to) = - ( t o / 4 T r ) e " , / t ° ) The values of a for so l i d and l i qu id Co at the melting point are within the experimental e r ro r , but the lack of data at room temperature for the same material enable us to conclude a convergence of the opt ica l propert ies at the melting point . ."However the facts that the absorption peaks typ ica l of a metal in the so l id state are less 46 pronounced and broader as the temperature increases and that these peaks are not present in the l i qu id s ta te , sug-gest that the opt ica l propert ies converge to a common value at the melting point . In such cases the L i f s h i t z - V a n der Walls should be negligible-' at. a so l i d i f y i ng in te r face . 129 C h a p t e r VI SUMMARY AND CONCLUSIONS Lead-antimony a l l o y s c o n t a i n i n g i r o n p a r t i c l e s were s o l i d i f i e d u s i n g a zone r e f i n i n g t e c h n i q u e o p e r a t i n g i n two modes, v e r t i c a l and h o r i z o n t a l w i t h r o t a t i o n . At a n o n - p l a n a r i n t e r f a c e the p a r t i c l e s were s e g r e g a t e d to i n t e r s t r u c t u r a l p o s i t i o n s l i k e c e l l w a l l s o r i n t e r d e n d r i t i c r e g i o n s f o r both s o l i d i f i c a t i o n modes. When the i n t e r f a c e moved v e r t i c a l l y downward l a r g e r p a r t i c l e s were p r e f e r -e n t i a l l y s e g r e g a t e d . The average s e g r e g a t i o n was h i g h e r f o r the d e n d r i t i c i n t e r f a c e compared to a c e l l u l a r s u r f a c e w i t h a l m o s t 90% of the p a r t i c l e s i n the i n t e r d e n d r i t i c zones. For a c e l l u l a r i n t e r f a c e a d v a n c i n g h o r i z o n t a l l y the degree o f s e g r e g a t i o n o f p a r t i c l e s was s i m i l a r t o the v e r t i c a l mode but the s i z e d i s t r i b u t i o n was random. When pure l e a d w i t h i r o n p a r t i c l e s were s o l i d i f i e d w i t h a p l a n a r i n t e r f a c e i n both modes no a c c u m u l a t i o n of p a r t i c l e s or d i f f e r e n c e i n d e n s i t y between quenched l i q u i d and s o l i d was found i n d i c a t i n g no p a r t i c l e s e g r e g a t i o n or p u s h i n g . The B o i l i n g and Cisse 1 t h e o r y as w e l l as Chernov et a l . t h e o r y p r e d i c t p u s h i n g a t the growth" v e l o c i t i e s employed. 130 With the h e l p o f a p h y s i c a l model i n which both modes were simulated,,a mechanism has been d e v e l o p e d to a c c o u n t f o r the s e g r e g a t i o n o b s e r v e d i n the m e t a l l i c a l l o y s . The d r i v i n g f o r c e f o r the p a r t i c l e s e g r e g a t i o n i s p r i n c i p a l l y the buoyancy f o r c e a c t i n g on the p a r t i c l e . In the v e r t i c a l c a s e , l a r g e r p a r t i c l e s ha v ring more mo me rit um c o l 1 i de w i t h the c e l l c e n t r e s or d e n d r i t i c t i p s and are d e f l e c t e d i n t o t h e i r f i n a l p o s i t i o n s . For p a r t i c l e s l a r g e r than the t h i c k n e s s of the .boundary l a y e r the p r e s e n c e o f n a t u r a l c o n v e c t i o n h e l p s them to f i n d t h e i r way to i n t e r s t r u c t u r a l p o s i t i o n s . In the h o r i z o n t a l mode, due t o r o t a t i o n and buoyancy forces^ t h e p a r t i c l e s sweep the i n t e r f a c e u n t i l they are t r a p p e d i n the i n t e r c e l l u l a r g r o o v e s . The s m a l l i r o n p a r t i c l e s t r a v e l d i s t a n c e s o f the o r d e r o f the c e l l dimen-s i o n s per r e v o l u t i o n which exposed them t o the i n t e r -s t r u c t u r a l r e g i o n s . Once the p a r t i c l e g e t s i n one o f t h e s e i n t e r s t r u c t u r a l r e g i o n s i t i s shown t h a t due t o t h e v i s c o u s f o r c e s the p a r t i c l e o f average s i z e i s s t o p p e d a l m o s t i n s t a n t l y , and cannot come out i n t o the l i q u i d and move f u r t h e r . S e g r e g a t i o n was a l s o o b s e r v e d i n a c a s t i n g e x p e r i m e n t . Most o f the p a r t i c l e s were t r a p p e d i n the b r a nches o f the d e n d r i t e s i n an e a r l y s t a g e o f the s o l i d i f i c a t i o n p r o c e s s . 131 T r a p p i n g by d e n d r i t e b r a n c h e s c o u l d r e s u l t as the p a r t i c l e s floated, towards the s u r f a c e . In a c a s t i n g 15.5% o f the p a r t i c l e s managed to remain i n the l i q u i d due to buoyancy f o r c e s and c o n v e c t i o n i n the i n t e r d e n d r i t i c l i q u i d . Only a few s m a l l p a r t i c l e s were found i n the d e n d r i t i c c o r e s s u g g e s t i n g t h a t t h e i r e n g u l f m e n t was enhanced by t h e i r low m o b i l i t y or the d e n d r i t e s may have n u c l e a t e d on them. D i f f e r i n g markedly from t h e m e t a l l i c s ystem, p u s h i n g o f Cr p a r t i c l e s was o b s e r v e d i n w a t e r . To s o l v e the a p p a r e n t c o n t r a d i c t i o n between the p r e s e n t o b s e r v a t i o n s where no r e j e c t i o n was d e t e c t e d i n m e t a l l i c s y s t e m s , and the t h e o r e t i c a l p r e d i c t i o n s t h a t r e j e c t i o n t a k e s p l a c e , the L i f s h i t z - Van der Waals f o r c e between s o l i d and p a r t i c l e s was c a l c u l a t e d u s i n g the Drude f o r m u l a e f o r the d i e l e c t r i c c o n s t a n t s . T h i s f o r c e i s shown to be a t t r a c t i v e w i t h a magnitude s i m i l a r to t h a t r e p o r t e d i n the l i t e r a t u r e . Cone!us i o n s : For the Fe-Pb m e t a l l i c system adopted i n t h i s i n v e s t i g a -t i o n : (1) S e g r e g a t i o n of p a r t i c l e s a t a n o n - p l a n a r i n t e r f a c e i s due t o n a t u r a l c o n v e c t i o n i n the l i q u i d and buoyancy f o r c e s a c t i n g on the p a r t i c l e s . (2) No e x p e r i m e n t a l e v i d e n c e o f p u s h i n g o f p a r t i c l e s by an a d v a n c i n g s o l i d - 1 i q u i d i n t e r f a c e was o b s e r v e d . (3) The L i f s h i t z - V a n der Waals f o r c e between p a r t i c l e and s o l i d has been c a l c u l a t e d to be a t t r a c t i v e i n t h i s system. A c c o r d i n g l y , p a r t i c l e s s h o u l d not be r e j e c t e d by an a d v a n c i n g i n t e r f a c e i n t h i s system as i s o b s e r v e d . 133 REFERENCES 1. M. V. Pikunov. Non F e r r o u s M e t a l s , T h e i r Treatment and Working, Metal 1 u r g i z d a t , Moscow, (1957) pp. 55-67. 2. V.H.S. Kud and W.R. W i l c o x . Ind. Eng. Chem. P r o c e s s . Des. D e v e l o p . , 1_2, 3, 1973. 3. G.F. B o i l i n g and J . C i s s e . J o u r n a l o f C r y s t a l Growth 10 (1971) pp. 56-66. 4. D. R. Uhlmann, B. Chalmers and K. A. J a c k s o n . J o u r n a l o f A p p l i e d P h y s i c s . 35, 10, (1964). pp. 2986-2993. 5. 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(1971) pp.369-379. 31. M.D. Maheshwari, T. M i k h e r j e e and T.T. I r a n i . I r o n -making and S t e e l m a k i n g , 9, 4, (1982) pp.168-177. 32. M. Myers and M.C. F l e m i n g s . Met. T r a n s . , 3_, (1972) pp.2225-2234. 33. M.M. B e l l . M..A. Sc. T h e s i s , U.B.C. (1971). 34. A. M i t c h e l l and M.M. B e l l . Can. Met. Q u a r t . , ] ! , 2, (1972) pp.363-369. 135 35. F. Reyes, Ph.D. T h e s i s , U.B.C. (1983). 36. A.A. M i t c h e l . Ironmaking and S t e e l m a k i n g ( Q u a r t e r l y ) 3, 1, (1974) pp.172-179. 37. R. C I i f t , J.R. Grace and M.E. Weber. " B u b b l e s , Drops and P a r t i c l e s " . Academic P r e s s . New York N.Y. ( 1 9 7 8 ) . 38. " Q u a n t i t a t i v e M i c r o s c o p y " Ed. by R.T. de H o f f and F.N. R h i n e s . McGraw H i l l Book Co. New York, N.Y. (1968). 39. C E . Schvezov and D. F a i n s t e i n - P e d r a z a . J . of C r y s t a l Growth, 34, (1976) pp.55-60. 40. B. Chalmers. 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The O p t i c a l P r o p e r t i e s o f L i q u i d M e t a l s i n " L i q u i d M e t a l s - C h e m i s t r y and P h y s i c s " Ed. by S.Z. Beer. Macel Dekker Inc. New York, N.Y. (1972) pp.331-371 . 47. M. S h i m o j i " L i q u i d M e t a l s " Academic P r e s s . London (1977). 48. J.M. Ziman. " P r i n c i p l e s o f the Theory o f S o l i d s 2nd Ed. Cambridge U n i v . P r e s s , Cambri dge , ( 1 972••). 49. F. A b e l e s . O p t i c a l P r o p e r t i e s o f M e t a l s i n " O p t i c a l P r o p e r t i e s o f S o l i d s " . Ed. by F. A b e l e s . 50. C. K i t t e . l . " I n t r o d u c c i o n a l a F i s i c a d e l E s t a d o S o l i d o " . Ed. R e v e r t e S.A. B a r c e l o n a (1965). 136 51. G.P. S h i g a and M. P e l l s . J . Phys. C. ( P r o c . Phys. S o c ) , 2, 2, (1969) p.1847. 52. O.C. M i l l e r . P h i l . Mag., 20, (1969) p p . I l l 5-1132. 53. A.E. C o r t e . J . of Geophys. R e s e a r c h . , 67, 3 (1962) pp.1085-1Q90. ~~ 54. V.H.S. Kuo and W.R. W i l c o x . Sep. S c i e n c e , 8, 37, (1973) pp.375-377. 55. R.H. Ewing. P h i l . Mag., 2_5, (1972) p. 779. 56. M.J. S t e w a r t and F. Weinberg. J . o f C r y s t a l Growth 1_2 , ( 1 972) pp. 21 7-227. 57. M.J. S t e w a r t and F. Weinberg. J . o f C r y s t a l Growth j_2, ( 1972) pp.228-238. 58. 0. D r i e d g e r , A.W. Neumann and P.J. S e l l . K o l l o i d -Z.U.Z. Polymere, 201, (1965) pp.52-57. 59. I b i d . 2_04, (1965) pp.101-105. 137 Appendix I THEORIES OF PARTICLE REJECTION AT A SOLID-LIQUID INTERFACE a) C o n d i t i o n s f o r P u s h i n g 138 It was seen above^ t h e e x i s t e n c e o f a s t e a d y -s t a t e r e j e c t i o n o f a p a r t i c l e i n a PLS system depends on the system t h r o u g h the i n t e r a c t i o n o f the t h r e e con-s t i t u e n t s . The r e s u l t o f the i n t e r a c t i o n s h o u l d be r e -f l e c t e d i n s p e c i f i c c o n t r i b u t i o n s to the t o t a l energy o f the system. The knowledge and a n a l y s i s o f t h i s i n t e r -a c t i o n and the e n e r g i e s i n v o l v e d s h o u l d e n a b l e p r e d i c t i o n s to b,e majdeoaSiQto when t r a p p i n g o r pus h i n g w i l l o c c u r . F i g . 2 shows some o f the s i t u a t i o n s t h a t can be ima g i n e d . In the f i g u r e the f r e e e n e r g i e s o f the PLS system as a f u n c t i o n o f the d i s t a n c e from the S-L i n t e r f a c e a r e s c h e m a t i c a l l y r e p r e s e n t e d . I t i s assumed t h a t t h e r e i s e q u i l i b r i u m and the phase t r a n s f o r m a t i o n i s not taken i n t o c o n s i d e r a t i o n . F f and F.. and F^ , are the energy o f the p a r t i -c l e i n t h e s o l i d and i n the l i q u i d r e s p e c t i v e l y , f a r from the i n t e r f a c e . F. , and F.. r e p r e s e n t two i n i t i a l e n e r g i e s which a r e h i g h e r and lower than the energy o f the f i n a l s t a t e , F^. As the p a r t i c l e a p proaches the i n t e r f a c e an i n t e r a c t i o n among the t h r e e c o n s t i t u e n t s d e v e l o p s . T h i s i n t e r a c t i o n s h o u l d be m a n i f e s t e d t h r o u g h a f o r c e o f a s t i l l unknown n a t u r e . T h i s f o r c e i n p r i n c i p l e s h o u l d be i n -dependent o f the e n e r g i e s o f the i n i t i a l and f i n a l s t a t e s where the i n t e r a c t i o n i s n e g l i g i b l e . Moreover, i t may be a t t r a c t i v e , r e p u l s i v e o r f i r s t a t t r a c t i v e and then r e -p u l s i v e or any o t h e r c o m b i n a t i o n . As a r e s u l t o f t h i s i n -t e r a c t i o n , c u r v e s a to e may be c o n s i d e r e d p o s s i b l e , a) and b) are the r e s u l t o f the p r e s e n c e o f a r e p u l s i v e f o r c e 139 Figure 2. Free energies of the PLS system vs. distance from the interface(schematic). i ' and i are two i n i t i a l s tates with higher and lower energy than the f i n a l state f. Curves a to e are the possible e f f e c t s of the close presence of the p a r t i c l e i n the f r o n t . In a and b the p a r t i c l e i s r e p e l l e d by the solid,and i n cand d the p a r t i c l e i s a t t r a c t e d by the s o l i d . I n case e there i s no i n t e r a c t i o n between p a r t i c l e and s o l i d . between p a r t i c l e and i n t e r f a c e . To move a p a r t i c l e with, e n e r g i e s ; F^ » o r F | t o t h e i n t e r f a c e r e q u i r e s ; a work t o be d o n e on t o t h e p a r t i c l e t o i n c r e a s e i t s e n e r g y . Curves, c ) and d) i l l u s t r a t e tfie. o p p o s i t e c a s e i n which, t h e f o r c e i s ; a t t r a c t i v e . In t h i s c a s e /moving t h e p a r t i c l e t o t h e i n t e r -f a c e r e l e a s e s e n e r g y , r e d u c i n g F- a n d . F - , . In c a s e ( e ' ) t h e e n e r g y i s c o n s t a n t i n t h e l i q u i d and i n c r e a s e s l i n e a r l y t o t h e f i n a l v a l u e i n t h e s o l i d . From t h e p o i n t o f v i e w o f p u s h i n g c a s e s ( a ) and ( b ) may l e a d t o t h e s t e a d y r e j e c t i o n o f a p a r t i c l e . . C a s e ( c ) s h o u l d l e a d t o t r a p p i n g . C a s e s ( d ) and ( e ) a r e d i f f i c u l t t o a s s e s s . In c a s e (d) t h e f o r c e i s a t t r a c t i v e b u t t h e f i n a l s t a t e has a h i g h e r e n e r g y t h a n t h e i n i t i a l one. I f t h i s c o n d i t i o n e x i s t s on a r e a l PLS s y s t e m p u s h i n g may o r may n o t o c c u r i n t h i s c a s e . In c a s e ( e ) t h e r e i s no i n t e r -a c t i o n i n t h e s y s t e m when t h e p a r t i c l e i s i n t h e l i q u i d a n d , i n t h e s o l i d t h e e n e r g y i n c r e a s e s l i n e a r l y t o t h e f i n a l v a l u e Ff-. Once more i t i s r e c a l l e d t h a t a r e p u l s i v e f o r c e is; n e c e s s a r y n o t o n l y t o a c c e l e r a t e t h e p a r t i c l e f r o m z e r o t o t h e growth, v e l o c i t y , b u t a 1 so t o w i t h s t a n d t h e d r a g f o r c e w h i c h o r i g i n a t e s f r o m t h e m o t i o n o f f l u i d r e l a t i v e t o t h e p a r t i c l e d u r i n g s o l i d i f i c a t i o n . In t h i s c a s e t h e S t o k e s d r a g f o r c e , (F = 6,frryRnv where n. i s t h e v i s c o s i t y , R>n i s 141 t h e p a r t i c l e r a d i u s ; and y i s t h e v e l o c i t y o f t h e l i q u i d ) does, n o t a p p l y b e c a u s e o f t h e p r e s e n c e o f an i n t e r f a c e i m m e d i a t e l y b e h i n d the. p a r t i c l e . This, i n t e r f a c e e f f e c t i v e l y 4 a c t s ; as a s i n k . C a r r i e r i n t r o d u c e d t h i s , f a c t and c a l c u l a -t e d the v a l u e o f t h e d r a g f o r c e f o r a p l a n a r s i n k w h i c h is, g i v e n by t h e f o l l o w i n g e x p r e s s i o n F = 6 m R b 2 v / d s . . . ( 1 ) where d g i s t h e d i s t a n c e between p a r t i c T e : a n d i n t e r f a c e . The f o r m o f t h i s f o r c e , t h a t t e n d s t o i n f i n i t e v a l u e s a s t h e d i s t a n c e d $ g o e s t o z e r o , may l e a d t o t h e c o n c l u s i o n t h a t a PLS s y s t e m . w i t h a ( d ) - o r ( e ) - l i k e e n e r g y d i a g r a m ca n n e v e r p u sh a p a r t i c l e s i n c e t h e r e p u l s i v e f o r c e b e g i n s t o be i m p o r t a n t when t h e p a r t i c l e has a l r e a d y been c a p t u r e d by t h e s o l i d . 5 Neuman and c o - w o r k e r s assumed t h a t a t v e r y low s o l i d i -f i c a t i o n r a t e s o n l y t h e t h e r m o d y n a m i c a s p e c t i s i m p o r t a n t , and any o t h e r c o n t r i b u t i o n such, as t h a t due t o d r a g f o r c e s , i s n e g l i g i b l e i n c o m p a r i s o n t o t h e p o t e n t i a l d r i v i n g f o r c e f o r m i g r a t i o n . T h i s d r i v i n g f o r c e was assumed t o he t h e s u r f a c e t e n s i o n o f t h e PLS s y s t e m . C o m p u t i n g t h e c h a n g e s i n f r e e , e n e r g i e s t h r o u g h t h e d i f f e r e n t s t e p s o f e n g u l f i n g o f a p a r t i c l e [ s e e F i g . 3) t h e n e t c h a n g e i n f r e e e n e r g y , f o r a p a r t i c l e o f u n i t a.rea x a n ; .be...given by 142 A G n e t aps; " a p l ••• C 2 i where a and a p - | are th.e i n t e r f a c i a l e n e r g i e s , o f P-S and P-L r e s p e c t i v e l y . Th..is. change is; i n d e p e n d e n t o f what hap-pens, i n between t h e i n i t i a l and f i n a l states;. I t i s assumed t h a t the i n t e r f a c i a l e n e r g i e s ; do not change w i t h the p r e -sence o f the p a r t i c l e v e r y c l o s e to the i n t e r f a c e . In t h i s c a se t h e r e i s no change i n the f r e e energy o f the system when the p a r t i c l e i s i n the l i q u i d . The f r e e energy b e g i n s to i n c r e a s e when the low; energy s u r f a c e a r e a between p a r t i c l e and l i q u i d i s r e p l a c e d by a hi g h energy s u r f a c e a r e a between p a r t i c l e and s o l i d , t h a t i s when the p a r t i c l e i s e n g u l f e d by the s o l i d . C o n s i d e r i n g t h a t the s u r f a c e a r e a o f a s p h e r i c a l s e c t o r i s 2 / 3 T R h,- where R i s the r a d i u s o f the sp h e r e and h. i s t h e h e i g h t o f th.e spherical sect'or):(that is,thecd'ep'thtthe p a r t i c l e has p e n e t r a t e d i n t o the s o l i d ) , the v a r i a t i o n i n f r e e energy i s p r o p o r t i o n a l to h. When h = 2 R , i . e . the whole p a r t i c l e i s i n the s o l i d , the f r e e energy r e a c h e s i t s maximum v a l u e . T h i s maximum v a l u e does not change f o r f u r t h e r depths i n t o the s o l i d . In t h i s c ase the v a r i a t i o n i n f r e e energy o f t h e PLS system w i t h d i s t a n c e from the S-L i n t e r -f a c e may be r e p r e s e n t e d by c u r v e (e) i n F i g . 2 ; t h a t i s , c o n s t a n t i n the l i q u i d and l i n e a r l y a p p r o a c h i n g the f i n a l v a l u e i n t h e s o l i d i n a d i s t a n c e equal to the d i a m e t e r o f the p a r t i c l e . 143 The conditions proposed by Neuman and his co-workers are that when A G n e t < 0 trapping, of the p a r t i c l e w i l l be favored, whereas i f AG n e t>0 r e j e c t i o n w i l l occur. These conditions 5-7 were checked experimentally using matrixes of naphthalene and bi pheni1 and organic p a r t i c l e s of acetal , nylon, poly-estryrene and t e f l o n . These ma t e r i a l s covered the range from h y d r o p h i l i c to hydrophobic p a r t i c l e s . The r e j e c t i o n of the p a r t i c l e s was q u a n t i f i e d through t h e i r m o b i l i t y at an i n t e r -face moving h o r i z o n t a l l y at a v e l o c i t y well below 8 ym s e c - 1 . The r e s u l t s which appear in Table I were in agreement with the thermodynamic p r e d i c t i o n s : i . e . AGne^.<0 corresponds to engulfment and AG n e t>0 r e j e c t i o n . In a d d i t i o n Omenyi and Neuman7 confirmed the importance of the surface thermodynamic e f f e c t s observing the behavior of glass spheres ( h y d r o p h i l i c ) and glass spheres covered with s i l i c o n (hydrophobic) which were pushed and trapped r e s p e c t i v e l y . One important aspect of t h i s work which i s unclear i s in the procedure to obtain the surface tension of the P-S i n t e r f a c e , To obtain the surface tensions they used P s 59 60 the f o l l o w i n g semi-empirical r e l a t i o n s ' obtained f o r s o l i d s with low i n t e r f a c i a l energies (of the order of values f o r water). 144 CO.. 015 a -2:C.0Va .a, •+ a, ) c o s a = s v • s v ' v ^— ...3Ca) y a ] y C Q . 0 1 5 / a s y a l y -1} a' s l = S V ;lV- ~ . ; . - 3 ( b ) 1 - 0 . 0 1 5 / a s v a l y and Young's; e q u a t i o n a s v " a s . l = °lv £ 5 § e y . . . 3 ( c ) where: i s t h e Young's c o n t a c t a n g l e , a s v » a s-| and a - j v a r e t h e i n t e r f a c i a l e n e r g i e s o f s o l i d / v a p o r , s o l i d / l i q u i d and l i q u i d and l i q u i d / v a p o r i n t e r f a c e s r e s p e c t i v e l y . T h e s e e q u a t i o n s a r e u s e d i n t h e f o l l o w i n g way. Va l u e s , o f a-| v and d e t e r m i n e d e x p e r i m e n t a l l y a r e p u t i n t o E q u a t i o n 3 ( a ) e n a b l i n g 0 S V t o be c a l c u l a t e d . S u b s t i t u t i n g a. g v and i n E q u a t i o n 3 ( b ) g i v e s a s-| • A l t e r n a t i v e l y , may a l s o be o b t a i n e d u s i n g E q u a t i o n 3 ( c ) r e p l a c i n g t h e v a l u e s o f ff. , a-, and e v . The c o n t a c t a n g l e s Q were m e a s u r e d a t o V I V Jr y room t e m p e r a t u r e u s i n g t h e s e s s i l e d r o p t e c h n i q u e e m p l o y i n g t h e m a t r i x and p a r t i c l e m a t e r i a l as s u b s t r a t e s . The v a l u e s o f a s v c a l c u l a t e d a r e t h e n e x t r a p o l a t e d t o t h e m e l t i n g p o i n t u s i n g e s t i m a t e d c o e f f i c i e n t s , d a s v / d T , f o r t h e v a r i a t i o n o f s u r f a c e e n e r g y with, t e m p e r a t u r e . What ' . i s ~ u n c l e a r i n t h e p r o c e d u r e t o o b t a i n a ps i s t h e v a l i d i t y o f t h e a s s u m p t i o n made t o use E q u a t i o n 3 ( b ) . They assumed t h a t " . . . r e l a t i o n s b e t w e e n t h e r m o d y n a m i c 1 4 5 Figure 3. Changes i n free energy during the engulfement of a cube or a sphere of u n i t area by a solid.The surface energies for the bulk media do not change due to the presence of a t h i r d i n t e r f a c e . 146 q u a n t i t i e s may he e x p e c t e d n o t t o d e p e n d s t r o n g l y on t h e s t a t e o f a g g r e g a t i o n . . . . " With, t h i s a s s u m p t i o n , t h e n , . E q u a t i o n 3 (b 1 h o l d s when i n s t e a d o f t h r e e p h a s e s i n t h r e e d i f f e r e n t s t a t e s o f a g g r e g a t i o n Csolid, l i q u i d and g a s ) , f o r which, i t was d e v e l o p e d , t h e r e a r e s t i l l t h r e e p h a s e s b u t two o f them s o l i d p h a s e s ( p a r t i c l e and m a t r i x ) and t h e r e m a i n i n g a g a s e o u s p h a s e . Then a^ v i s r e p l a c e d by a i n E q u a t i o n 3 ( b ) and t h e r e s u l t i s a^^. P i k u n o v 1 p r o p o s e d a d i f f e r e n t c o n d i t i o n f o r p u s h i n g i n w h i c h t h e i n t e r f a c i a l e n e r g i e s a .p a^ v and a a r e i n v o l v e d . In a t y p i c a l p u s h i n g c o n f i g u r a t i o n t h e e s t i m a t e d p a r t i c l e - i n t e r f a c e s e p a r a t i o n i s assumed t o be much l e s s t h a n 1 ym, t h a t i s t h e p a r t i c l e and i n t e r f a c e a r e n e a r l y i n " c o n t a c t " . I f S i s t h e s u r f a c e a r e a o f t h e p a r t i c l e i n " c o n t a c t " w i t h t h e i n t e r f a c e t h e e n e r g y of t h e s y s t e m i s SXap-| + o l s ) w h i l e t h e e n e r g y of t h e p a r t i c l e - s o l i d i n t e r -f a c e i s So . The d i f f e r e n c e between t h e two e n e r g i e s i s ps assumed t o be t h e d r i v i n g f o r c e f o r p u s h i n g , t h e r e f o r e when cr < a , + af - . . . . 4 ( a ) ps pi IS. t h e p a r t i c l e w i l l be t r a p p e d , and when aps. > a p l + a l s - - ^ ( b ) the p a r t i c l e w i l l he r e j e c t e d , were not pro y e d e x p e r i m e n t a l l y , a r e i n c l u d e d i n form a l t h e o r i e s t i e s and they w i l l be p r e s e n t e d 147 These r e l a t i o n s , however, T h i s and o t h e r c o n d i t i o n s p r e d i c t i n g c r i t i c a l v e l o c i -i n the next s;ection. . 4 b) Theory o f Uhlmann e t a l . In t h i s t h e o r y the d r i v i n g f o r c e f o r p u s h i n g i s c o n s i d e r e d to be the d i f f e r e n c e i n s u r f a c e energy & e F 0 °ps ~ ^ a p l + a l s ^ a s P r o P o s e c ' by Pik u n o v . However, t h i s i n t e r f a c i a l e nergy i s not c o n s i d e r e d ' e f f e c t i v e u n t i l the p a r t i c l e i s v e r y c l o s e to the i n t e r f a c e . To take t h i s i n t o a c c o u n t an e x p r e s s i o n i s assumed f o r the dependence o f the d r i v i n g energy w i t h d i s t a n c e between the p a r t i c l e and the i n t e r f a c e g i v e n by ' i d ' A a - A a Q 1 n 0 ' L d_ where d i s the p a r t i c l e / i n t e r f a c e d i s t a n c e and d 0 i s an assumed minimum d i s t a n c e between p a r t i c l e and i n t e r f a c e where they a r e i n c o n t a c t . A minimum d i s t a n c e d Q i s n e c e S ' s a r y i n t h e d e r i v a t i o n to a v o i d i n f i n i t e v a l u e s o f A a. The v a l u e o f the c o n s t a n t n i s some p o s i t i v e number between 4 and 5. The v a r i a t i o n i n s u r f a c e f r e e energy g i v e n by E q u a t i o n 5 c o r r e s p o n d s to a c h e m i c a l p o t e n t i a l which, i s w r i t t e n as 148 Figure 4. The e f f e c t i v e surface energy of the PLS system as a function of distance, d i s a minimum distance at which p a r t i c l e and s o l i d are considered to be i n contact. 149 . . .6 where V q i s th.e atomic yolame o f the l i q u i d . Such, p o t e n -t i a l p rovides; the. d r i v i n g f o r c e to c a r r y m a t e r i a l i n t o the l i q u i d f i l m between t h e p a r t i c l e and the i n t e r f a c e . Two f o r m u l a t i o n s are c o n s i d e r e d : a) d i f f u s i o n i n the r e g i o n o f c o n t a c t a l o n e and b) d i f f u s i o n w i t h v i s c o u s d r a g . a) In approach (a) two b a s i c e q u a t i o n s a r e p r o p o s e d . From mass c o n s e r v a t i o n , assuming t r a n s p o r t by d i f f u s i o n i n the f l u i d f i l m , the f o l l o w i n g e q u a t i o n i s o b t a i n e d : r dr i r a d r J D * • *' From the e q u i l i b r i u m c o n d i t i o n a t the i n t e r f a c e , a change i n f r e e energy a s s o c i a t e d w i t h the c u r v a t u r e of the i n t e r f a c e i n the c o n t a c t r e g i o n i s compensated by a change i n c h e m i c a l p o t e n t i a l i n the f i l m g i v e n by E q u a t i o n 5. The e q u i l i b r i u m c o n d i t i o n i s g i v e n a s: L a t H r i / R - ^ - V ^ r 1 ^ = 0 •••8 where L , + i s the l a t e n t heat o f f u s i o n per atom a„ i s a at o l e n g t h o f the o r d e r o f a m o l e c u l a r d i a m e t e r and O/R - l / p s . ) i s t he net c u r v a t u r e o f the i n t e r f a c e ; 1/R i s the c u r v a t u r e o f the i r r e g u l a r i t i e s ; o f the p a r t i c l e s and 1 / p * is. the 150 c o n t r i b u t i o n o f tlie s e p a r a t i o n between p a r t i c l e and i n t e r -f a c e , d-j i s the s e p a r a t i o n a t the c e n t e r o f the c o n t a c t r e g i o n (see F i g . 4 ) . The s o l u t i o n o f E q u a t i o n 7 w i t h u g i v e n by f o r m u l a 6 p e r m i t s the r e p l a c e m e n t o f LaQ and 1 / P s i n E q u a t i o n 8. T h i s l e a d s to La. where 1 R L r. 2d. nr. vk.T o-1 4 n+1 V Dd, o 1 rl = 0 o l a t e n t heat per u n i t volume e f f e c t i v e c o n t a c t r a d i u s , t h a t i s the u r a d i u s at which s e p a r a t i o n becomes. i h f l h ' i t g v = growth v e l o c i t y U s i n g E q u a t i o n 9 t h e c r i t i c a l v e l o c i t y can be d e t e r m i n e d i n the f o l l o w i n g way. T h i s e q u a t i o n g i v e s two d e f i n i t e b r a nches o f ( d ^ / d Q ) n v e r s u s v (see F i g . 5 ) . The upper group i n which ( d 1 / d Q ) n d e c r e a s e s w i t h v ji.Sj s t a b l e . T h i s i s a s s o c i a t e d w i t h the case i n which the i n t e r f a c e i s f l a t and the. p a r t i c l e i s f a r from the i n t e r f a c e . The lower branches, a r e uns.tab.le a g a i n s t s m a l l s e p a r a t i o n f l u c t u a t i o n s . T h e r e f o r e the c r i t i c a l v e l o c i t y i s the l a r g e s t growth r a t e i n the s t a b l e upper c o n f i g u r a t i o n . M a x i m i z i n g E q u a t i o n 9. t h e c r i t i c a l v e l o c i t y , a t which, p a r t i c l e s a r e t r a p p e d , i s g i v e n by 151 \ (n + 1) C.La 0V oD/kTR 2) . . . IQ b) When t h e drag f o r c e g i v e n by f o r m u l a (;1 ) i s i n -c l u d e d i n the a n a l y s t s . E q u a t i o n 9 t r a n s f o r m s to La 2d-nr. 1 n y.kT 4 n+1 V J ) d , o 1 6n.R R v d r 2 h s o . . .11 where t h e l a s t term i s the c o n t r i b u t i o n to the f r e e energy due t o t h e drag f o r c e which i s t r a n s m i t t e d to the i n t e r -f a c e t h r o u g h the i r r e g u l a r i t i e s o n l y . R Q i s the p a r t i c l e r a d i u s and h i s t h e depth o f the c o n t a c t r e g i o n i n t o the s o l i d . T h i s i s assumed t o be i n d e p e n d e n t o f t h e p a r t i c l e r a d i u s . The c r i t i c a l v e l o c i t y then becomes d„hLa„d, s o 1 6nR QR 2n 1 + 1 + 6 n R 0 n ( n + l )V QD" d s h d ] k T 0.5 .. .12 which r e d u c e s t o E q u a t i o n 10 when n 0. In E q u a t i o n 12 d s , h , d-| and n a r e unknown; n i s assumed t o be 5 and d i s -t a n c e s are o f the o r d e r o f 1Q~7. R e p l a c i n g the r e m a i n i n g p arameters w i t h t h e s e c o r r e s p o n d i n g to the w a t e r - i c e system the c r i t i c a l y e l o c i t y i s one o r d e r o f magnitude l a r g e r than e x p e r i m e n t a l y a l u e s o b t a i n e d by Uhlmann. R e d u c t i o n o f the d i f f u s i o n c o e f f i c i e n t s f o r b u l k l i q u i d by a f a c t o r o f 400 i s r e q u i r e d to f i t t h e o r y w i t h e x p e r i m e n t . 152 i/d, V Figure 5 Schematic representation of separation vs. growth v e l o c i t y for various p a r t i c l e s i z e s . The upper branches are stable and the lower are unstable against f l u c t u a t i o n i n growth velocity.The c r i t i c a l i s the maximum v e l o c i t y . Figure 6. The B o i l i n g and Cisse scheme of pushing. The symbols used i n the theory are represented. 153 On th.e o t h e r hand and c o n t r a d i c t i n g what may he. e x p e c t e d , the V a l u e s g i y e n by E q u a t i o n 11 a r e almost t h r e e o r d e r s o f magn i t u d e l e s s than t h o s e o b t a i n e d from E q u a t i o n 12. For a p a r t i c l e o f r a d i u s R Q = 2 um i n w a t e r - i c e the c r i t i c a l v e l o c i t i e s c a l c u l a t e d from E q u a t i o n 11 and 12 a r e 0.6 um s e c " ' and 200 ura s e c " ' r e s p e c t i v e l y . 3 c) B o i l i n g and Cisse" T h eo r y The main d i f f e r e n c e between t h i s t h e o r y and the p r e v i o u s t h e o r y o f Uhlmann i s t h a t d i f f u s i o n i s c o n s i d e r e d to be the - l i m i t i n g ; c ase f o r f l u i d to move i n t o the l i q u i d f i l m between the p a r t i c l e and i n t e r f a c e . T h e r e f o r e d i f f u s i o n i s i n c l u d e d i n the v i s c o u s d r a g . The r e p u l s i v e f o r c e f o r p u s h i n g o r i g i n a t e s from an i n t e r f a c e c u r v a t u r e e f f e c t o n l y . The drag f o r c e i s r e c a l c u l a t e d t a k i n g i n t o a c c o u n t the c u r v a t u r e o f t h e i n t e r f a c e i n the r e g i o n of c o n t a c t . I t s maximum v a l u e c o r r e s p o n d s .to an i n t e r f a c e w i t h s p h e r i c a l shape b e h i n d t h e p a r t i c l e . I f the r a d i u s o f the [ i n t e r f a c e i s R/a, where R i s the p a r t i c l e r a d i u s and a i s a c o e f -f i c i e n t l e s s than o r equal t o 1, the drag f o r c e i s : Fta] = 6ffin.VR2/d(.l - a ) 2 . . . 13 where V = growth, v e l o c i t y d = P-S. d i s t a n c e , a t t h e c e n t e r o f the. c o n t a c t r e g i o n M e c h a n i c a l e q u i l i b r i u m i s f e a s i b l e when 154 F ( a l a 2 cr, „ - ... 1 4 2 > u i s R T h i s i n d i c a t e s t h a t the. maximum f o r c e t r a n s m i t t e d to the 2 s o l i d t h r o u g h the s m a l l c o n t a c t a r e a •nrQ i s compensated by 2 the c u r v a t u r e e f f e c t (.see F i g . 6 ) . In the ar e a T r r Q the .: t r a n s p o r t m a t e r i a l may be d i f f u s i o n o r a flo w p r o c e s s . In the l i m i t , assuming a d i f f u s i o n mechanism, f o r the l e a s t advantageous sequence g i v e n by a random walk a n a l y s i s rl < 4bD/V ...15 < 4kT/3^nV Where i t was used the S t o k e s - E i n s t e i n r e l a t i o n D = kT/3Tuy,a and b, the l a y e r h e i g h t f o r growth, was taken as t h e i n -t e r a t o m i c d i s t a n c e a . In o r d e r to make the i n t e r f a c e r e s p o n s i v e to the p a r t i -c l e , a was assumed to have the e x p r e s s i o n a = exp-g(.d- a 0 ) / a 0 ...16 where g. i s a pa r a m e t e r , f o r g>> d / a Q t h e r e i s i n t i m a t e con-t a c t and f o r d >> g. a Q , a t e n d s to z e r o , then t h e r e i s no i n t e r a c t i o n . O p e r a t i n g w i t h E q u a t i o n s 13 to 16 the f o l l o w i n g 155 e x p r e s s i o n f o r ttie y e . l o c i t y t s o b t a i n e d n 2;V 2R 3 = w ( a ] 4 k T a ] s aj^ ...17 where ¥ ( a ) i s a f u n c t i o n o f a a c c o r d i n g to Y(CX) = •2 a (1-«•)'' (6-1 no;) with, a maximum-va 1 ue o f f ( a ) . m , v = 0.34 f o r max 3 = 1 . E q u a t i o n 17 i s the s i m p l e t h e o r y v a l i d f o r s m a l l . smooth s p h e r i c a l p a r t i c l e s . For l a r g e p a r t i c l e s g r a v i t y and the number o f i r r e g u l a r i t i e s a t which the i n t e r a c t i o n i s c a r -r i e d out s h o u l d be c o n s i d e r e d . The number o f c o n t a c t p o i n t s N i s d i r e c t l y i n t r o d u c e d as a f a c t o r i n the second member o f E q u a t i o n 17>. T h i s i s . : s u p p o r t e d by the e x p e r i m e n t a l o b s e r v a t i o n t h a t the c r i t i c a l v e l o c i t y f o r a s m a l l p a r t i c l e at a g r a i n , g r a i n boundary and t r i p l e j u n c t i o n a r e r e l a t e d by V(g) i V ( g.b.)//2~ V(g.) = V ( t . j . ) / / 3 ~ when the number o f c o n t a c t p o i n t s and a l s o the g r a v i t y are i n -c l u d e d the f o l l o w i n g f o r m u l a s are o b t a i n e d f o r the c r i t i c a l v e l o c i t y : R < R b 2 w 2 n 3 T ~ 4 ¥ ( a ) ,-r-i , . . n v R _ _ i _ J k T a ] s a Q ...18(a) R>Rt b 2,.2D3 . 2 Y ( a ) • . .._3n -. 4¥.(«) ,, l o 7 . » n V R + - ^ g A p a ^ V R R b kT a u a Q ...18(b) 156 R » R b nVR 3 ~ N2(xkTa l s/R b^gAp ...18(c) where Rfa i s the bump r a d i u s , g i s the a c c e l e r a t i o n o f g r a v i t y and Ap i s the d e n s i t y d i f f e r e n c e between p a r t i c l e and m e l t . A comparison o f the model p r e d i c t i o n s w i t h e x p e r i m e n t a l measurements w i l l be made i n the next s e c t i o n . N e v e r t h e l e s s , t h i s t h e o r y does not p r o v i d e enough i n -f o r m a t i o n to p r e d i c t when p u s h i n g w i l l be p r e s e n t f o r a g i v e n p a r t i c l e and s o l i d i f y i n g m a t e r i a l . I f i t i s assumed t h a t the i n t e r f a c e always c u r v e s b e h i n d a f o r e i g n p a r t i c l e l e a v i n g a t h i n f i l m o f l i q u i d i n between, t h i s t h e o r y p r e d i c t s p u s h i n g always o c c u r s . Moreover, as the o n l y p a r t i c l e p arameter a p p e a r i n g i n t h i s t h e o r y i s the p a r t i c l e r a d i u s , p u s h i n g s h o u l d be i n d e p e n d e n t o f the type o f p a r t i c l e f o r a g i v e n moving i n t e r f a c e . T h i s i s c l e a r l y c o n t r a d i c t o r y to what i s o b s e r v e d , as d i s c u s s e d i n s e c t i o n 2. d) The D i s j o i n i n g P r e s s u r e and Chernov e t a l . Theory So f a r i t has been seen t h a t the models p r e s e n t e d used thermodynamic q u a n t i t i e s , l i k e i n t e r f a c i a l e n e r g i e s whose v a l u e s are supposed to be u n a l t e r e d by the f a c t t h a t \\ \\ \ \ \V h (L) //////// (S) 157 Figure 7. A p a r t i c l e ( s o l i d , l i q u i d or gas) i n "contact" with a s o l i d . T h e d i s j o i n i n g pressure i n a gas bubble as a f u n c t i o n of separat ion h i s measured by pres s ing the bubble against the s o l i d , n(h) 0 \ \ OC \ \ h Figure 8. Two t y p i c a l behaviour of the d i s j o i n i n g pressure f o r a bubble i n a po lar l i q u i d (1) , and i n a non-polar l i q u i d (.2). Branches oc and fi give s tab le f i lms whi le branch # i s uns tab le . 158 the f i l m between p a r t i c l e and s o l i d i s v e r y t h i n , o f the o r d e r o f 10"^,- cm- or 10"^ cm 8. In a d d i t i o n , none of the t h e o r i e s p r e d i c t i n g p u s h i n g ( e x c e p t the Uhlmann e t a l . t h e o r y ) i n v o l v e s p r o p e r t i e s o f a l l . t h r e e media a t the same time as might be e x p e c t e d i n a t h e o r y . The Chernov e t a l . t h e o r y p a r t i a l l y a ttempts to i n c l u d e t h i s i n t h e i r t h e o r y u s i n g the c o n c e p t o f " d i s j o i n i n g p r e s s u r e " i n t r o d u c e d many y e a r s ago (see < F r e n k e l . Y. I . , Ch.V, r e f . 9 ) . Measurements of t h i s p r e s s u r e made by p r e s s i n g a gas bubble ( F i g . 7) a g a i n s t d i f f e r e n t s o l i d s and a v a r i e t y o f l i q u i d s gave 3 4 -2 v a l u e s o f the o r d e r o f 10 to 10 dyn cm f o r d i s t a n c e s -5 between bubble and s o l i d a f the o r d e r o f 10 cm = 1,000 °A. They o b s e r v e d two markedly d i f f e r e n t p r e s s u r e v a r i a t i o n s w i t h d i s t a n c e f o r d i f f e r e n t l i q u i d s as shown i n F i g . 8. The f u l l l i n e c o r r e s p o n d s to a p o l a r l i q u i d l i k e w a t e r , the dashed l i n e i s f o r a n o n - p o l a r l i q u i d l i k e a m e t a l . I t can be o b s e r v e d t h a t the d i s j o i n i n g p r e s s u r e i n c r e a s e s s t e a d i l y as the i n t e r f a c e i s approached s t a r t i n g from i n -f i n i t y (g branch) u n t i l a maximum i s i r e a c h e d . A f t e r t h i s p o i n t the p r e s s u r e drops g i v i n g an u n s t a b l e f i l m (y branch) then r i s e s s h a r p l y as the d i s t a n c e between bubble and the i n t e r f a c e becomes v e r y s m a l l (^ b r a n c h ) . The dashed l i n e c o r r e s p o n d s to a n o n - p o l a r f i l m and does not show an u n s t a b l e r e g i o n where the p r e s s u r e i s n e g a t i v e . T h i s d i s -j o i n i n g p r e s s u r e c o n t r i b u t e s to the c h e m i c a l p o t e n t i a l o f the f i l m ; the amount g i v e n by n(h.)°,, so t h a t 159 u-j = M - - ir(h.)n ... 19 00 where w^ i s the c h e m i c a l p o t e n t i a l o f the bul k l i q u i d and n i s the at o m i c volume. The p r e s s u r e i s c o n s i d e r e d p o s i t i v e when the f i l m tends to t h i c k e n and t h e r e f o r e i n t h i s case a p a r t i c l e i s pushed away from the i n t e r f a c e . The m i g r a t i o n o f s o l i d p a r t i c l e s o b s e r v e d d u r i n g s o l i d i f i c a t i o n s u g g e s t s t h a t the same p r e s s u r e i s p r e s e n t i n the f i l m between s o l i d or l i q u i d p a r t i c l e s and a s o l i d - l i q u i d i n t e r f a c e . T h i s p r e s s u r e i s b u i l t up from t h r e e main i n t e r a c t i o n s Van der Waals ( d i s p e r s i o n ) Debye ( e l e c t r o s t a t i c ) S t r u c t u r a l In m e t a l s o n l y Van der Waals and s t r u c t u r a l i n t e r a c t i o n s a r e p o s s i b l e . The f i r s t c o n t r i b u t i o n f o r a s m a l l d i s t a n c e h between p a r t i c l e and the f r o n t i s g i v e n by the e x p r e s s i o n 8 ' 1 0 ' 1 ^ n(h) = B 3 / h 3 ...20 B 3 i s a c o n s t a n t f o r the system P-L-S i n c o n s i d e r a t i o n . When Bg>0 t h e r e i s p u s h i n g , w i t h B3<;0 t h e r e i s t r a p p i n g . 11-14 In the r i g u r o u s L i f s h i ' t z - V a n der Waals t h e o r y what i s c a l l e d d i s j o i n i n g p r e s s u r e i s o r i g i n a l l y c a l c u l a t e d f o r a system s i m i l a r to t h a t shown i n F i g . 9 as 5 the f o r c e per u n i t a r e a between the b o d i e s 1 and 2. T h i s f o r c e i s 160 g i v e n by the e q u a t i o n n = F = ...21(a) 8 u h co wi t h e,(i§) - e , ( iO e 9 ( i C ) - eo ( i c ) e-| (i§) + e 3 ( U ) £ z ( i 5 ) + e 3 ( U ) ...21(b) 0 where i s the d i e l e c t r i c c o n s t a n t o f the ith-medium as a f u n c t i o n o f the i m a g i n a r y p a r t o f the complex f r e q u e n c y to. These d i e l e c t r i c c o n s t a n t s a r e r e l a t e d to the i m a g i n a r y p a r t s e"(to) o f the complex d i e l e c t r i c c o n s t a n t s t h r o u g h the K r a m e r - K r o n i g r e l a t i o n s The p r e s e n c e o f e"(co), which d e s c r i b e s the a b s o r p t i o n p r o p e r t i e s o f a s u b s t a n c e , g i v e s the name o f d i s p e r s i o n i n t e r a c t i o n t o t h i s f o r c e , to p l a y s the r o l e o f a c h a r -a c t e r i s t i c f r e q u e n c y f o r the t h r e e media P-L-S, and depending on i t s s i g n the f o r c e w i l l be e i t h e r a t t r a c t i v e or r e p u l s i v e . In t h i s t h e o r y to>o g i v e s Bg>o and t h e f o r c e i s a t t r a c t i v e . Chernov's c o n v e n t i o n i s opposed to t h i s , f o r B 3>o t h e d i s j o i n i n g p r e s s u r e i s p o s i t i v e , the f i l m tends t o t h i c k e n and t h e r e f o r e the p a r t i c l e i s r e p e l l e d . co 0 In E q u a t i o n 21(a) and ( b ) , f o r example, i f media 1 and 2 are the same, i . e . , s o l i d and p a r t i c l e , e-j =• the f o r c e i s always a t t r a c t i v e . I f the media s e p a r a t i n g s o l i d and p a r t i c l e i s vacuum £3 =• 1 and the f o r c e i s always a t t r a c t i v e as w e l l . But when the f i l l i n g medium i s a l i q u i d and the two o t h e r b o d i e s are d i f f e r e n t m a t e r i a l s the f o r c e may be a t t r a c t i v e o r r e p u l s i v e , d e p e n d i n g .; on t h e v a l u e o f to. I t i s n e c e s s a r y to mention t h a t t h e r e i s no c a l c u l a t e d or measured v a l u e o f the c o n s t a n t s Jo or Bg f o r a s o l i d - l i q u i d system o f the same m a t e r i a l and a t the m e l t i n g p o i n t , t h a t i s , f o r an a d v a n c i n g s o l i d i f y i n g i n t e r f a c e . Moreover, t h e r e i s no t a b u l a t e d v a l u e s o f t h e s e c o n s t a n t s f o r a r e p u l s i v e c ase ( i . e . , to<o). The measured v a l u e s f o r i d e n t i c a l 1 and 3 m a t e r i a l s a r e o f the o r d e r o f -1 4 -1 5 10 - 10 e r g . f o r B^ i n r e a s o n a b l e agreement w i t h the c a l c u l a t e d v a l u e s 1 0 ' 1 7 " 1 8 o f co. For d i s t a n c e P-S l a r g e r than , the wave . l e n g t h ..at; which the i n t e r a c t i o n . i s p erformed but. s h o r t e r , than 1 0 ~ 4 cm, the L i s h i t z - V a n der Waal f o r c e i s a f u n c t i o n o f the f o u r t h power of the s e p a r a t i o n P-S. In the s p e c i a l case i n which the l i q u i d f i l m i s a metal the f o r c e i s always a t t r a c t i v e and depends on the f i f t h power o f the s e p a r a t i o n . T h i s f o r c e i s g i v e n by the e x p r e s s i o n fi c 2 F = 0.0034 ...23 a 3 l where a ^ i s the d.c. c o n d u c t i v i t y o f the m e t a l . 162 Figure 9. Three media,solid(.1). - l i q u i d or gas (31 and s o l i d , l i q u i d or gas-(.2) l ead ing to an important i n t e r a c t i o n at very c lose dis tances (.10-^) descr ibed by the L i f s h i t z ^ V a n der Waals theory . r r — Figure 10. The Chernov scheme of pushing. The symbols used i n the thoery are represented. 163 The s t r u c t u r a l component o f the d i s j o i n i n g p r e s s u r e i s a t t r i b u t e d to the o r d e r p r e s e n t i n the l i q u i d near the c r y s t a l . When two s o l i d s w i t h t h e i r a s s o c i a t e d o r d e r e d l i q u i d l a y e r are br o u g h t t o g e t h e r an " e n t r o p y r e p u l s i o n " a r i s e s as a r e s u l t o f t h e i n t e r a c t i o n between the two l a y e r s . In a p o l a r l i q u i d i t i s easy to imagine t h i s as a d i p o l e - d i p o l e i n t e r a c t i o n a t the s u r f a c e o f the c r y s t a l s which i s t r a n s m i t t e d t h r o u g h the l i q u i d . In w a t e r , the s t r u c t u r a l i n t e r a c t i o n i s s t r o n g enough to e x p l a i n , by i t s e l f , the r e j e c t i o n o f p a r t i c l e s a t an a d v a n c i n g water-21 2 r i c e i n t e r f a c e and a l s o the motion o f a l o a d e d w i r e i n i c e . ' c T h i s i n t e r a c t i o n a r i s e s from the f a c t t h a t water i s p o l a r 19 20 and a s s y m e t r i c and i c e i s n o n - p o l a r . ' T h i s i n t e r a c t i o n may be p r e s e n t i n any o t h e r m a t e r i a l w i t h the same p r o p e r t i e s . M e t a l s , however, are not i n c l u d e d i n t h i s group o f m a t e r i a l s . A l t h o u g h , as p o i n t e d out i n r e f . 10, any i n t e r f a c i a l o r d e r -i n g i n t e r a c t i o n s h o u l d be e f f e c t i v e t h r o u g h a Van der Waals; f o r c e . T h e r e f o r e i t may be e x p e c t e d t h a t t h i s f o r c e would be i n c l u d e d i n the g e n e r a l e x p r e s s i o n 21. T h e r e f o r e i f the Van. der Waals f o r c e s can. account f o r the r e j e c t i o n o f p a r t i c l e s a t a s o l i d - l i q u i d i n t e r f a c e the s i g n o f the c o n s t a n t ui may d e t e r m i n e when p u s h i n g o c c u r s i n a g i v e n PLS system i n which t h e r e i s no e l e c t r o s t a t i c i n t e r a c t i o n . S i m i l a r to the two p r e v i o u s t h e o r i e s , Chernov e t a l . 164 proposed .two e q u a t i o n s f o r the e q u i l i b r i u m . For thermo-dynamic e q u i l i b r i u m o f the i n t e r f a c e the f o l l o w i n g e q u a t i o n i s p r o p o s e d fials(R]1 + R2.] ) + AS(T a ,-T)-ASm [ C ( r ) - r ] -ji(h)fi = 0 ::.24 where Q, = a t o m i c volume o f the l i q u i d = S-L i n t e r f a c i a l energy R.j , R 2 = main c r y s t a l s u r f a c e c u r v a t u r e r a d i i AS = e n t r o p y o f f u s i o n co -T--1 = i n t e r f a c e t e m p e r a t u r e f o r p l a n e i n t e r f a c e T = i n t e r f a c e t e m p e r a t u r e f o r the cave r e g i o n b e h i n d the p a r t i c l e m = l i q u i d u s s l o p e C ( r ) = c o n c e n t r a t i o n o f s o l u t e a t a d i s t a n c e r from a v e r t i c a l c e n t e r p l a n e o f the p a r t i c l e as shown i n F i g u r e 10. C°° = c o n c e n t r a t i o n o f s o l u t e f a r from the , p a r t i c l e where the i n t e r f a c e i s f l a t . In E q u a t i o n 24 the c o n t r i b u t i o n s o f the Gibb-Thomson e f f e c t , due to t e m p e r a t u r e , i m p u r i t y and d i s j o i n i n g p r e s s u r e a r e i n c l u d e d i n t h a t o r d e r . The s o l u t i o n o f E q u a t i o n 24 f o r the i n t e r f a c e shape i n pure m a t e r i a l s i s a p p r o x i m a t e d by a p a r a b o l o i d and a p l a n e as f o l l o w s : Z ( r ) H -R + h£ - v r /2R r < r g 165 25 r>r o o where Z ( r ) i s t h e d i s t a n c e f r o m a . h o r i z o n t a l p l a n e , p a s s i n g t h r o u g h t h e c e n t e r o f t h e p a r t i c l e , t o t h e s o l i d - 1 i q u i d i n t e r f a c e , r o i s t h e r a d i u s o f t h e c o n t a c t r e g i o n , ho i s t h e p a r t i c l e - i n t e r f a c e d i s t a n c e a t t h e c e n t e r o f t h e c o n t a c t r e g i o n , k> = R/R| = R/R 2. R i s t h e p a r t i c l e r a d i u s and i s t h e d i s t a n c e o u t s i d e t h e c o n t a c t r e g i o n . T h e s e s y m b o l s a r e shown i n F i g . 10. The d r a g f o r c e o n t o t h e p a r t i c l e i s c a l c u l a t e d i n -c l u d i n g t h e f a c t t h a t t h e s i n k ( t h e i n t e r f a c e ) b e h i n d t h e p a r t i c l e i s c u r v e d . F o r t h e s h a p e o f t h e i n t e r f a c e E q u a t i o n 25 i s u s e d . W i t h t h e same e q u a t i o n t h e d i s j o i n i n g p r e s s u r e o v e r t h e w h o l e p a r t i c l e i s c a l c u l a t e d ; t h i s g i v e s t h e r e p u l s i v e f o r c e o n t o t h e p a r t i c l e . E q u a t i n g t h e d r a g f o r c e t o t h e d i s j o i n i n g f o r c e t h e e q u a t i o n f o r t h e m e c h a n i c a l e q u i l i b r i u m i s o b t a i n e d i n t h e f o l l o w i n g f o r m : , , . ,2 67rnVR2 ( l - v ) 2 h 0 L-l - v ( 3 - y ) -h ( r Q ) + v h 2 < r o>j * B 3 R ( l .-v) h| v -h ^ C r " ) 26 where t h e f i r s t member i s t h e d r a g f o r c e and t h e s e c o n d i s 166 the r e p u l s i v e f o r c e due to the d i s j o i n i n g p ressure.vr'With adequate s i m p l i f i c a t i o n , the c r i t i c a l v e l o c i t i e s are un-ambiguously o b t a i n e d from E q u a t i o n 26 f o r s m a l l and l a r g e p a r t i c l e s . The r e s p e c t i v e f o r m u l a s g i v e n f o r the " e x a c t " s o l u t i o n o f E q u a t i o n 26 a r e : 2 Small P a r t i c l e s P><A /I 0.14 a 1/3 V = -r^— B~ . . .27 c nR B 3R 2 Large P a r t i c l e s R>A / l V c + 0.15 B 3 / r i R . ) ( G A S / B 3 f i ) 1 / 4 ...28 where X and 1 a r e c h a r a c t e r i s t i c l e n g t h s g i v e n by the r e l a t i o n s X = ( a a ^ S / G A S ) 1 / 2 and 1 = ( B 3 f t / G A S ) 1 / 4 . F o r B 3 = 1 0 " 1 5 e r g , a = 3 x l 0 " 2 3 c m 3 , AS = 2 k ; 3 x l 0 " 1 6 e r g d e g " 1 , 2 1 2 2 G = 10 deg cm" , X '/I = 5x10" cm. That i s the t r a n s i -t i o n from s m a l l to l a r g e p a r t i c l e s o c c u r s a t R= 500 ym. -4/3 At t;h'is r a di us the c r i t i c a l v e l o c i t y changes from a R to R"1 dependence on the p a r t i c l e r a d i u s . One i m p o r t a n t f a c t o f t h i s t h e o r y i s t h a t E q u a t i o n s 27 and 28 do not c o n t a i n unknown p a r a m e t e r s ; i e x c e p t - f o r the v a l u e o f B 3 which i s e s t i m a t e d from t h o s e o b t a i n e d f o r systems i n which the f o r c e i s a t t r a c t i v e as i t was mentioned above. In the next e x p e r i m e n t s . s e c t i o n the t h e o r i e s are compared w i t h Appendix-I I CALCULATION OF u 168 APPENDIX I I given by formula 19 i s : co e - [ ( i - 6=3(12) -^ 2(iE) - ^ 3 ( i ^ ) eT(ic) + e3(i5) e 2(iC) + 63(1?) d5 19 0 substituting the d i e l e c t r i c constants for 2 CO the integral i s : 1 ' 2 2 / _d£ (O = T A(0_ A(JO I : r r r where and 4 p 1 3 p23 / ~ - - ...A-l 0 < ? i - p i 3 ) ( e 2 + V 3 > 2 _ 2 2 Aco„ • • - co . - co P1J PT PJ 2 _ 1 , 2 , 2 , Using the fact that the integrant of A-l i s an even function, we can evaluate the same integral within -«>,<» l i m i t s , i . e . : 2/ fU)-d£ = / f(?)dg = . ZiriX Res f(0 0 where the residues of f(5) in the upper simi-space should be considered. The two p01 es with\pqs 11ive imaginary parts are: Evaluating the residue by the formula: Res f U ) = a = lim f U ) 9 s o we obtain: Res f U ) = •  1 2 1 H l 3 ( V 3 " < W Similarly Res f ( s ) i f 2 ' 2 2 ? = ? 2 2i 7 w p 2 3 ( o ) p 1 3 - U p 2 3 ) Substituting in formula A-l and operating i t i s f i n a l l y obtained 2 - 1 / 2 .2 .2 TT ' "023 ' "013 w = A t 0P i 3 A w P 2 3 a 2 2- / 2 2 x CO _ 0 CO n o I CO o o ~ CO n o J p23 pi 3 p23 pl3' 2 4irNe To obtain the values of co . = n with n = valence, N = p rri no atoms/unit volume the following procedure i s followed: Liquid Pb 46 Hodgson obtained the following r e l a t i o n between constants f l y A 2 c o p = 1.344xl0" 4 p Q e l f 1 where y = c o l l i s i o n frequency P q = e l e c t r i c a l r e s i s t i v i t y from optical constants, for lead the values g i v e n 4 7 are: fiy = 2.86 eV and p Q = 98.2 y ohm-cm. Substituting values f l u , ^ 1 ) ' = 14.7 eV or ^ - 2.23xl0 1 6 sec" 1 X P b ( 1 ) = 845.3 A° P Solid Pb The following values are used: n = 4 45 m* = e f f e c t i v e mass = 1.12 m e = 4.8GxlO" 1 0 ues oo m = 9.10x10 gr M M - Na.d _ 6.022xl0 2 3 x 11 _ Q N Q V L N nN = N f = TT - — - 3.19x10 r MPb 207.19 Na = Arogadro's number d = density M = atomic weight n Pb Therefore, substituting values f J b ( s ) = 12,52 eV or </ b ( s ) = 1 .goxlO^sec"1 P P Pb(s) *p = 992.1 A° 22 171 Sol)id Fe 1/N = 0.01195 m3 at 400°C (Metals Handbook, Vol. 1, 9th Ed. ASM) ri = 2.1 ) 5 Q ) ref. .tt m* = 12 m ) with these values *J*M =4.5eV or ^ s > = 6 . 8 2 x ] Q 1 5 s e c - l ./e(s) = 2 7 g 4 A o P Substituting these values of co in co P to = 1.904xl0 1 5sec _ 1 and for from Equation 19 (a) B, = ^ = 2.54xl0" 1 4 erg. NOTE: tt This anomalous very high value of eff e c t i v e mass i s typical of the t r a n s i t i o n metals, i . e . Fe, Co, Ni, Pd, Pt. 

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